UndergraduateLectureNotesinPhysics

EditorialBoardNeilAshbyUniversityofColorado,Boulder,Colorado,USA

WilliamBrantleyDepartmentofPhysics,FurmanUniversity,Greenville,SouthCarolina,USA

MatthewDeadyPhysicsProgram,BardCollege,Annandale-on-Hudson,NewYork,USA

MichaelFowlerDeptofPhysics,UnivofVirginia,Charlottesville,Virginia,USA

MortenHjorth-JensenDept.ofPhysics,UniversityofOslo,Oslo,Norway

MichaelInglisEarth&SpaceSci,SmithtownSciBld,SUNYSuffolkCountyCommunityCollege,LongIsland,NewYork,USA

HeinzKloseHumboldtUniversity,Oldenburg,Niedersachsen,Germany

HelmySherifDepartmentofPhysics,UniversityofAlberta,Edmonton,Alberta,Canada

UndergraduateLectureNotesinPhysics(ULNP)publishesauthoritativetextscoveringtopicsthroughoutpureandappliedphysics.Eachtitleintheseriesissuitableasabasisforundergraduateinstruction,typicallycontainingpracticeproblems,workedexamples,chaptersummaries,andsuggestionsforfurtherreading.

ULNPtitlesmustprovideatleastoneofthefollowing:

Anexceptionallyclearandconcisetreatmentofastandardundergraduatesubject.Asolidundergraduate-levelintroductiontoagraduate,advanced,ornon-standardsubject.Anovelperspectiveoranunusualapproachtoteachingasubject.

ULNPespeciallyencouragesnew,original,andidiosyncraticapproachestophysicsteachingattheundergraduatelevel.

ThepurposeofULNPistoprovideintriguing,absorbingbooksthatwillcontinuetobethereader'spreferredreferencethroughouttheiracademiccareer.

SerieseditorsNeilAshbyProfessor,UniversityofColorado,Boulder,CO,USAWilliamBrantleyProfessor,DepartmentofPhysics,FurmanUniversity,Greenville,SC,USAMatthewDeadyProfessor,BardCollegePhysicsProgram,Annandale-on-Hudson,NY,USAMichaelFowlerProfessor,DepartmentofPhysics,UniversityofVirginia,Charlottesville,

VA,USAMortenHjorth-JensenProfessor,UniversityofOslo,Oslo,NorwayMichaelInglisProfessor,SUNYSuffolkCountyCommunityCollege,LongIsland,NY,

USAHeinzKloseProfessorEmeritus,HumboldtUniversityBerlin,Berlin,GermanyHelmySherifProfessor,DepartmentofPhysics,UniversityofAlberta,Edmonton,Alberta,

CanadaMoreinformationaboutthisseriesathttp://www.springer.com/series/8917

AlessandroBettini

ACourseinClassicalPhysics1—Mechanics1sted.2016

AlessandroBettiniDipartimentodiFisicaeAstronomia,UniversitàdiPadova,Padova,Italy

ISSN2192-4791 e-ISSN2192-4805

ISBN978-3-319-29256-4 e-ISBN978-3-319-29257-1DOI10.1007/978-3-319-29257-1

LibraryofCongressControlNumber:2016934941

©SpringerInternationalPublishingSwitzerland2016

Thetextisprimarilybasedonthebook"Meccanicaetermodinamica",A.Bettini,Zanichelli,1995.Theauthorownallrightsintheformerpublications.

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PrefaceThisisthefirstinaseriesoffourvolumes,allwrittenatanelementarycalculuslevel.Thecompletecoursecoversthemostimportantareasofclassicalphysicssuchasmechanics,thermodynamics,statisticalmechanics,electromagnetism,waves,andoptics.Thevolumesresultfromatranslation,anindepthrevisionandupdateoftheItalianversionpublishedbyDecibel-Zanichelli.Thisfirstvolumedealswithclassicalmechanics,includinganintroductiontorelativity.

ThelawsofPhysics,andmoreingeneralofNature,arewritteninthelanguageofmathematics.Thereaderisassumedtoknowalreadythebasicconceptsofcalculus:functions,limits,andthedifferentiationandintegrationoperations.Weshallhowever,withoutmathematicalrigor,givethenecessaryinformationonvectorsandmatrices.

Physicsisanexperimentalscience,meaningthatitisbasedontheexperimentalmethod,whichwasdevelopedbyGalileoGalileiintheseventeenthcentury.Hetaughtus,inparticular,thattotrytounderstandaphenomenononemustsimplifyasmuchaspossibletherelevantworkingconditions,understandingwhichoftheaspectsaresecondaryandeliminatingthemasfaraspossible.Theunderstandingprocessisnotimmediate,butratheritproceedsbytrialanderror,inaseriesofexperiments,whichmightlead,withabitoffortuneandalotofthinking,todiscoverthegoverninglaws.Inductionofthephysicslawsprocessgoesbackfromtheobservedeffectstotheircauses,and,assuch,cannotbepurelylogic.Onceaphysicallawisfound,itisnecessarytoconsiderallitspossibleconsequences.Thisisnowadeductiveprocess,whichislogicalandsimilartothemathematicalone.Eachoftheconsequences,thepredictions,ofthelawmustthenbeexperimentallyverified.Ifonlyonepredictionisfoundtobefalsebytheexperiment,evenifthousandsofthemhadbeenfoundtrue,itisenoughtoprovethatthelawisfalse.Thisimpliesthatwecanneverbecompletelysurethatalawistrue;indeedthenumberofitspossiblepredictionsdoesnothavelimits,andinanyhistoricalmomentnotallofthemhavebeencontrolled.However,thisisthepricewemustpayinchoosingtheexperimentalmethod,whichhasallowedhumankindtoadvanceinthepastfourcenturiesmuchmorethaninalltheprecedingmillennia.

ClassicalMechanicsisoneofthebigintellectualconstructionsofPhysics.Itslawsarewellestablishedaswellastheirlimitsofvalidity.Consequently,itcanbeexposedinanaxiomaticway,asachapterofmathematics.Wecanstartfromasetofpropositionswhoseaxiomsareassumedtobetruebydefinition,anddeducefromthemanumberoftheoremsusingonlylogics,asfromthe

EuclidpostulatestheEuclideangeometrytheoremsarededuced.Weshallnotfollowthispath.Thereasonisthat,whileitallowsashorterand

quickertreatmentandisalsologicallymoresatisfactoryforsomebody,italsohidestheinductivetrial-and-errorhistoricalprocessthroughwhichthepostulatesandthegenerallawshavebeendiscovered.Thesearearrival,ratherthanstartingpoints.Thispathhasbeencomplex,laborious,andhighlynonlinear.Errorshavebeenmade,hypotheseshavebeenadvancedthatturnedouttobefalse,butfinallythelawswerediscovered.Theknowledgeofatleastafewofthemostimportantaspectsofthisprocessisindispensabletodevelopthementalcapabilitiesthatarenecessarytoanybodycontributingtotheprogressofnaturalsciences,whethertheypursueapplicationsorteachthem.Thisisoneofthereasonsforwhichweshallreadanddiscussseveralpagesofthetwoscientiststhatbuiltthefoundationsofphysics,GalileoGalileiandIsaacNewton.Asecondreasonisthatreadingthegeniusesisalwaysanenlighteningexperience.

TheGalileiandNewtonmechanicsthatweshalldiscussinthisbookisacoherentsetoflawsabletodescribeagreatnumberofphysicalphenomena.Theselaws,however,havealimitedvalidity.Onetypeoflimitationsdoesnothaveafundamentalnature.Someofthelaws,asforexamplethelawsoffrictionortheelasticforceare,consciouslywecansay,approximate.Inotherwords,theyprovideadescriptionthatweknowtobevalidonlyinafirstapproximationandprovidedthatthevaluesofcertainquantitiesarewithinsomedefiniteintervals(forexample,fortheelasticforce,fornottoolargestrains).Weshallalwaysclearlystatethoselimits.

Thelimitsofthesecondtypeareofafundamentalnature.AfirstlimitoftheGalilei-Newtonlawsismetwhenthevelocitiesareveryhigh,highenoughtogetclosetothespeedoflight.

Thelatterissohigh,300,000km/s,thatthespeedsofallobjectsofacommonexperience,planetsincluded,areextremelysmallincomparison.However,wecanreachvelocitiesclosetothatoflightinexperimentswithmicroscopicparticles,likeatomicnucleiandelectrons.Intheuniversetherearedoublestarsanddoubleblack-holes,whichareextremelydenseandrotateabouteachotheratveryhighspeeds,closetothespeedoflight.WeobservethatintheseconditionsthepredictionsofNewtonianmechanicsareincontradictionwithexperience.Newtonianmechanicsisanapproximationvalidatvelocitiessubstantiallysmallerthanthespeedoflight.ThetheorythatgeneralizesNewtonianmechanics,includinghigh-speedphenomena,isrelativisticmechanics,whichwasdevelopedbetweentheendofthenineteenthandthebeginningofthetwentiethcenturiesby,principally,HendrikLorentz,HenryPoincaré,andAlbertEinstein.Wediscussthebasicelementsofrelativistic

mechanicsinChap.6.Theyarenotnecessaryforunderstandingofthefollowingones.

Newtonorrelativisticmechanics,dependingonthevelocitiesoftheproblem,iscalledclassicalmechanics.However,noteventhisistrueineverycircumstance;thelawsofclassicalmechanicsdonotdescribecorrectlytheverysmall-scalephenomena,likevibrationsandrotationsofmolecules,thoseoftheelectronsinsideatoms,thenuclear,andsubnuclearphenomena.Asamatteroffact,thebodiesatthesemicroscopicscalesbehaveinacompletelydifferentwaythanthoseofeverydayexperience.Thetheoryabletodescribealltheknownphenomenabothatsmallandlargescaleisquantumphysics.Itslimitforlargescalesisclassicalmechanics.Thestudyofquantumphysicsnotonlyrequiresmathematicalinstrumentsmoreadvancedthanclassicalphysics,but,evenmoreimportantly,cannotbeprofitablystudiedwithoutanin-depthknowledgeofclassicalphysics.Consequently,thiscourseislimitedtoclassicalphysics.Weshallhoweverwarnthereaderofthelimitofvalidity,whenevernecessary.

Inthisbookwedealwiththemechanicsofamaterialpointandofextendedbodies,inparticularoftherigidones.Themechanicsoffluidswillbeoneoftheobjectsofstudyinthesecondvolume,togetherwiththeirthermalproperties.Mechanicaloscillationsaretreatedhereonlyintheirmostelementaryaspects.Adeeperdiscussionwillbegiven,togetherwithelectricoscillationsinthefourthvolume.

Westartthefirstchapterwithintroductoryelements:themeasurementofphysicalquantities,themeasurementunitsandtheirinternationallyadoptedsystem,theInternationalSystem,referenceframes,andbasicconceptsonvectorsandmatrices.Thesecondpartofthefirstchapterdealswithkinematics,whichisthemathematicaldescriptionofmotion,withoutreferencetoitscauses.Thesecondchaptercontainsthefundamentallawsofthematerialpoint(thesimplestbody)andthebasicconceptsofmass(boththeinertialandthegravitationalmasses),offorce,ofmomentum,ofmomentofaforce,andofangularmomentum.Weintroducealsotheconceptsofworkofaforce,ofenergy,ofpower,andtheenergyconservationprinciple.Weworkontheseargumentsmainlyconsideringthetwomostusualexamplesofforce,weightandfriction.Atthispointwehaveacquiredthebasiclawsofmechanics.Historically,thesearetheresultoftheworkofG.GalileiandI.Newton.Itisimportanttohavesomeknowledgeofhowthesegreatauthorscametoestablishthelawsofmechanics.Forthispurposeafewoftheirfundamentalpages,describingexperimentsandmathematicalarguments,arereproducedanddiscussed.Thereaderwillseealsohowbothauthorsexposetheconceptsinascientificsuperblanguage.

Thethirdchapterdescribesthedifferentforces,givestheirmathematicalexpressions,anddiscussestheirlimitsofvalidity.Wediscussimportantexamplesofmotion,inparticularthecircularandtheoscillatoryones.Weknownowthatthedifferentforcesthatweseeinnatureandthatlookatfirstsightverydifferentcanbereducedtoaverylimitednumberoffundamentalforces.Theforcespresentatmacroscopiclevel,thelevelofclassicalmechanics,aredifferentmanifestationsoftwobasicones:thegravitationalandtheelectromagneticforces.Thelatterwillbestudiedinthethirdvolumeofthiscourse,theformerinthefourthchapterofthisfirstbook.Asamatteroffactthestudyofgravitationhasenormoushistoricandculturalimportance.Itunderlinesourcomprehensionoftheuniverseinwhichwelive.Forthisreasonwerecallthemostimportantstepsinthehistoricaldevelopmentoftheuniversalgravitationtheory.

Thedescriptionofanymotiondependsontheframetowhichitisreferred.Inparticularitisdifferentintwoframesmovingonerelativetotheother.Thestudyofthisissueistheobjectofthefifthchapter,inthelimitofvelocitiesmuchsmallerthanthatoflight.Weshallmeetwiththeextremelyimportantprincipleofrelativity,auniversallyvalidprincipleestablishedalreadybyGalilei.TherelationsbetweenreferenceframesatspeedcomparabletothatoflightandthecriticalanalysisoftheconceptsoftimeandspaceintervalsleadingtotherelativisticmechanicsaredealtwithinChap.6.

Inthelasttwochapterswestudythemechanicsofextendedbodies.WestartChap.7withsystemsmadeofonlytwodifferentmaterialpoints.Weshowthatinanycaseinwhichaforceactsonabody,thisisduetoanotherbody,whichinturnisacteduponbyaforceduetothefirstone.Inotherwordstheforcesarealwaysduetotheinteractionbetweenbodies.Havingstudiedtheissueontwo-bodysystems,weproceedinthesecondpartofthechapterwiththestudyofmaterialsystemsinfullgenerality,findingthefundamentallawsoftheirmotion.Inthelastchapterwestudytheprincipalaspectsofthemotionofparticular,andimportantly,materialsystems,namelyrigidbodies.Theirmotionisdescribedbywell-defineddifferentialequations.Theirsolutionisanimportantmathematicalproblem,whichishoweveroutsidethescopeofthiscourse.

Eachchapterofthebookstartswithabriefintroductiontoascopethatwillgivetothereaderapreliminaryideaoftheargumentshe/shewillfind.Thereisnoneedtofullyunderstandtheseintroductions,atthefirstreading,asalltheargumentsarefullydevelopedinthefollowingpages.

Attheendofeachchapterthereaderwillfindanumberofqueriesonwhichtocheckhis/herlevelofunderstandingoftheargumentsofthechapter.Thedifficultyofthequeriesisvariable;someofthemareverysimple,somemore

1.

2.

3.

4.

complex,afewaretruenumericalexercises.Ontheotherhand,thebookdoesnotcontainasequenceoffullexercises,consideringtheexistenceofverygoodtextbooksdedicatedspecificallytothat.

Theanswerstoalargemajorityofthequeriesareincluded.However,thesolutionofnumericalexercises(withoutlookingattheanswers)ismentalgymnasticsthatisabsolutelynecessaryforunderstandingthesubject.Onlytheefforttoapplywhathasbeenlearnedtospecificcasesallowsustomasterthemcompletely.Thereadershouldbeconsciousofthefactthatthesolutionofnumericalexercisesrequiresmentalmechanismsdifferentfromthoseengagedinunderstandingatext.Thelatter,indeed,hasbeenalreadyorganizedbytheauthor;solvingaproblemrequiresmuchmoreactiveinitiativefromthestudent.Thisisjustthetypeofinitiative,acreativeactivitythatisneeded,foradvancingscientificknowledgeanditstechnicalapplicationsaswell.Consequently,thestudentshouldworkonexercisesalone,withoutlookingatthesolutionsinthebook.Evenfailedattemptstoautonomouslyreachthesolution,providedtheyareundertakenwithsufficientpersistence,giveimportantreturns,becausetheydevelopprocessingskills.Ifafterseveralfailedattemptsthesolutionhasnotyetbeenreached,itisabetterpracticetomomentarilyabandontheexercise,ratherthanlookingatthesolution,goingtoanotherone,andcomingbacklater.

Thefollowingworkingschemeismethodologicallyadvisable:

Examineatdepththeconditionsposedbytheproblem.Ifitispossible,makeadrawingcontainingtheessentialelements.

Solvetheproblemusingletters,notnumbers,intheformulas,thendevelopthemuntiltherequestedquantitiesareexpressedintermsoftheknownones.Onlythenshouldyouputnumbersintheformulas.

Confirmthecorrectnessofthephysicaldimensions(seeSect.1.3).

Whennecessarytransformallthedataintothesamesystemofunits(preferablySI,seeSect.1.2).Usescientificnotation,forexample2.5×103ratherthan2500,2.5×10−3ratherthan0.0025.Ingeneraltwoorthreesignificantfiguresareenough.

5.

Onceyouhavethefinalresult,alwaysverifyifitisreasonable.Forexamplethemassofamoleculecannotturnouttobe30mg,thespeedofabulletcannotbe106m/s,thedistancebetweentwotownscannotbe25mm,etc.

AcknowledgmentsThepagesfromIsaacNewton’s,PhylosophyaeNaturalisPrincipiaarefromtheEnglishtranslationfromLatinbyAndrewMotte(1729)modernizedbytheauthor.

ThepagesfromG.Galilei’sDialogueconcerningtwochiefworldsystemsareatranslationintoEnglishbytheauthorfromtheEdizioneNazionaledelleOpere,editedbyAntonioFavaro;Florence,tip.Barbèra,1890–1909.

ThepagesfromG.Galilei’sDialoguesandmathematicaldemonstrationsconcerningtwonewsciencesareadaptedfromtheEnglishtranslationfromItalianandLatinbyHenryCrewandAlfonsodeSalvio;McMillan1914.

Figure4.18isfromtheNationalAeronauticsandSpaceAdministrationathttp://www.compadre.org/Informal/images/features/Jupitmoons12-20-072.jpg

Figure4.21isfromtheEuropeanSpaceAgencyathttp://www.esa.int/var/esa/storage/images/esa_multimedia/images/2007/05/globular_cluster_ngc_28082/9535369-4-eng-GB/Globular_Cluster_NGC_2808.jpg

Figure4.22isfromtheNationalAeronauticsandSpaceAdministrationathttp://hubblesite.org/newscenter/archive/releases/2007/41/image/a//

SymbolsandUnitsTable1 Symbolsfortheprincipalquantities

Acceleration a,asAngularacceleration α,αAngularfrequency ωAngularmomentum l,LDensity(mass) ρDynamicfrictioncoefficient μdForce FFrequency νGravitationalfield GGravitationalmass mgGravityacceleration gImpulse iInertiaradius ρInertialmass miKineticenergy UKMass m,MMomentofaforce τMomentofinertiaabouta-axis IaMomentum pNewtonconstant GNNormalconstraintreaction NPeriod TPlaneangle θPolarangle θ,ϕPolarcoordinates(space) ρ,θ,ϕPositionvector rPotential ϕPotentialenergy UpPower wPressure pReducedmass μSpringconstant kStaticfrictioncoefficient μs

Time tTension TTotalangularmomentum LtotTotal(mechanical)energy UtotTotalmoment MTotalmomentum PYoungmodule EWeight FwWork WMeanvalue,ofx <x>Angularvelocity ω,ΩVelocityoflight(invacuum) cVelocity v,υVelocitydividedbylightvelocity βUnitvectorofv uυUnitvectorsoftheaxes i,j,kVolume V

Table2BaseunitsintheSI

Quantity Unit SymbolLength metre/meter mMass kilogram kgTime second sCurrentintensity ampere AThermodynamictemperature kelvin KAmountofsubstance mole molLuminousintensity candela cd

Table3Decimalmultiplesandsubmultiplesoftheunits

Factor Prefix Symbol Factor Prefix Symbol

1024 yotta Y 10−1 deci d

1021 zetta Z 10−2 centi c

1018 exa E 10−3 milli m

1015 peta P 10−6 micro µ

1012 tera T 10−9 nano n

109 giga G 10−12 pico p

mega M femto f

106 10−15

103 kilo k 10−18 atto a

102 hecto h 10−21 zepto z

10 deka da 10−24 yocto y

Table4.Fundamentalconstants

Quantity Symb. Value UncertaintySpeedoflightinvacuum c 299792458ms−1 Definition

Newtonconstant GN 6.67384(80)×10−11m3kg−1s_2 120ppm

Astronomicalunit a.u. 149597870700 DefinitionAvogadronumber NA 6.0221415(10)×1023mole−1 170ppb

Table5.Solarplanetsorbits

Planet Meandistancefromsun(a.u.)

Siderealperiod(tropicalyear)

Anglewithecliptic

Eccentricity

Mercury 0.387099 0.24085 7°00'14'' 0.2056Venus 0.723332 0.61521 3°23'39'' 0.0068Earth 1 1.00004 0 0.0167Mars 1.523691 1.88089 1°50'59'' 0.0934Jupiter 5.202803 11.86223 1°18'19'' 0.0484Uranus 19.181945 84.01308 0°46'23'' 0.0472Neptune 30.057767 164.79405 1°46'26'' 0.0086Pluto 39.51774 248.4302 17°08'38'' 0.2486

Tropicalyear=timeintervalbetweentwoconsecutivepassagesofthesunatthespringequinox

Table6.Dataonsomebodiesofthesolarsystem

Body Meanradius(Mm)

Radius(Earthradiuses)

Mass(Earthmasses)

Meandensity(kg/m3)

Mercury 2.44 0.38 0.055 5430Venus 6.05 0.95 0.815 5250Earth 6.37 1 1 5520Moon 1.74 0.27 0.012 3360Mars 3.38 0.53 0.108 3930Jupiter 71.49 11.19 317.9 1330Saturn 60.27 9.46 95.18 710Uranus 25.56 3.98 14.54 1240

Neptune 24.76 3.81 17.13 1670Pluto 1.12 0.176 0.0026 1990Sun 696 109.3 330,000 1400

Table7.Greekalphabet

alpha α Α iota ι Ι rho ρ Ρbeta β Β kappa κ Κ sigma σ,ς Σgamma γ Γ lambda λ Λ tau τ Τdelta δ Δ mu μ Μ upsilon υ Υ,epsilon ε Ε nu ν Ν phi ϕ,φ Φzeta ζ Ζ xi ξ Ξ chi χ Χeta η Η omicron ο Ο psi ψ Ψtheta θ,ϑ Θ pi π Π omega ω Ω

Contents1Space,TimeandMotion

1.1MeasurementofPhysicalQuantities

1.2TheInternationalSystem(SI)

1.3SpaceandTime

1.4Vectors

1.5OperationswithVectors

1.6ScalarProductofTwoVectors

1.7VectorProductofTwoVectors

1.8BoundVectors,Moment,Couple

1.9Matrices

1.10Velocity

1.11AngularVelocity

1.12Acceleration

1.13TimeDerivativeofaVector

1.14MotiononthePlane

1.15FromAccelerationtoMotion

1.16FreeFallMotion

1.17Scalars,Pseudoscalars,VectorsandPseudovectors

Problems

2DynamicsofaMaterialPoint

2.1Force,OperationalDefinition

2.2ForceIsaVector

2.3TheLawofInertia

2.4TheNewtonLawsofMotion

2.5Weight

2.6Examples

2.7CurvilinearMotion

2.8AngularMomentumandMomentofaForce

2.9TheSimplePendulum

2.10TheWorkofaForce.TheKineticEnergyTheorem

2.11CalculatingWork

2.12AnExperimentofGalileionEnergyConservation

2.13ConservativeForces

2.14EnergyConservation

2.15ATheoremConcerningCentralForces

2.16Power

Problems

3TheForces

3.1ElasticForce

3.2HarmonicMotion

3.3IntermolecularForces

3.4ContactForces.ConstraintForces

3.5Friction

3.6ViscousDrag

3.7AirDragandIndependenceofMotions

3.8DampedOscillator

3.9ForcedOscillator.Resonance

3.10EnergyDiagramsinOneDimension

3.11EnergyDiagramsforRelevantForces

Problems

4Gravitation

4.1TheOrbitsofthePlanets

4.2ThePeriodsofthePlanetsandtheRadiiofTheirOrbits

4.3TheKeplerLaws

4.4TheNewtonLaw

4.5TheMoonandtheApple

4.6TheGravitationalForceoftheHomogeneousSphere

4.7MeasuringtheNewtonConstant

4.8TheGravitationalField

4.9GalileiandtheJovianSystem

4.10Galaxies,ClustersandSomethingElse

4.11EllipticOrbits

4.12TheNewtonSolution

4.13TheConstantsofMotion

Problems

5RelativeMotions

5.1CovarianceofthePhysicalLawsUnderRotationsandTranslations

5.2UniformRelativeTranslation.RelativityPrinciple

5.3Non-uniformTranslation.PseudoForces

5.4RotationandTranslation.PseudoForces

5.5MotioninaRotatingFrame

5.6TheInertialFrame

5.7Earth,asaNon-inertialFrame

5.8TheEötvösExperiment

Problems

6Relativity

6.1DoesanAbsoluteReferenceFrameExist?

6.2TheMichelsonandMorleyExperiment

6.3TheLorentzTransformations

6.4CriticismofSimultaneity

6.5DilationofTimeIntervals

6.6ContractionofDistances

6.7AdditionofVelocities

6.8Space-Time

6.9Momentum,EnergyandMass

6.10Mass,MomentumandEnergyforaSystemofParticles

6.11Forceandacceleration

6.12LorentzCovarianceofthePhysicsLaws

6.13WhatIsEqualandWhatIsDifferent

Problems

7ExtendedSystems

7.1InteractionEnergy

7.2CentreofMassandReducedMass

7.3DoubleStars

7.4Tides

7.5ImpulseandMomentum

7.6TheAction-ReactionLaw

7.7Action,ReactionandLinearMomentumConservation

7.8SystemsofParticles

7.9TheCenterofMass

7.10LinearMomentumConservation

7.11ContinuousSystems

7.12AngularMomentum

7.13AngularMomentumConservation

7.14EnergyofaMechanicalSystem

7.15CenterofMassReferenceFrame

7.16TheKönigTheorems

7.17ElasticCollisions

7.18InelasticCollisions

Problems

8RigidBodies

8.1RigidBodiesandTheirMovements

8.2AppliedForces

8.3EquilibriumoftheRigidBodies

8.4RotationAboutaFixedAxis

8.5ConservationAngularMomentumAboutanAxis

8.6WorkandKineticEnergy

8.7CalculatingInertiaMoments

8.8TheoremsontheMomentsofInertia

8.9TorsionBalance

8.10CompositePendulum

8.11Dumbbell

8.12AngularMomentumAboutaFixedPole

8.13KineticEnergy

8.14RotationAboutaFixedAxis.ForcesontheSupports

8.15RollingMotion

8.16RollingonanInclinedPlane

8.17Gyroscopes

8.18CollisionsBetweenMaterialSystems

8.19TheVirtualWorksPrinciple

Problems

Solutions

Index

(1)

©SpringerInternationalPublishingSwitzerland2016AlessandroBettini,ACourseinClassicalPhysics1—Mechanics,UndergraduateLectureNotesinPhysics,DOI10.1007/978-3-319-29257-1_1

1.Space,TimeandMotion

AlessandroBettini1

DipartimentodiFisicaeAstronomia,UniversitàdiPadova,Padova,Italy

AlessandroBettiniEmail:alessandro.bettini@pd.infn.it

Physicsisanexperimentalsciencethatgivesaquantitative,mathematicaldescriptionofnaturalphenomena.Thismeansthatphysicallawsaremathematicalrelationsamongstphysicalquantities(suchasposition,velocity,force,energy,etc.).Theserelationsaretobeconsideredtrueonlyiftheycorrespondtoexperience.Physicallawsmustalwaysbeexperimentallyverified.Experimentisthesolejudgeofscientifictruth.Consequently,anyphysicalquantitymustbemeasurable,namelythesetofoperationsthatmustbeperformedtomeasureitmustbedefined.First,asystemofunitsofmeasurementmustbedefined.Weshallseeinthefirstthreesectionshowthisisdone.Thechoiceofunitsisaprioriarbitrary;thephysicallawsdependonNature,notonourchoices,Inpractice,however,havingstandardizedchoicesisextremelyimportanttomaketheresultsunderstandabletoeverybody.Internationalagreementshavedefinedthesystemofunitstobenamed,inFrench,SystèmeInternational(InternationalSystem).

Someofthephysicalquantities,likemassandtemperature,arerepresentedbyasinglenumberandarecalledscalar.Other,likevelocityandforce,aremorecomplex;specifyinghowbigtheyareisnotsufficient,alsotheirdirectionmustbegiven.Mathematically,anorderedsetofrealnumbersrepresentsthem;theyarevectorquantities.WeshallstudyinthesectionsfromSects.1.5to1.8theelementarymathematicalpropertiesofvectorsandoftheoperations(sum,difference,products)amongstthem.InSect.1.9weshallintroducesome

elementsthatwillbeusefulinthefollowingonanothermathematicalobject,matrices.

Inthesecondpartofthechapterweshallmovetophysics,dealingwiththekinematicsofthepoint-likeparticle,namelythestudyofitsmotion,independentlyofitscauses.Weshallintroducetheconceptsofvelocity,angularvelocityandacceleration.Thesearevectorquantities,ingeneraldependingontime.Section1.13isagainofmathematicaltype,presentingaformulathatwillbeveryusefulinthefollowing,thetimederivativeofavector.

Afewtypesofmotionareparticularlyimportant:circularmotion,studiedinSects.1.11and1.12,motiononaplaneinSect.1.14andfreefallofweightsinSect.1.16.

1.1 MeasurementofPhysicalQuantitiesPhysicsgivesaquantitativedescriptionofnaturalphenomena(or,better,oftheknownpartofthem).Measurementoftherelevantphysicalquantitiesleadstodiscoveryofthephysicallaws,whicharemathematicalrelationsamongstthosequantities(forexample,thelawofthefreefall,theKeplerlaws,etc.).

Allnaturalphenomenatakeplaceinspaceandhaveatemporalduration;someofthemhappenbefore,othersafterwards.Consequently,spaceandtimearefundamentalconcepts.Physicalobjectsarecharacterizedbyquantitieslikelength,area,volume,color,hardness,mass,temperature,etc.Alltheseconceptsresultfromacommonexperienceandarepresentincommonlanguage.However,Physicsmustgivetoeachquantityarigorousdefinition,inordertobeabletogiveitnumericalvalues.Inthisdefinitionprocess,theconceptmaybecomeratherdifferentfromacommonlanguage.

Considerforexamplethelengthofanobjectorthedistancebetweentwoplaces.Ifwewanttodesignateanumberwemustfirstdefineaunitoflength.Indeedwesay:“Thatbaris5mlong”,or,ifweareinEngland:“Thatcityis20milesaway”.Themeasureofthelengthofanobjectistheratiobetweenitslengthandthelengthofanotherobjectwehavechosenasunit.“Abaris5mlong,”meansthatitslengthisequaltothatof5one-meterlongrulesinaline.“Themassofabodyis8kg,”meansthatitisequaltothatofeightbodiesof1kgtogether.

Themeasurementofphysicalquantityistheratiobetweenthatquantityanditsmeasurementunit.

Themeasurementoperationallowsassociatingtoeachphysicalquantityanumber.Thesymbolsthatappearinthephysicallawsrepresentingthevariousphysicalquantitiesarejustthesenumbers.Forexample,whenwewriteF=ma

wemeanthattheratiobetweentheforceweareconsideringandtheforcetakenasunit,isequaltotheratioofthemassoftheobjectandthemassoftheobjecttakenasunit,timestheratiobetweenthedesignatedaccelerationandtheunitacceleration.

Everyphysicalquantitymustbemeasurableanditsdefinitionmustbepreciseandrigorous.Theoperationaldefinitionisthemosteffectivewaytodefineaphysicalquantity.Thisisdefinedasthesetofoperationsneededtomeasurethatquantity.

Thisprocedurehastwoimportantimplications.Thefirstimplicationisthatquantitiesthatarenot,eveninprinciple,measurablearenotphysicalquantities.Thisdoesnotimplythatsuchquantitiescannotbeused.Indeedtheyareoftenusefulinthemathematicaldevelopmentsofatheory.Anytheory,however,ifithastobeaphysicaltheory,mustleadtopredictionsthatareexperimentallytestable.Theexperimentallytestablepredictionsaremathematicalrelationsamongstphysicalquantities,meaningmeasurableones.Theauxiliary,non-measurable,quantities,shouldnotappearinthefinaltheoreticalexpression.

Thesecondimplicationisthatscalematters:quantitiesmaybesmallorlarge.Considerforexamplethelength.Ifwewanttomeasuredistances,say,frommillimeterstokilometerswecanusegraduatedbarsorrules(likeyardsticks,measuringtapes,calipers,gauges,etc.).Ifweneedtomeasuredistancesoftensorhundredsofkilometers,asforexamplebetweentwomountaintopsortheheightofMountEverest,theprocedureisverydifferentandwemustperformtriangulations.Ifthedistancesareverymuchlarger,asthoseofthegalaxies,theprocedureschangecompletelyagain.Anddifferentproceduresarerequiredtomeasuresmalldistancessuchasthediameterofanatomorofanatomicnucleus.Ineveryrangeofordersofmagnitude,thesetofprocedurestomeasurealengthisdifferent.Toberigorouswewouldneedtotalkofmanydifferentlengths.Thiswouldleadtoaterribleconfusion.Fortunately,weexperimentallyverifythat,inthelargeintervalsinwhichtwoormoremethodsworkcontemporarily,theresultsareequal,andwecandefineasinglelengthconcept.However,theaboveargumentstellustobecareful.Supposethataphysicallawiswellexperimentallyverifiedforobjectsofsizesbetweenmetersandkilometers.Wetendtothinkthesamelawtobevalidalsoforobjectsmuchsmallerandmuchlargerthanthat.Butwehavenoguaranteethattheextrapolationistrue.Onthat,asalways,onlyexperimentcanjudge.ForexampletheNewtonlawsvalidatthespeedsofordinaryexperiencearenolongervalidatspeedscomparabletothespeedoflight.Thelawsofclassicalmechanicsaresimplynotvalidforatomsandsmallerobjects.

Letusgobacktothemeasuringoperation.Foreachquantityweneeda

measurementunit.Thechoiceisinprinciplearbitrarybutisfarfrombeingsoinpractice.IfeveryCountry,forexample,wouldchooseadifferentunitforlengthsorareas,theexchanges,notonlythescientificones,butalsothecommercialones,wouldbeextremelycomplex.Theunitsmustbestandardized.TheissueissoimportantthatbothunitsandproceduresaremadecompulsorybylawinthemajorityofCountries.

1.2 TheInternationalSystem(SI)ThemoderninternationalstandardizationofunitsstartedwiththeFrenchRevolution.In1791theDecimalMetricSystemwasofficiallyannounced,butittookalmostacenturyforitssubstantialdiffusionandacceptance(and,mostimportant,Napoleontoimposeit;inBritain,wherehisreachwasinsufficient,theImperialSystemisstillused,asitisalsointheUSA).InMay1875,atthe“Metreconvention”,therepresentativesof17NationssignedaninternationaltreatyinParis.Nationalandinternationallaboratorieswerecreatedwiththemissiontodevelopmeasurementstandardsandprocedures.Thisisaveryimportantsectorofphysics,knowasmetrology.

InternationalOrganizationswerecreatedtofosterinternationalstandardizationofweightsandmeasuresintheworld.TheInternationalConferenceofWeightsandMeasures,CGPMforbriefusingtheinitialsinFrench,whichmeetseveryseveralyears,isthemaindecision-makingbody.Itdecidesontheevolutionoftheinternationallyadoptedsystemofunits,namedinFrenchSistèmeInternationalorSIforbrief.In1971theEuropeanCommunityissuedadirectivetothememberstatesforthelegaladoptionoftheSI.

Therearetwoclassesofunits:baseunitsandderivedunits.Thebaseunitsaregivenbydefinition.Eachderivedunitisobtainedusingaphysicslaw,namelyamathematicalexpressionthatlinksittoquantitiesofthebasicunits.Thechoiceofthebasicunits,andeventheirnumber,is,fromalogicpointofview,arbitrary.Thechoicesarebasedonconvenience,takingquantitiesforwhichmeasurementscanbeasmuchaspossiblepreciseandreproducible.

Letusconsideranexample.Takethe“physicallaws”:(1)theareaSofarectangleofsidesoflengthsaandbisproportionaltotheproductofthelengths,(2)theareaAofacircleisproportionaltothesquareofthelengthRofitsradius,(3)thespacescoveredbyabodymovinginabsenceofanyforceisproportionaltothetimetemployedandtoitsvelocityυ.Themathematicalexpressionfortheselawswouldbe

(1.1)wherek,k′andk″arepurelynumericalconstants.Theydependonthechoiceofmeasurementunits.Wemighttakebothlengthandareaasbasequantitiesandasunitsthemeterandsquarefootrespectively.Thekandk′wouldthenhavedefinitevalues.Ourmeasuringsystemwouldbesimplertakinglengthasthebaseunit,say1m,andtheareaasderived.Still,however,somearbitrarinessremains.Forexample,wecanchoosetheunitsinsuchawaytohavek=1or,differently,tohavek′=1.Inthefirstoptiontheunitaryareaisthesquareof1mside,inthesecondoneitisthecircleof1mradius.Thesecondchoicegivesk=1/π,andappearsfunny.Thechoicek=1appearstobetheobviousone,andistheuniversallyusedone,but,inprinciple,itisnotnecessary.

Similarly,inthethirdequationwemakek″=1bychoosingasmeasuringunitofvelocitythevelocityofabodythatcoverstheunitlengthintheunitoftime.

Asalreadymentioned,theinternationallyacceptedsystemofunitsistheSI.Itistheeasiesttouseandthemostrationalone.IntheSIthebaseunitsareseven:length,mass,time,electriccurrentintensity,thermodynamictemperature,amountofsubstance,luminousintensity.Foreachofthem,thenameoftheunit(e.g.“meter”)anditssymbol(e.g.“m”)arefixed,asinTable1.1.Noticethattheinitialofthenameofaunitisalwayslowercase,includingwhenitisthenameofascientist(e.g.“ampere”).Mostimportant,theSIgivesapreciseandcleardefinitionforeachunit.Noticethatthesemaychangewithtime,asaconsequenceoftheprogressofmetrology,afterformalapprovalbytheCGPM.Weshallgiveherethedefinitionsofthefirstthreeunits,whicharetheonlyonesneededinthistextbook.Theotheroneswillbedefinedintheothervolumesoftheseries,whenneeded.

Table1.1 Thebasequantities,theirunitsandsymbols

Quantity Unit SymbolLength metre/meter mMass kilogram kgTime second sCurrentintensity ampere AThermodynamictemperature kelvin KAmountofsubstance mole molLuminousintensity candela cd

Themetre(meteristhedistancetravelledbylightinvacuuminatimeintervalof1/229792458ofasecond.

Thekilogramisthemassoftheinternationalprototypekilogram(locatedinthePavillondeBreteuilatSèvres).

Thesecondisthedurationof9192631770periodsoftheradiationcorrespondingtothetransitionbetweenthetwohyperfinelevelsofthegroundstateoftheCesium133atom.

TheSIdefinesthenamesandsymbolsofallthederivedunits.Weshallintroducethemwhenwemeetthemforthefirsttime.TheSIfurtherdefinesnamesandsymbolsofmultiplesandsubmultiplesoftheunits.Thisisdoneinstepsingeneralofthreeordersofmagnitudes,ofoneorderforthefirstthree,asinTable1.2.Withtheexceptionofda,h,andk,allmultipleprefixsymbolsareuppercase;allsubmultipleprefixsymbolsarelowercaseletters.

Table1.2 Decimalmultiplesandsubmultiples

Factor Prefix Symbol Factor Prefix Symbol

1024 yotta Y 10−1 deci d

1021 zetta Z 10−2 centi c

1018 exa E 10−3 milli m

1015 peta P 10−6 micro µ

1012 tera T 10−9 nano n

109 giga G 10−12 pico p

106 mega M 10−15 femto f

103 kilo k 10−18 atto a

102 hecto h 10−21 zepto z

10 deka da 10−24 yocto y

Thederivedmeasurementunitsaredefined,asmentioned,usingaphysicallawinordertohaveadefinitionassimpleaspossible.Hence,theunitforareasisthesquareof1mside,theunitofvolumeisthecubeof1mside,theunitofvelocityisthevelocityofabodytravelling1minonesecond,etc.

The(mean)accelerationisthechangeofvelocityΔυdividedbythetimeintervalΔtinwhichthatchangehappens,namely .Theaccelerationunitistheaccelerationofabody,thevelocityofwhichvariesbyaunit(1m/sor1ms−1)intheunitoftime(1s).Itisconsequentlythemeterpersecondpersecond(m/s2orms−2).

Letusnowobserve,asanexample,thatalltheplanefigures,triangles,rectangles,circlesetc.areexpressedasanumericalfactortimestheproductof

twolengths.Namely,allareashaveaphysicaldimensionoflengthsquared.Ifwechangetheunitoflength,forexamplefrommetertocentimeter,themeasuresofalltheareaschangebythesamefactor:1002intheexample.Thephysicaldimensionsofvelocityarelengthdividedbytime,ofaccelerationoflengthdividedbytimesquared,etc.Thecorrespondingmathematicalexpressionsarecalleddimensionalequationswhichareofthetype

(1.2)Dimensionalequationsareveryusefulinpractice.Consideranyrelationship

amongstphysicalquantities,forexampleF=maorA+B=C.Allthetermsmusthavethesamephysicaldimensions.Otherwise,achangeofunitswillcausethedifferenttermstochangeindifferentways;thevalidityoftherelationwoulddependonthechoiceofunits,whichisarbitrary.Thisistheso-calledhomogeneityprinciple.Itisveryusefultocheckanalyticalexpressionsobtainedwithmoreorlesscomplexcalculations.Ifwefindthatsomeofthetermshavedifferentdimensions,wemustconcludethatwehavemadesomemistake.

Noticethattherearealsophysicalquantitieshavingnildimensions,namely[L0T0M0],theyarepurenumbers.Animportantexampleistheangle.Inradians(rd)itistheratiobetweenthearcofacircumferenceanditsradius.Ifwechangetheunitoflength,theratiobetweentwoofthemdoesnotchange.

Finallynoticethataphysicallawmaycontainmathematicalfunctions,forexample .Theseexpressionsmakesenseonlyifboththefunctionsthemselves(x,y,z)andtheirarguments(α,β,γ)havenophysicaldimensions.Allofthemmustbepurenumbers.

1.3 SpaceandTimeOurstudybeginswiththestudyofthemotionofbodies.Motionofabodymeansthatitspositioninspacevariesintime.Thenotionofmotionisrelative:apassengerinaplanesittinginhischairhasafixedpositionrelativetotheplane,butmovesat,say800km/hrelativetoapersonstandingonearth.Thelattermovesat800km/hrelativetothepassenger,intheoppositedirection.Todescribethemotionwethenneedareferenceframe.

Wenormallylivestandingonearthandsucharethelaboratoriesinwhichwedoourexperiments.Letusthenstartbychoosingareferenceframefixedontheearth.Thepossiblechoicesarestillinfinite.

Thepositionofabodyisdefinedwhenweknowwereitis.Thesimplestcaseiswhenwedealwithaparticle,abodythatissosmallthatitcanbe

consideredpoint-like.Itiscalledamaterialpoint.Letusseehowwecandefinethepositionofamaterialpoint.Foranextendedbodythepositionsofallitspointsshouldbesimilarlydefined.

Toknowthepositionofapointinspaceweneedthreenumbers,oneforeachofitsdimensions.Todefineitspositiononagivensurface,twonumbersareneeded(asforexamplelongitudeandlatitudeontheearthsurface).Toknowthepositiononagivencurve,onenumberisneeded.

LetusstartbyconsideringapointPthatcanmoveonlyonastraightline(seeFig.1.1a).Todefineitsposition:(1)wechooseoneofthetwodirectionsandcallitpositive,(2)wechooseapointonthelineandcallittheoriginoftheco-ordinates(OinFig.1.1a),(3)wechooseaunitlength.Theorientedline,withanoriginandameasuringunitiscalledaco-ordinateaxis.ThepositionofthegenericpointPisgivenbyarealnumber,calledtheco-ordinateofthepoint(xinthefigure),whichisthedistanceofPfromO,takenaspositiveifPisontherightofO,negativeifitisontheleft.

Fig.1.1 Orthogonalco-ordinateframes.aOnedimension,btwodimensions,cthreedimensions

LetusnowassumethatpointPcanmoveonaplane(Fig.1.1b).Wenowneedtwoco-ordinateaxes,whichshouldnotbeparallel.Itisusuallyconvenienttotakethemperpendicular,theoriginatthepointinwhichtheycrossandthesameunitlengthforboth(noneofthesechoicesiscompulsory,theyarejustgenerallythemostconvenient).ThepositionofPisgivenbyitstwoco-ordinates,whichisanorderedpairofrealnumbers(x,y).

Considernowapointinspace.ThereferenceframeshowninFig.1.1ciscalledaCartesianrectangularright-handedframe,afterRenéDescartes(1596–1650).Itismadeofthreeco-ordinateaxes,calledx,yandz.Theycrossinasinglepoint,theoriginoftheframe.Alltheanglesbetweenthe(three)pairsofaxesareright.Thelengthunitsonthethreeaxesareequal.Finallywemust

(1)

(2)

(3)

(4)

(1)

choosepositiveorientationsoftheaxes.Therearetwobasicpossibilities.Letusassumethatwehavealreadydefinedthepositivedirectionsofxandy.Wehavetwopossiblechoicesforthepositivedirectionofz.Figure1.1cshowsoneofthem;anobserverstandingwithhisfeetonthexyplanelyingalongthezaxisandlookingdown,willingtomovethexaxisontheyaxisbya90°rotation,seesthisrotationhappeninganticlockwise.Thesecondpossibilityistheoppositesignofz.Thetwoframesarecalledright-handedandleft-handedrespectively.

Nowconsidertheinversionoftheaxes.Ifwestartfromaright-handedframeandinvertoneaxis,thatisamirrorreflectionandwegetaleft-endedframe.Thesamehappensifweinvertallthreeaxes.Theinversionoftwoaxesgives,onthecontrary,thesameresultasarotationof180°aroundthethirdaxis:theinitialandfinalframehavethesame“handness”.

Todefinethereferenceframewehavemadeaseriesofchoices,whichwerecall:

choiceoftheorigin

choiceofthedirectionsoftheaxes

choiceofthepositivedirections(left-handedorright-handed)

choiceoftheunits.

Whileeachofthesechoicesisarbitrary,wecanaskwhetherthereisany

privilegedchoice,orifthereisonethatisbetterposed,arethephysicslawsindependentofthesechoices?Theanswerscannotcomefromlogicsormathematics,butonlyfromanexperiment.Letusconsidereachofthem.

Arethephysicslawsindependentoftheoriginoftheaxes?Tocheckthepoint,letusbuildtwoidenticalapparatuses.Leteachofthemcontaininclinedplaneswithballsrollingonthem,pendulums,flywheels,gears,etc.,allidentical.Wepositionthetwoapparatusesintwodifferentlocations.Wepreparethemtobeinexactlythesameinitialstate:thependulumsareoutof

(2)

(3)

equilibriumatthesamedistance,thespheresareatthesameheightsontheinclinedplanes,thegearsandtheflywheelsareinthesamepositions.Weletthemgocontemporarilyandobservetheirevolutions.Dothetwosystemsevolveinthesameway?Dotheyassumethesameconfigurationsatthesametimes?Asamatteroffacttheanswerisnotalwaysyes.However,everytimesomedifferenceisnoticed,itispossibletoidentifythereasonforthatinsomephysicalconditionthatisdifferentinthetwolocations.Forexample,thegravitationalaccelerationmightbeabitdifferentinthetwositesandconsequentlytheperiodsofthependulumsareabitdifferenttoo.Inanycase,experimentsshowthat,onceallthelocaleffectsareeliminated,oraccountedfor,theapparatusesevolveinthesamemanner,i.e.goingthroughthesameconfigurationsatthesameinstants.

Theveryimportantconclusionis:Thephysicallawsareindependentonposition.Inotherwordsallpositionsareequivalent,orspaceishomogeneous.Letusrepeatthatthisisanexperimentalconclusion.Noexperimentuptonowhasfounditwrong.Onecanstatethatthephysicallawsareinvariant,meaningthattheydonotvary,underspacetranslations.

Arethephysicslawsindependentofdirectionsoftheaxes?Wenowtakeourtwoidenticalapparatusesandrotateonetotheother.Forexample,inonecasethez-axisisvertical,intheotherisat45°withthevertical.Dothetwosystemsevolvethroughthesamestates?Certainlynot!Indeed,forexample,pendulumsoscillatearoundaverticalaxisinonecase,aroundaninclinedoneintheother.Inthiscaseaprivilegeddirectionexists,thedirectionofweight.But,thinkamoment.Ifwewerefarfromearth,orinabsenceofweight,theprivilegeddirectionwouldnotexist.Thatdirectionisnotapropertyofthespace,butisthe“local”effectofabody,theearth.Inotherwords,ifwewanttocomparethetwoexperimentsinthesameconditions,weshouldalsorotatetheearthinthesecondcase.Ifalltheexternalconditionsareproperlytakenintoaccount,alltheexperimentsshowthatthephysicallawsareindependentonthedirectionsoftheaxes.Inotherwords,noprivilegeddirectionexists,or,spaceisisotropic.Stillinotherwords,physicslawsareinvariantunderrotations.

Arethephysicslaws,independentoftheorientation,left-handedorright-handed?Experimentshaveshownthatallphysicslawsatthemacroscopic

(4)

levelareindependentofthechoice.Butthisisnotrueratamicroscopiclevel.Aclassofradioactivephenomena,likebetadecays,isduetoafundamentalforcecalledweakinteraction.Itslawsdistinguishbetweentheleftandrightcases.Namely,notallthephysicslawsareinvariantunderinversionoftheaxes.

Arethephysicslawsindependentofthescaleoflength?Thistimewebuildtwoapparatusesthatareidenticalbutforhavingalltheirdimensionsdifferent,scaledbythesamefactor.Dothetwoevolveinthesamemanner?TheanswerwasdiscoveredbyGalileoGalilei(Italy,1564–1642)andisNO.

Considerforexampleabeammadeofacertainhomogeneousmaterial.The

beamhasacertainlength,anditscrosssection,whichweassumetoberectangular,hasacertainwidthandacertainheight.Welayitontwosupportsneartoitsextremesonahorizontalplane.Supposethebeamtobeinequilibrium.Wenowtakeabeamgeometricallysimilartothefirstonebuttentimeslonger,tentimeswider,tentimeshigher.Againwelayitontwosupportsneartoitsextremes.Weobservethatthebeambreaksdowninitsmiddlepoint.Thereasonisthefollowing.Theweightofthebeamisaforceappliedinitsmiddlepointdirecteddownward.Theweighttendstobreakthebeam,thecohesionforcesbetweenmoleculestendtokeepittogether.Theweight,whichisproportionaltothevolume,isforthesecondbeamonethousandtimeslargerthanforthefirstone.Theresistancetofractureisproportionaltotheareaofthecrosssectionandforthesecondbeamisonehundredtimeslargerthanforthefirstone.Consequently,aboveacertaindimensionthebeambreaksdownunderitsownweight.Forthesamereasontheanimalscannotbetoobig.Thebonesofthelegsofahypotheticalhorsetentimesbiggerthantherealoneswouldbreakundertheirownweights.Weknownowthatthefundamentalreasonforthatisthatsubstancesaremadeofmoleculesandatoms,whichhaveadefinitesize.Certainlywecannotbuildoneoftheabove-consideredapparatusessosmalltobemadeofafewmolecules.

Asanotherexample,considertheheavenlybodies.Starsandthelargestplanetemitlight,thesmallerplanet,likeearth,donot.Onlyifthesize,hencethemassofthebody,islargeenough,thepressureandtemperatureinitscore,whichareduetotheactionofthegravitationalforcesbetweenitsparts,arelargeenoughtofirethethermonuclearfusionreactionsthatproducelight.

Inconclusion:ThephysicslawsareNOTinvariantunderchangesofscale.Wenowcomebacktomathematicsofreferenceframes.Inthefollowingwe

shallneedtouseanothertypeof,equivalent,co-ordinates,thesphericalpolarcoordinates.Figure1.2ashowssuchco-ordinatesontheplane,Fig.1.2binspace,togetherwiththeorthogonalcoordinatesinbothcases.Ontheplane,thetwopolarco-ordinatesofthegenericpointPareitsdistancefromtheoriginρ,whichisanon-negativenumber,calledaradius,andtheangleϕ,betweenthex-axisandthesegmentOP,calledanazimuth.Itismeasuredinanticlockwisedirectionandvariesbetween0and2π,namely

Fig.1.2 Polarco-ordinates.aTwodimensions,bthreedimensions

(1.3)Wecaneasilyseefromthefigurethattherelationsbetweenpolarand

rectangularco-ordinatesare

(1.4)andtheinverseones

(1.5)Figure1.2bshowspolarco-ordinatesinthreedimensions.Thefirstco-

ordinateofthegenericpointPisagainitsdistancerfromtheorigin(radius),thesecondco-ordinateistheangleϕbetweentheplanethroughthezandPandtheplanexz(azimuth),thethirdco-ordinateistheangleθbetweenthesegmentOPandthezaxis(zenithangle).Againrisanon-negativenumber.Theangleθvariesfrom0toπ,coveringinsuchawaythesemi-planeshowninthefigure.Thissemi-planerotatesaroundzwhenϕvariesbetween0and2π.Hence

(1.6)Therelationswiththeorthogonalco-ordinatesare

(1.7)andtheinverseones

(1.8)

IfthepointPisonthexyplane,namelyifθ=0,Eq.(1.8)become

whichareequaltoEq.(1.4).Toknowthemotionofabodyweneedtoknowitspositionindifferenttime

instants.Consequentlywemostmeasurethetime.Moreprecisely,wemeasureintervalsoftime,ratherthananabsolutetime.Inpracticewechooseacertaininstantanddefineitastheoriginoftimes,forwhicht=0.Wenextchooseatimeintervalanddefineitastheunitoftime.IntheSIitisthesecond.Inprincipleweshouldalsochooseoneofthetwodirectionsaspositive,butthechoiceisobvious.Itis,wecansay,imposedbyNature:thepositivedirectionoftimeisfrompasttofuture.Consequently,thetimeofaneventisnegativeifithappenedbeforet=0,positiveifafterthat.

Wenowask:istheoriginoftimesarbitrary?Asalwayswemustapplytotheexperiment.Letusgobacktooneofourexperimentalapparatusesandletusrepeattheexperimentstartingfromthesameinitialstate,forexampleinthemorning,thenintheafternoon,andagaininthenight,etc.Foreachtrialwetaketheoriginoftimeastheinitialinstant.Weobservethat,onceallthespuriouselementsaretakencareof(e.g.lightintheday,darkinthenight)alltheexperimentsevolveinthesameway.Theoriginoftimesisarbitrary,timeishomogeneous.Thephysicslawsareinvariantundertranslationsintime.Inaddition,similarlytospace,nofundamentaltimeintervalexists.

WehavesaidthatthechoiceofthepositivedirectionoftimeisimposedbyNature.Severalbookshavebeenwrittenonthisissue,the“arrowoftime”.Weshallnotenterinthisdiscussion.Weonlystateherethatinthepurelymechanicalphenomenanoarrowoftimeexists.Supposewehitabilliardballandshootamovieofitsmotionhittingotherballs,thewalls,etc.Ifwenowplaythemoviebackwardsweobserveaperfectivelylegitimateevolution.Wecannotknowifitisbackwardsofforwards.But,wait;thisisnottrueforever.Indeed,ifthemovieislongenoughweseethat,whenplaidforwards,thespeedsoftheballsgraduallyslowdownandfinallytheystop.Ifitisplaidbackwards,theballsareinitiallysteadyandstartmovingalone.Thenaturalarrowoftimeistheoneinwhichthekineticenergydiminishes.Whenwestudythermodynamicsinthesecondvolumeofthiscourseweshallseehowitexplainsthearrowoftime.

1.4 VectorsManyphysicalquantities,suchastemperatureandatmosphericpressure,arerepresentedbyasinglenumber.Thisisnotthecaseofotherones,suchasvelocity,acceleration,force,etc.Forexampletoknowthevelocityofacarisnotsufficienttoknowhowfastitmoves(thatisthenumberwereadonthespeedometer),butalsoinwhichdirection(towardsSouth,Northorother).Anotherexampleisadisplacementinspace.Toknowitweneedtoknowhowlongitisandinwhichdirectionithappens.Thesephysicalquantitiesarerepresentedbyvectors.

Avectorisamathematicalentity.Todefineit,letusstartconsideringlinesegments.Asegmentiscalledoriented,ifoneofitstwosensesischosenaspositive.Twoorientedsegmentsaresaidtobeequipollentiftheyhavethesamelength,thesamedirectionandthesamesense.Avectoristheclassofalltheorientedsegmentsequipollenttoagivenone.Itisgraphicallyrepresentedwithanarrow.Itischaracterizedbythelength,calledmagnitude,thedirectionandthesense.Differentlyfromtheorientedsegment,itisnotcharacterizedbyitsposition.Thevelocitiesoftwocarsmovingat100km/hheadingWest,onenearParis,onenearLondonarethesame.

Onceareferenceframeischosen,wecanrepresentavectorbyanorderedtripleofrealnumbers,whichareitscomponentsinthatreference.However,anorderedtripleofrealnumbersisnotnecessarilyavector.Tobesothefollowingimportantpropertymustbesatisfied.Indeed,ifwechangethereferenceframe,forexamplerotatingtheaxes,thecomponentsofthevectorchange,butthevectordoesnot.Vectorisadefiniteobject;itscomponentsarethewaytoseeitinoneoranotherframe.Tosatisfytheseproperties,thevectorcomponents,namelytheorderedtriplesinthetwoframes,mustbeconnectedbywell-definedrelations,whichweshallnowfind.

Figure1.3showsareferenceframeandapointPofco-ordinatesx,y,z.ConsidertheorientedsegmentfromtheoriginOtoPandthecorrespondingvectorr(namelytheclassofequipollentorientedsegments).Itiscalledapositionvector,andis,wecansay,theprototypeofallvectors.Itscomponentsinthe(Cartesian)referenceframeweareconsideringaresimplythecoordinatesofP,i.e.theorderedtriple(x,y,z).Letusnowtakeanotherreferencewiththesameoriginandaxesrotatedbyanangleθ.ThepointPdoesnotmoveandrdoesnotchange.Butitscomponents(x′,y′,z′)aredifferent.Thegeneralrelationbetweenthetwotripletsisrathercomplex.Forsimplicityweshallconsidertwoframeswiththesameoriginandthesamez-axis,asshowninFig.1.4.

Fig.1.3 Theorthogonalco-ordinatesandthepositionvector

Fig.1.4 ArotationofaCartesianreferenceframearoundthecommonzaxis

LetusconsiderthepointPinthefigureofco-ordinates(x,y)inoneframe,(x′,y′)intheother.Wemustexpressx′andy′asfunctionsofx,yandθ.Onerelationisobvious,z′=z.Inpractice,wearereducedtotwodimensions.

WenowdrawperpendicularsfromPtoalltheaxes.WealsodrawthesegmentABperpendiculartoPQ.Thefigureshowsthatx′isthesumoftwolengthsalongthex′axisandy′thedifferenceoftwolengthsalongAB.Weobtain

(1.9)

where,tobecomplete,weincludedalsothethirdco-ordinate.Noticethattheserelationsareboththerelationsbetweenthecomponentsofthepositionvectorinthetwoframesandtherelationsbetweentheco-ordinatesinthetwoframes.Asamatteriffacttheyanalyticallydefinetherotationoftheaxes.

Wenowstatethatavectorisanorderedtripleofrealnumbersthatunder

rotationsofthereferenceframetransforms(changes)inthesamewayasthetriplerepresentingthepositionvector,namelyasco-ordinates.

Figure1.5arepresents,inaplaneforsimplicity,agenericvectorA,whichwecanthinkasofdrawnstartingfromtheorigin,becausealltheequipollentsegmentsarethesamevector,anditscomponentsinthetwoframes.

Fig.1.5 ComponentsofthevectorAintwoframesdifferentforaarotationbatranslation

Bydefinition,therelationsamongstitscomponentsareequaltoEq.(1.9),namely

(1.10)

Theinverserelations,namelytheexpressionsof(A′x,A′y,A′z)asfunctionsof(Ax,Ay,Az)andθcanbeobtainedintwoways:invertingthesystem(1.10)or,whichissimpler,thinkingthatthefirstreferenceisobtainedfromthesecondbyarotationofanangle–θ.Consequentlywehave

(1.11)

Wehaveconsideredtwoframesdifferingforarotationoftheaxes,withacommonorigin.Considernowtwoframesdifferingforatranslation,namelywithparallelaxesanddifferentorigins,asshowninFig.1.5b,againforsimplicityinaplane.WeseethatthecomponentsofthevectorAinthetwoframesareequal.

1.5 OperationswithVectorsAquantityrepresentedbyanumber,liketemperatureorpressure,iscalleda

scalar.Scalarsareinvariantunderrotationsoftheaxes.Intworeferenceframesrotatedonetotheotherascalarhasthesamevalue.Noticethatnoteveryquantityisscalar.Forexamplethexcomponentofavectorisnot,becauseitchangesunderrotations.

WeshallrepresentavectorwithitscomponentsinagivenframewithA=(Ax,Ay,Az).

GiventhevectorAandthescalarktheirproductisthevectorkA=(kAx,kAy,kAz).NamelythecomponentsofkAarektimesthoseofA.Tobesure,wemustverifythatthejustgivendefinitionagreeswiththedefinitionofvector.Indeed,itisimmediatetocheckthattheorientedtriple(kAx,kAy,kAz)transformslikeavector.

Geometrically,kAisthevectorwiththesamedirectionasA,themagnitude|k|timestheoneofAandthesenseofAoroppositedependingonkbeingpositiveornegativerespectively.

TheproductofAtimesthereciprocalofitsmagnitudeisavectorwiththedirectionofAandunitarymagnitude.Avectorofunitarymagnitudeiscalledaunitvectororversor.WeshallusethesymboluAfortheunitvectorofA.

TheproductofthevectorAand–1iscalledtheoppositeofA.IthasthesamemagnitudeanddirectionofAandoppositesense.

ConsidernowtwovectorsAandB,whichinagivenreferenceframehavethecomponents(Ax,Ay,Az)and(Bx,By,Bz)respectively.ConsiderthetripleofnumbersthatarethesumsofthehomologouscomponentsofAandB,namely(Ax+Bx,Ay+By,Az+Bz).Isitavector?Letuscheck.KnowingthatAareBvectorsweknowthat

Bysummingmembertomemberwehave

Weseethattheanswerispositive.Wecanthendefineasthevectorsumoftwovectorsthevectorwithcomponentsequaltothesumsoftheirhomologouscomponents.Noticethatthejustfoundpropertiesareimmediateconsequencesofthecomponenttransformationsbeinglinearoperations.

Itisimmediatetoverifythatthesumofvectorshastheusualpropertiesof

thesum,namelycommutative

(1.12)andassociative

(1.13)Figure1.6showsthegeometricmeaningofthevectorsum.InFig.1.6athe

sumismadeputtingthetailofBontheheadofA;thesumisthevectorfromthetailofAtotheheadofB,asoneimmediatelyunderstandsthinkingtothecomponents.Forthecommutativeproperty,wemighthavedoneviceversa,namelystartfromBandputtingthetailofAontheheadofB.Weshouldhavereachedthesamepoint.

Fig.1.6 Thesumoftwovectors

Figure1.6bshowsanequivalentwaytosum,theparallelogramrule.Weputbothvectorswiththetailsinthesamepointandwedrawtheparallelogramhavingthemassides.

ThevectordifferencebetweenthetwovectorsAandBisthevectorofcomponentsequaltothedifferencesbetweenthehomologouscomponentsor,equivalently,thesumofAand–B.ThegeometricalmeaningisshowninFig.1.7.

Fig.1.7 Thedifferencebetweentwovectors

Thepropertiesofvectorsums,orcomposition,whichwehavejustdiscussedlookstobeobvious,buttheyarenot.Indeedtheyarevalidifthespaceisflat,notifithasanycurvature.Tomakethingssimpler,considertwodimensions.Aplanesurfaceisflat,butnotasphericaloneorasaddleshapedone.Asamatter

offactthesurfaceonwhichwelive,thesurfaceoftheearth,isflatonlyifweconsiderdistancessubstantiallysmallerthantheearthradius,whichhasameanvalueR=6371km,andonlyinafirstapproximation.

Letusconsiderthefollowingexampleofvectoraddition.ConsideravectorwiththetailinAat45°inlatitudeand0°inlongitudeandtheheadBonthesamemeridianat46°latitude.Thelengthofonedegreealongameridianiseverywhere10000km/90=111km.ThesecondvectorhasthetailinBandtheheadonthesameparallel100kmtowardsWest,sayinC.Nowwecommutetheoperations.Westartwithavector100kmlongfromAto,say,Donitsparallelat100kmtoWest.Thenweadda111mlongvectortotheNorthwithtailinDandhead,say,inC′.WillC′beequaltoC?TheanswerisNO.Thisisbecausethedistancebetweentwomeridiansalongaparallelisdifferentatdifferentlatitudes.Indeedtheradiusoftheparallelatthelatitudeλisr(λ)=Rcosλ,thatis7071kmat45°and6947kmat46°,whichis1.8%shorter.Consequently,C′is1.8kmWestofC.

QUESTIONQ1.2.Repeatthecalculationwiththesamevectorlengthsstartingat65°latitude.

QUESTIONQ1.1.Repeatthecalculationwithvectorlengthsof1kmstartingat45°latitude.

Thequestionwhetherornotspacehasacurvatureshouldbeansweredexperimentally,andexperimentsshowthisbeingthecase.Inparticular,themeasurementofthemeancurvatureoftheUniverseovercosmologicaldistancesisoneoftheimportantobjectsofcontemporarycosmology.Allmeasurementsarecompatiblewithzeromeancurvature,withintheiruncertainties.However,weshouldmentionthatspacecurvatureexistsinanothercontext.Generalrelativitydescribeslocalgravitationaleffectsintermsofamodificationofgeometryinthespacesurroundingamassiveobject.Itspredictionsareconfirmedbyobservations.Weshallnotdealwiththistopicinthisbook.

1.6 ScalarProductofTwoVectorsTheretwowaystotaketheproductoftwovectors,calleddotproductandcrossproductrespectively.Westartherewiththeformer.

ConsiderthetwovectorsAandB.Theirdotproductisindicatedwithadotbetweenthem,namely .Inagivenreferenceframethedotproductis,bydefinition,thesumoftheproductsofthehomologouscomponents

(1.14)Thedotproducthastheimportantpropertytobescalar,namelyinvariant

underrotationsoftheaxes.Itisconsequentlyalsocalledascalarproduct.Letusshowtheproperty,namelythat

Forsimplicity,letusconsideronlyarotationaroundthez-axis.ThecomponentsofAintherotatedframeasfunctionsofitscomponentsinthestartingonearegivenbyEq.(1.10)andsimilarlyforB.Wecanwrite

Weseethattheproductisinvariant.Itiseasytoshowthatboththecommutativeanddistributivepropertiesare

validforthedotproduct.

(1.15)Weshallseenowthegeometricmeaningofthescalarproduct.Wecanprofit

fromitbeinginvarianttochooseconvenientaxes.WetakexinthedirectionofAandyintheplanedefinedbyAandB(Fig.1.18a).Ifθistheanglebetweenthevectors,thecomponentsareA=(A,0,0)and .Theirdotproductisthen

(1.16)Inwords,thescalarproductoftwovectorsistheproductoftheirmagnitudes

timesthecosineoftheanglebetweenthem.Therearealsotwootherinterpretationsthatmaybeuseful.Thescalarproductistheproductofthemagnitudeofthefirstvectortimestheprojectionofthesecondvectoronthefirstone(Fig.1.8b),or,thesamewithinvertedroles(Fig.1.8c).

Fig.1.8 Geometricmeaningsofscalarproduct

Thedotproductiszeroifthevectorsareperpendicular,positiveiftheangleisacute,andnegativeifobtuse.

Aparticularandinterestingcaseistheproductofavectorbyitself

(1.17)Bydefinitionthesquareofavectoristhedotproductofthevectortimes

itselfandisequaltothesquareofitsmagnitudeandalsotothesumofthesquaresofitscomponents.ThelatterpropertyisanimmediateconsequenceofthePythagoreantheorem.Itisalsocalledthenormofthevector.Thenormisobviouslythesameinanyreference.

Figure1.9showsaCartesianreferenceframeinwhichthreeimportantvectorsaredrawn,theunitvectorsofthecoordinateaxes,i,jandk.Theyhaveunitmagnitudeandaremutuallynormal.Consequently

Fig.1.9 TheunitvectorsoftheCartesianaxes

(1.18)

Thecomponentsofanyvectorcanbewrittenintermsofthethreeunitvectors.Indeed,thexcomponentofthevectorAisitsdotproductwithi,becausethemagnitudeofthelatteris1,andsimilarlyfortheothercomponents.Wethencanwritethevectoras

(1.19)namelyasthesumofthreevectorshavingthedirectionsoftheaxes.Thesearecalledthevectorcomponents.

Inparticularthepositionvectorcanbewrittenas

(1.20)

1.7 VectorProductofTwoVectors

GiventhetwovectorsA=(Ax,Ay,Az)andB=(Bx,By,Bz),theircrossproductisdefinedastheorderedtripleofrealnumbers

(1.21)

Wenowshowthatthecrossproducttransformsasavectorunderrotationsoftheaxesandisalsocalledthevectorproduct.Weshowthatforthex′component,thedemonstrationfortheothertwoareexactlythesame.

Thevectorproductisnotcommutativeandtheorderofthefactorsmatters.Wehaveimmediatelyfromthedefinitionthat

(1.22)Invertingtheorderofthefactorstheproductchangessign.Thepropertyis

calledanticommutative.Itiseasytoseethatthevectorproductisdistributivetothesum

(1.23)Wenowseethegeometricmeaningofthecrossproductusingthesame

frameasintheprevioussection.WedrawthetwovectorsasstartingfromthesamepointandtakethexaxisinthedirectionandsenseofA,theyaxisintheplaneofthetwovectorsandthezaxistocompletetheright-handedreference(Fig.1.10).ThecomponentsareA=(A,0,0)and .Thecrossproducthasonlythezcomponentdifferentfromzero

Fig.1.10 Thevectorproductoftwovectors

(1.24)Hence,thecrossproductisinthepositivedirectionofthez-axisifθisinone

ofthefirsttwoquadrants(Fig.1.10a),inthenegativeoneifinthethirdandfourthones(Fig.1.10b).

Inconclusion,thegeometricmeaningofthevectorproduct,independentlyofthereferenceframe,isthefollowing.Itsmagnitudeisequaltotheareaoftheparallelogramhavingthetwovectorsassides.Alternatively,wecanalsosaythatitsmagnitudeisthemagnitudeofthefirst(A)timestheprojectionofthesecondonthenormaltothefirst(Bsinθ)orviceversa.Thedirectionoftheproductisperpendiculartotheplaneofthetwovectors.Itssenseistheoneseeingthefirstfactorgoingtothesecondthroughthesmallerangleinanticlockwisedirection.

Noticethatwehavefollowedherethesameconventionweusedtodefinethepositivedirectionofthez-axis.Inaleft-handedframe,thesenseofthevectorproductwouldhavechangedtoo.

Thecrossproductiszeroifoneofthevectorsiszeroorifthetwoareparallel.Inparticulartheproductofavectortimesitselfiszero.

Eachoftheunitvectorsoftheaxesisthecrossproductoftheothertwo

(1.25)Theexpressionsofthistypecanbeeasierrememberedthinkingthateachof

themisobtainedfromthepreviousonebycyclicpermutation.Wenowdefinethescalartripleproductofthreevectors,intheorderA,BandC.Itisthedotproductofthefirstvectorstimesthecrossproductofthesecondtimesthethird:

(1.26)Toseethegeometricalmeaning,wetakethethreevectorsstartingfromthe

samepointasinFig.1.11.

Fig.1.11 Thescalartripleproduct

Wecanconsiderthemasthesidesofaparallelepiped.AsweknowthemagnitudeofB×Cisequaltotheareaoftheparallelogramhavingthetwo

vectorsassides,whichisafaceoftheparallelepiped.ItsdirectionisthenormaltothatplaneandthepositivesenseistheonethatseesBgoingtoC,rotatingthroughthesmallerangle,inanticlockwisedirection.LetassumethatAliesonthesamesideoftheplanemadebyBandCasB×C.ThedotproductofAtimesB×CistheproductoftheprojectionofAonthedirectionofB×ChenceonthedirectionperpendiculartotheplaneofBandCtimesthemagnitudeofB×C.Butthisprojectionisjusttheheighthoftheparallelepiped.Inconclusionthetripleproductisequaltothevolumeoftheparallelepipedhavingthethreevectorsassides.Inthiscaseweareconsideringthatthisistrueinabsolutevalueandsign.ItistheoppositeofthisvolumeifcaseAliesontheoppositesideoftheplanemadebyBandCthatB×C.

Thefollowingpropertiesareimmediatelydemonstrated:thetriplescalarproductiszeroifthethreevectorsarecoplanar,hence,inparticular,iftwoorthreeareparallel.Thetripleproductdoesnotvaryifthefactorsarecircularlypermuted

(1.27)Obviouslyalso

(1.28)Asecondtripleproductisthetriplevectorproduct,whichisthecross

productofthefirstvectortimesthecrossproductofthesecondandthirdones.Bydirectverificationoneshowsthat

(1.29)

1.8 BoundVectors,Moment,CoupleTheforcesarevectors.However,tocompletelycharacterizeaforceweneedalsotoknowitsapplicationpoint.Ifwepushanobjectwithourfinger,wenotonlyexertonitanactionofacertainintensityandinacertaindirection,butalsowedothatinacertainpoint.Ifwechangethatpoint,theeffectoftheforcewouldchange.Avectorwithanassociatedapplicationpointiscalledaboundvector.Thelinewiththedirectionoftheforcethroughtheapplicationpointiscalledthelineofaction.

Figure1.12showsthevectorAanditspointofapplicationP.Itmaybeaforceforexample.WearbitrarilychooseapointΩ,whichwecallthepole.ThemomentofAaboutΩisdefinedasthevectorproductofthevectorleadingfromthepoletotheapplicationpointofA,namely

Fig.1.12 ThemomentofvectorAaboutthepoleΩ

(1.30)Letusseeitsgeometricalmeaning.Thedirectionofthemomentofthe

vectorAisperpendiculartotheplanedefinedbythesegmentΩPandA.ToseeitspositivedirectionweimagineAtobeaforceandΩParigidbar.Ifweseetheforceturningthebarinananticlockwisedirection,weareonthepositivesideofthemoment.Themagnitudeofthemomentisgivenbytheproductofmagnitudeofthedistance(hinthefigure)ofthepoleΩfromtheactionlineofA.Inparticular,ifΩliesontheactionlinethemomentiszero.

TheimportanceofthemomentswillbeclearwhenwestudythemechanicsoftheextendedbodiesinChap.7.Wenowconsiderasimpleandparticularlyimportantcase,thecoupleofvectors.Acoupleisapairofboundvectorsequalinmagnitudeinequalandoppositedirection.Thedistancebetweenthetwoactionlinesiscalledthearmofthecouple.

Averyimportantpropertyofthecoupleisthattheirmomentisindependentofthepole.Thismaybecalledthemomentofthecoupleoracoupletorque.Thetwotermsaresynonymous.

ConsiderforsimplicitythepoleΩlyingintheplaneofthecouple,asinFig.1.13(buttheargumentisvalidingeneral).ThetwovectorsareAand–A.P1andP2theapplicationpointsrespectively.Thetotalmoment,i.e.thesumofthetwomomentsaboutΩis

Fig.1.13 Acoupleofboundvectors

whichisindependentofthepole.Wecanalsoseethatthemagnitudeofthecouplemoment(ortorque)istheproductofthemagnitudeAofthevectorstimesthearmdofthecouple,namely

(1.31)Itsdirectionisperpendiculartotheplaneofthecouple,positiveontheside

seeingthecouplerotateinananticlockwisedirection.

1.9 MatricesMatricesareproperlystudiedinmathematicscourses.Inthistextbookonlyafewsimpleconceptsanddefinitionswillbeneededandarerecalledhere.

AmatrixAisanarrayofnumbersorderedinrowsandcolumns,sayMlinesandNcolumns

(1.32)

Thematrixissaidtobesquareifthenumbersofrowsandcolumnareequal;thisnumberiscalledtheorderofthematrix.Thegenericelementofthematrixisaijwherethefirstindexi(i=1,…,M)istherowindex,thesecondj(j=1,…,N)thecolumnindex.

Matriceswiththesamenumbersofrowsandcolumncanbeadded.ThesumS=A+BofsuchmatricesAandBisthematrixhavingaselementsthesumsofthecorrespondingelementsofAandB,namelysij=aij+bij.

IfthenumberofcolumnsofthematrixAisequaltothenumberofrowsofmatrixBtheproductP=ABisdefinedasfollows.BeMthenumberofrowsandNthenumberofcolumnsofA,NthenumberofrowsandLthenumberofcolumnsofB.TheproductmatrixPhasMrowsandLcolumnsanditsgenericelementis

(1.33)

Wecanusetheconceptofmatrixproducttore-writeEq.(1.10)forthetransformationofavectorbetweentworeferenceframesincompactform:

(1.34)

Weseethatvectorsarerepresentedbyamatrixwithonecolumnandthreerows,whiletherotationisrepresentedbyathree-by-threematrix.

Continuingwiththedefinitions,theminorAijofthegenericelementaijisdefinedasthematrixoneobtainsfromAsuppressingrowiandcolumnj(i.e.therowandthecolumntowhichtheelementweareconsideringbelongs).

Forsquarematrices,sayA,thedeterminantcanbedefined.Itisanumber,indicatedwith||A||orwithdetA.Thedefinitionisrecurrent.Iftheorderofthematrixisone,itsdeterminantisitsonlyelement.Iftheorderistwo,

(1.35)

Ifthematrixorderisthreeorlarger,onestartschoosingarow(oracolumn).Itcanbeshownthatthechoiceisarbitrary.Wethenchoosethefirstrow.Thenwemultiplyeachelementoftherowtimesthedeterminantofitsminor,keepingitasitis,ifthesumoftheindicesiseven(11,13,15,…),changingitssign,ifitisodd(12,14,16,…).Finallywesumallthesenumbers.Thedeterminantofthe3×3matrix

is

(1.36)

Itiseasytoshowthatiftwo(ormore)rowsortwocolumnsareequal,orsimplyproportional,thedeterminantisnull.Itisalsoshownthatthedeterminantsoftwomatricesdifferingonlyfortheexchangeoftwocontiguousrowsortwocontiguouscolumnsareequalandopposite.

Thescalartripleproductofthreevectors,sayA,BandC,canbeusefullyexpressedasthedeterminantofa3×3matrixoftheircomponents

(1.37)

thatisEq.(1.26).Thejustmentionedpropertiesofthedeterminantcorrespondtotheknownpropertiesofthetripleproduct:itisnulliftwofactorsareequalorparallel,i.e.withproportionalcomponents;invertingtwofactorsthetripleproductchangessign.

Finally,alsothevectorproductoftwovectorscanbewrittenformallyasthedeterminantofthematrixhavinginthefirstrowtheunitvectorsoftheaxes,andsecondandthirdrowsthecomponentsofthetwovectorsintheorder.Indeed

(1.38)

thatisEq.(1.21).

1.10 VelocityWeshallnowstudythemotionofthesimplestbody,thematerialpointorparticle.Thisisthecasewhenitsdimensionsaresmallcomparedtothedistancesfromotherobjects.Thisisclearlyanidealizationbutitworksofteninpractice.Forexampletheplanetsarecertainlynotpoint-like,howeverinthemathematicaldescriptionoftheirmotionsaroundthesuntheycanbeconsideredassuchinagoodapproximation,aslongaswedonotconsidertherotationsabouttheiraxes,orthevariationsofthedirectionsofthoseaxes,orthetidesontheirsurfaces.Ashipcanbeconsideredapointwhensheisfarfromshore,butwhensheentersaharborherdimensionmustbepreciselyknown.

Aswehavealreadystated,themotionhastobestudiedinagivenreferenceframe.Theparticledescribesinitsmotionacurve,whichiscalledthetrajectory,asshowninFig.1.14a.Thepositionvectorisafunctionoftimer(t)or,inotherwords,theco-ordinatesarethreefunctionsoftimex(t),y(t),z(t).Ifweknowthesefunctionswecompletelyknowthemotionoftheparticle.Wesaythatthesystemhasthreedegreesoffreedom.

Fig.1.14 aThetrajectoryofaparticle,bthevelocity

Letusconsiderthepositionvectorattheinstantoftimet,r(t)asrepresentedinFig.1.14aandanimmediatelyfollowinginstantt+Δt,r(t+Δt),whereΔtisashorttimeinterval.InthistimeintervaltheparticlehasmovedbyΔs,whichisastepinthespacehavingamagnitudeandadirection,namelyitisavector.LookingatthefigureoneimmediatelyseesthatΔsisequaltothedifferencebetweenthetwovectorsr(t+Δt)andr(t).ThisisthevariationofthevectorrinthetimeintervalΔt.Hence

(1.39)TheaveragevelocityinthetimeintervalΔtisthevectorobtainedbydividing

thedisplacementbythetimeintervalinwhichithappens:

(1.40)or,forthecomponents

(1.41)Velocityisthelimitfor oftheaveragevelocity,namely

(1.42)Inwords,thevelocityisthetimederivativeofthepositionvector.Its

componentsarethederivativesofthecoordinates

(1.43)Inthelimit thedirectionofΔsbecomestangenttothetrajectory,in

everypointofthetrajectorythedirectionofvelocityisthatofthetangentinthatpoint(Fig.1.14b).

Thephysicaldimensionsofvelocityarethoseofalengthdividedbyatime;

theunitisconsequentlythemeterpersecond(m/sorms−1).Themotionissaidtobeuniform,ifthemagnitudeofvelocitydoesnotvary

intime.Inauniformmotionhowever,thevelocityisnotnecessarilyconstant,becauseitsdirectionmayvary.Thedirectionofvelocitydoesnotvaryifthemotionisrectilinear.Henceamotionwithconstantvelocityisrectilinearuniform.

ExampleE1.1Themotionofaparticleisknownwhenitsthreeco-ordinatesasfunctionsoftimeareknown.Considerthemotiongivenbytheequations

whereaandbareconstants.Theco-ordinatesyandzarealwayszero.Consequentlythemotionisalong

thex-axis,hencerectilinear.Intheinitialinstant(t=0)theparticleisinthepositionx(0)=c.Itiscalledtheinitialposition.Astimevariesthepositionvariesinproportion,bbeingtheproportionalityconstant.Theparticlemovesinthepositivexdirection(increasingx)ifb>0,inthenegativeone(decreasingx)ifb<0.Thevelocityhasonlyonecomponentdifferentfromzero,Hence,themotionisalsouniform.

ExampleE1.2Considerthemotiongivenbytheequations

Nowthemotiontakesplaceinthexyplane,becausethezco-ordinateisalwayszero.Theinitialpositionis

Inordertofindtheequationofthetrajectorywemaytaketheratioofthedistancestravelledinthesametimealongyandx.Wefind

whichisaconstant.Thismeansthatthetrajectoryisthestraightlinethroughthepoint(c1,c2)andmakingwithx-axistheanglearctan(b2/b1).

Hence,themotionisrectilinearasshowninFig.1.15.

Fig.1.15 GeometryofthemotionofE1.2

Thecomponentsofvelocityintheplaneofthemotionare and

Thevelocityvectoristhenv=(b1,b2,0),havingthesame

directionasthe(rectilinear)trajectory.Itisconsequentlyrectilinearuniform.

ExampleE1.3Considerthemotion

(1.44)andletuscalculatethevelocity.Thereisonlyonenon-zerocomponent,namely

Thevelocityisnotconstantbutincreases(decreases)linearlywithtimeifb>0(b<0).Themotionisrectilinearbutnotuniform.

Aswehavealreadyseen,themotionofthebodiesisalwaysrelativetotheassumedreferenceframe.Consequentlyalsothevelocityisrelativetotheframe.InChap.5weshallstudyindetailtherelationsbetweenthekinematicquantities(position,velocity,acceleration,etc.)indifferentframesinrelativemotion.Weanticipatehereasimpleconcept,therelativevelocity.

Thevelocityofabodyrelativetoanotheroneisthevectordifferencebetweentheirtwovelocities.Indeed,letr1bethepositionvectorofthefirstbodyandr2thatofthesecond.Thepositionofthesecondbodyrelativetothefirstisthevector .Thetimederivativeofthisvectoristhevelocityof2relativeto1,whichisthevelocityof2seenbyanobservertravellingwith1.Callingitv12wehave

(1.45)Thevelocityofapassengerwalkingonthedeckofashiprelativetothe

vesselisthedifferencebetweenthevelocityvectorsofthepassengerandoftheshiprelativetothesea.

Noticethatthepositionof1relativeto2istheoppositeofthepositionof2relativeto1.Thesameistrueforthevelocities.

ExampleE1.4Considertwoships,AandB,whichatacertaininstantareinthepositionshowninFig.1.16.Theirvelocitiesarev1andv2respectively.ThetwocoursesinterceptinthepointoP.WilltheshipscollideinPiftheymovewithconstantvelocities?

Fig.1.16 Motionrelativetotheseaandofonesheeprelativetotheother

Theanswerisimmediateinaframefixedwithoneofthetwovessels,forexamplewithAasinFig.1.16b.Inthisframe,alltherelevantvelocities,includingthatofthesea,areobtainedfromthoserelativetotheseabysubtractingv1.HenceAdoesnotmove(bydefinition)andBmoveswithvelocityv2−v1.ThevectorRleadingfromAtoBisthesameinthetwoframes(theydifferbyatranslation).ShipB,asseenbyA,movesonthecourseshowninthefigure.HencetheminimumdistanceshewillpassfromAisAC,namelythedistanceofAfromthestraightlineBtravels.Inconclusion,theywillpassclosebutwillnotcollide.

NoticeonpurposethatapassengerAseesBmovingsideway,notinthedirectionofbow.Indeed,wehaveastrangeimpressionwhenwecrosscloselyanothership,particularlyoffshore,whenanyreferencetogroundismissing.Shelookstobetravellinginanot“natural”direction.

1.11 AngularVelocityAnimportantmotionisthecircularone,inwhichthetrajectoryisacircle.LetRbeitsradius.Itisalwaysconvenienttochoosethereferenceframetakingprofitfromthesymmetryoftheproblem,ifanyispresent.Wetaketheorigininthe

centerofthecircleandthez-axisperpendiculartoitsplane.ThemotionistheninthexyplaneasshowninFig.1.17a.

Fig.1.17 aThecircularmotion,baninfinitesimalmovement,cangularvelocityω

Wefurtherchoosetheoriginoftimeinthemomentinwhichthepointcrossesthepositivex-axis.Letϕ(t)betheanglebetweenthepositionvectorandthexaxisattimet,takenaspositiveinanticlockwisedirectionandlets(t)bethelengthofthearcsubtendedbyϕ(t),takenwiththesamesignasϕ,namelys(t)=Rϕ(t).Letdsbetheinfinitesimalmovementindt(Fig.1.17b).Theinfinitesimalchangesofsandϕarelinkedbytherelationds=Rdϕ,whereinournotationdsisthemagnitudeofdsifthemotionisanticlockwise(asinFig.1.17),andisoppositeifclockwise.Theangularvelocitymeasurestherateofchangeoftheangle.Wethenconsiderthetimederivative

(1.46)Thisquantityhasmagnitudeandasign,dependingonthesenseofrotation.

Infact,itisthezcomponentoftheangularvelocity,whichisavector.ItsmagnitudeistheabsolutevalueofEq.(1.46),itsdirectionisperpendiculartotheplaneofthemotion,takenpositiveonthesideseeingthemotionisanticlockwise.Thisisthez-axisinFig.1.17c.

Thephysicaldimensionsoftheangularvelocityaretheinverseoftime;itsunitisradianspersecond(rad/s)

Inacircularmotion,themagnitudesofvelocityυ=|ds|/dtandthemagnitudeoftheangularvelocityωarerelatedby

(1.47)Therelationbetweenthecorrespondingvectors,asimmediatelyseenfrom

Fig.1.17cis

(1.48)Letusconsiderthecaseinwhichthemagnitudeυofthevelocityisconstant.

Themotioniscircularanduniform,thearcsandthecorrespondinganglesareproportionaltothetimestakentotravelthem,namely(where,asusualthesignispositiveifthedirectionisanticlockwiseandviceversa).Hencewehavetheequationsofmotioninpolarco-ordinates:

(1.49)TheequationsofmotioninCartesianco-ordinatesare

(1.50)Asanexercisewecancheckthatthetrajectoryisindeedacircle.Takingthe

squaresofthemembersandsummingwehave whichistheequationofacircumference.NoticethatthetwoCartesianco-ordinatesxandyarenotindependentbutifweknowoneweknowalsotheother.Infacttheparticleisboundtotravelontoaprefixedtrajectory.Thesystemhasonedegreeoffreedom.Thisisevidentinpolarco-ordinates,Eq.(1.49).Twoofthemareconstant.

WenowexpresstheCartesiancomponentsofvelocity

(1.51)Thecomponentsofthevelocityvectorchangeintime:whentheparticle

movesonthecircleitsdirectioncontinuouslyvariesevenifitsmagnitudeisconstant.Indeed,themagnitudeis

(1.52)whichisaconstant.

Asafurtherexercise,letuscheckthatthevelocityisalwaystangenttothetrajectory,i.e.,perpendiculartothepositionvectoreverywhere.Toseethatwetaketheirscalarproductandget

Wenowmakethefollowingobservationthatwillbeusefulinthefollowing.Inthecasewehavenotedthatwehavetwovectors:thepositionvectorandthevelocity.Thexandycomponentsofthefirstvectorareproportionaltothecosine(Eq.1.50)andthesineoftheangularco-ordinaterespectively,thoseofthesecondtotheoppositofitssineandtoitscosinerespectively(Eq.1.51).Whenthishappensthetwovectorsareperpendicular.

Boththeco-ordinatesandthecomponentsofvelocityareproportionaltothecircularfunctionscosωtorsinωt,whichareperiodic.Infactthemotionis

periodic,meaningthatifpositionandvelocityhavesomevaluesintheinstantttheyhaveagainthesamevaluesattheinstantst+T,t+2T,etc.,foreveryt.ThetimeTiscalledtheperiodofthemotion.Itisinverselyproportionaltotheangularvelocity

(1.53)

1.12 AccelerationThemotionofabodyinwhichthevelocityvarieswithtimeinmagnitudeordirectioniscalledaccelerated.IfthechangeofvelocityinthetimeintervalΔtisΔv,theaverageaccelerationinthattimeintervalistheratio

(1.54)Theinstantaneousaccelerationattimetisitslimitfor ,namelythe

timederivativeofthevelocity

(1.55)Intheparticularcaseoftherectilinearmotion,whenthedirectionofthe

velocityisconstant,theaccelerationdirectionisalsoonthelineanditsmagnitudeandsignare

(1.56)

ExampleE1.5ConsideragainthemotionofExampleE1.3,namely

Themotionisalongthex-axiswithvelocity .Thexcomponentof

theacceleration,theonlydifferentfromzero,isthen .The

accelerationisconstantinmagnitudeanddirection.Suchmotionsarecalleduniformlyaccelerated.

Wenowconsiderauniformcircularmotioninwhichthevelocityvectorhasaconstantmagnitudeandvariesindirectionwithconstantangularvelocity.Inordertofindtheacceleration,considertheauxiliarydiagramofFig.1.18a(we

assumeananticlockwiserotationdirection).Theaxesofthefigurearethexandycomponentsofthevelocityvectorthatwethinkofashavingitstailintheorigin.Itisanalogoustothepositionvectorinthexyplane.Theanalogyiscompletebecausebothvectorsrotatewithconstantangularvelocityω.Inotherwords,theheadofthevelocityvectorAdescribesacircularlyuniformmotioninthevelocityplane,havingaradiusequaltoitsmagnitudeυ.

Fig.1.18 Uniformcircularmotion

Clearlythe“velocity”ofpointAisjusttheaccelerationofparticlePbecausethedisplacementofAinthetimeintervaldtisdvandconsequentlyits“velocity”is .Thisvectoristangenttothecircleandconsequentlyperpendiculartothevelocity(Fig.1.18a).Moreprecisely,thedirectionofaccelerationisobtainedfromthatofthevelocitybyarotationof90°inananticlockwisedirection.GoingbacktotherepresentationofthemotioninthexyplaneinFig.1.18b,theacceleration,whichwejustsawtobeat90°fromthevelocityanticlockwise,isradialdirectedtowardsthecenter.Itisthencalledcentripetalacceleration.

Weimmediatelyfindthemagnitudeoftheacceleration.WedenotebyαtheanglebetweenthevectorvandtheabscissaaxisinFig.1.18a,anddαitsvariationinthetimedt.Consideringthatthevectorrotateswithconstantangularvelocityω,wehavedα=ωdt.Ontheotherhandthechangeinvelocityis

andwegettheimportantrelation

(1.57)Summingup,ifthevelocityvariesonlyinmagnitude,theaccelerationis

paralleltovelocity,ifthevelocityvariesonlyindirection,theaccelerationis

perpendiculartothevelocity,directedtowardsthecenterofthetrajectory.WeshallseeinSect.1.14thatinthegeneralcaseinwhichbothmagnitudeanddirectionofvelocityvary,accelerationhastwocomponentsoneparallelandoneperpendiculartovelocity.

1.13 TimeDerivativeofaVectorInthestudyofuniformcircularmotionwehavedealtwiththepositionvectorrandthevelocityv.Bothareconstantinmagnitudeandvaryindirectionwithtime,rotatingattheangularvelocityω.Wehaveseenthatthemagnitudesoftheirtimederivativesarerespectivelyωrandωυ,namely,inbothcasesthemagnitudeofthevectortimesω.Inbothcasesthedirectionofthederivativevectorisat90°forwardtotheoriginalvector.Theresultisvalidalsoiftheangularvelocityisnotconstant.Indeed,wedidnotusethisassumption.

Wenowgeneralizetheargumentasfollows,withreferencetoFig.1.19.ConsiderthevectorfunctionoftimeA(t),constantinmagnitude,varyingonlyindirection.Atthegenerictimeinstantthevectorrotateswithangularvelocityω,notnecessarilyconstant.Letupbetheunitvectorrotatedbyπ/2relativetoAinthedirectionoftherotation.ThetimederivativeofAis

Fig.1.19 Arotatingvectoranditstimederivative

(1.58)orbetter

(1.59)Thisimportantformulathatweshalluseofteninthefollowingisdueto

Siméon-DenisPoisson(1781–1842)andiscalledaPoissonformula.ItisvalidifthemagnitudeAisconstant.

InthegeneralcaseinwhichthevectorAvariesbothindirectionandmagnitude,itstimederivativeisimmediatelyobtainedbywritingthevectorastheproductofitsmagnitudeanditsunitaryvector

ButthevectoruAisconstantinmagnitude,beingunitary,andwecanusethePoissonformulaforitsderivative.Weget

(1.60)whichisanimportantresultthatweshalluseofteninthefollowing.

1.14 MotiononthePlaneWenowconsiderageneralmotioninaplane.Weindicatewithuttheunitvectortangenttothetrajectoryinitsgenericpointinthedirectionofthevelocityinthatpoint.Ingeneralutvariesintime.Figure1.20showsthesituationintwoconsecutiveinstants.

Fig.1.20 aTheaccelerationvectorintwodifferentpointsofthetrajectory;btheosculatingcircle

Ineveryinstant,i.e.ineverypointofthetrajectory,ingeneralthevelocityisdifferent.Weindicatewithuntheunitvectornormaltothetrajectory.Itspositivedirectionisthedirectionobtainedbyrotatingutby90°inthedirectionoftheinstantaneousrotationofthevelocityvector.Thisgeometricallymeansthatunisdirectedtowardsthecurvaturecenter.Thelattermaylieontheleftortherightofthetrajectorydependingonthecase.Toobtaintheaccelerationwetakethederivativeofthevelocityexpressedastheproductofmagnitudetimesunitvector,v=υut.

(1.61)Asanticipated,theaccelerationhastwocomponents.Oneistangenttothe

trajectoryandequaltothetimederivativeofthemagnitudeofvelocity.Itisnull

ifthemotionisuniform,positiveifitisaccelerated,negativeifdecelerated.Theothercomponentisnormaltothetrajectoryinanycasetowardsthe“interior”ofthecurve.Itiszerowhenthedirectionofthevelocitydoesnotvary,evenifinstantaneously,asintheflexpointsofthetrajectory.

WecanexpressthenormalcomponentoftheaccelerationintermsofthecurvatureradiusofthetrajectoryinthepointPunderconsideration.Figure1.20bshowsthesituation.ConsiderallthecirclestangenttothecurveinPhavingradiusesbetween0andinfinity.Oneofthesegiveslocallythebestapproximationofthecurve.Itiscalledanosculatingcircle,fromtheLatinwordosculum,meaningkiss.ItsradiusRiscalledthecurvatureradiusofthecurveinthepointP.Itsreciprocalisthecurvature.Inaninflexionpointthecurvatureradiusisinfiniteandthecurvatureisnull.

NowwecanapproximatethesmallcurvesegmentaroundPwiththearcoftheosculatingcircleandthinkofthepointasmovingonthatarcwithangularvelocityω=υ/R.Inconclusion,thetwocomponentsoftheaccelerationare

(1.62)Weseethatthenormalcomponentoftheaccelerationisproportionaltothe

curvatureandtothesquareofthevelocity.

1.15 FromAccelerationtoMotionFigure1.21representsthetrajectoryofamaterialpointPinagivenCartesianreferenceframe,itspositionvectorr(t),itsvelocityv(t)anditsaccelerationa(t),thatareallfunctionsoftime.

Fig.1.21 Trajectory,positionvector,velocityandacceleration

Werecalltheirexpressions

(1.63)

(1.64)

(1.65)

(1.66)Inwords,thevelocityisthetimederivativeofthepositionvectorandthe

accelerationisthetimederivativeofthevelocityorthesecondtimederivativeofthepositionvector.Weshallseeinthenextchapterthataccelerationisproportionaltotheforce.

Weconsidernowtheinverseproblem,namelytofindthevelocityandthelawofmotiononcetheaccelerationa(t)isgiven.Asthevelocityisthetimederivativeofthepositionvector,thelatterisgivenbytheintegralofthevelocityontimefromtheinitialinstantt0tothetimetconsidered,namely

Ingeneral,wewanttoknowthepositionofPatthetimetandrewritetheexpressionas

(1.67)

Weseethatknowledgeofthevelocityv(t)isnotsufficient.Weneedalsotoknowthepositionofthebodyatacertaininstantt0.Thisinstantcanbeany,butgenerallyweknowhowthemotionbegan,namelyweknowtheinitialposition.Itiscustomarytochoosethatinstantastheoriginandt0=0.

Toaquestionlike“Acarhasbeentravellingataconstantspeedof100km/h.Whereisitafter2h?”Wecanonlyanswerithastravelled200km.Wecanknowitspositiononlyifweknowfromwereitstarted.

Equation(1.67)correspondstothreeintegrals

(1.68)

Ifwewanttoknowthevelocityforagivenacceleration,weproceedinthesamemannerbyintegrating

(1.69)

or,intermsofthecomponents

(1.70)

Again,weneedtodeterminetheintegrationconstants,namelythevelocityatacertaininstant,whichisusuallytheinitialone.

Oncethevelocityisknownweneedtointegrateagaintohavethelawofmotion.Forthatweneedtoknowboththeinitialpositionandtheinitialvelocity.

1.16 FreeFallMotionThestudyofthefreefallmotionofbodiesnearthesurfaceofearthisanimportantexampleoftheuseofthejustdevelopedformalism.With“freefall”wemeananidealizedsituationinwhichtheairresistancecanbeneglectedandthebodiesmoveonlyundertheactionofgravityWeanticipatethatundertheseconditionstheverticalandhorizontalmotionsareindependentfromoneanother,asweshallstudyinSect.3.7,andthatanyfreebodymoveswithaconstantacceleration,g,whichisverticallydirecteddownwardsandhasamagnitude(approximately)g=9.8m/s2.Wechooseareferenceframewiththez-axisverticalupward,andthexandy-axesinahorizontalplane,forexampletheground.Theaccelerationofthebody,thatweshallconsiderpoint-like,P,hasthe

components

(1.71)Themotionofthebodydependsontheinitialconditions.Ifforexamplewe

dropthebodyfromacertainheightwithnullvelocityitwillmoveverticallydownwithuniformacceleration.Ifwelaunchitverticallyupwardsitwillgraduallyslowdown,stopandthenfalldown.Ifwelaunchitatananglewiththehorizontalitwilldescribeacurvedtrajectory,etc.Letusstudythesemotions.

Letusstartfromthesimplestcase.Wedropthebodyattheheighthabovegroundwithnullvelocityatt=0.Theinitialconditionsare

Thexcomponentofthevelocityatthegenerictimetis

Thexcomponentofthevelocityisidenticallyzero(i.e.iszeroateveryinstantoftime)becausethexcomponentsofbothaccelerationandinitialvelocityarezero.Asimilarargumentleadsimmediatelytoconcludethatalsox(t)=0.Thesameistruefortheycomponentsofvelocityandpositionvectors.Noticethattheinitialconditionsx(0)=0andy(0)=0dependonthereferenceframe.Itsoriginhasbeenchoseninsuchawaytohavethepointfromwhichwedroptheparticleonthez-axis.Adifferentchoicewouldhaveledtotheinitialconditions,say,x(0)=a,y(0)=b.Thetwoco-ordinatesasfunctionsoftimewouldhavebeenx(t)=a,y(t)=b.Themotionobviouslyisthesame.

Wehavefoundthatthemotionisalongthez-axis.Astheaccelerationisconstant,itisuniformlyaccelerated(accelerationmayhavebothsigns,ifwewanttobespecificwecansayaccelerated,iftheaccelerationispositive,delayedifitisnegative).Letusnowfindthevelocityinthezdirection.

(1.72)

Velocityisalwaysnegative.Indeedthebodymovesalwaysinthezdirectionwehavechosenasnegative.Wenowintegrateoncemoretofindthepositionasafunctionoftime

(1.73)

whichisthelawofthemotion.Knowingcompletelythemotion,wecanlook

forinterestingproperties,forexamplethetimetakentoreachtheground.Thisistheinstantinwhichz=0,hence andthevelocityinthatinstant

(1.74)Considernowthesameinitialconditionswiththedifferencethattheinitial

velocityhasanonzeroverticalvalueυ0.Withthesameargumentsasbefore,weobtain

(1.75)

(1.76)Weshouldnowdistinguishthetwocasesofpositive(downwards)and

negative(upwards)initialvelocity.Ifυ0<0,thevelocityisalwaysnegative.Tofindtheinstanttinwhichthe

bodyisattheheightzwesolveEq.(1.76),obtaining

WehavetwosolutionsbecauseEq.(1.76)isofseconddegreeint.However,inthecaseweareconsidering,oneofthem,theonewiththenegativesign,isalwaysnegativeandconsequentlydoesnothavephysicalmeaning.Wemustchoosethesolutionwithapositivesign,becausethemotionstartsatt=0.

Thetimeofarrivalatground,thedurationofthefall,isthetimeatwhichz=0,namely

whichisshorterthaninthecaseofnullinitialvelocity.Obviouslytheexpressionsfoundinthelattercaseareparticularcases.

Ifυ0>0,fromEq.(1.75)weseethatthevelocityispositive,namelyupwards,forawhile,butitdiminisheswithincreasingtime.Itiszerointheinstanttm=υ0/g,andnegativeinlatertimes.Indeed,thebodyreachesthemaximumheightattm,namely (seeFig.1.22a).Inthiscasebothrootsfort(z)havephysicalmeaningsprovidedt≥0.Indeed,thebodygoestwicethroughthesameheight,ifitisz≥h,firstgoinguplatergoingdown.Ifz<honesolutionisnegativeandagaindoesnothavephysicalmeaning.

Fig.1.22 Freefalltrajectorieswithinitialvelocityaverticalupward,batanangleαwiththehorizontal

Whymayithappenthatamathematicalsolutionshouldbediscardedonphysicalgrounds?Thereasonisthattheequationsstatingthe“initialconditions”donotgiveinformationonthesystembeforethe“initial”instant.Inthiscasethebodystoodstill,sayinourhand.Butitwouldhavebeenpossiblethatitwasmovingupwardsinsuchawayastoreachz=hatt=0withvelocityequaltoυ0.Thediscardedsolutionwouldhavemadesense.

Wenowsupposethattheinitialpositionisagainattheheighthaboveground,butthatthevelocityv0isatanangleαwiththehorizontal.Thisiswhathappenswhenshootingwithacannonfromthetopofatower.Wechoosethezverticalupwardsasbeforeandthexhorizontalintheplaneofzandoftheinitialvelocity.Theinitialconditionsare

Themotionisintheplanexz,asshowinFig.1.22b.Wefind,asusual,thevelocityusingEq.(1.69)andtheinitialconditions.

(1.77)Weseethatthehorizontal,x,componentofthevelocityisconstantandequal

toitsinitialvalueandthattheverticalone,z,decreaseslinearlyintime,exactlyasinthecasewehaveconsidered.

Weintegrateoncemoreandusetheinitialconditionstoobtainthelawofmotion,finding

(1.78)or

(1.79)Wenowknowcompletelythemotion.If,forexample,wewanttoknowthe

shapeofthetrajectorywemusteliminatetfromtheequationsfortheco-

ordinates.Fromthefirstonewehave ,which,substitutedinthesecondequation,gives

(1.80)whichistheequationofaparabola.Thedistancexfatwhichthebodytouchestheground,namelytherangeoftheweapon,isthevalueofxcorrespondingtoz=0.WethenputthisvalueinEq.(1.80)andsolveforx.Wefind

(1.81)

Thenegativerootsolutionisfort<0andcorrespondstotheintersectionoftheparabolaontheleftofthetower.ItisshowndottedinFig.1.22bandshouldbediscarded.Thepositiverootisthesolutionforwhichwesearched.

Wenowfindthedurationoftheshot,whichisthetimetfatwhichthebodytouchesground.Withx=xfthefirstofthe(1.79)solvedfortgives

(1.82)

Wenowfindthemaximumheightzmreachedbythebody.Thiscanbedoneindifferentways.Oneisnoticingthatthisistheheightatwhichυz=0.FromEq.(1.78)weseethatahappeningat ,whichwassubstitutedinthesecondEq.(1.79),gives

ThesameresultcanbereachedfindingthemaximumofthesecondEq.(1.79).

Itisinterestingtoconsiderthespecialcaseα=0.Wewantthetimetftakenbythebullettoreachground.Equation(1.79)become

Thebullethitsthegroundintheinstant ,which,aswesee,is

independentofυ0.Thisimpliesthatforwhateverinitialvelocity,evenifenormous,thetimetakentofallfromtheheighthisalwaysthesameandisthenequaltothefreevertical(thespecialcaseυ0=0).Inotherwords,thevertical

andhorizontalmotionsareindependent.Thelawofindependenceof(thecomponentsof)motionwasdiscoveredby

G.Galilei.Inthe“DialogueconcerningthetwoChiefWorldSystems”hewrites(translationbytheauthor):

…supposehavingonthetopofatowerahorizontallyarrangedculverin(arelativelylightcannon)andfiringpoint-blankshots,namelyparalleltothehorizon;thenforlittleormuchgunpowderchargegiventoit,suchthatthecannonballwouldfallatadistanceofeitheronethousandsarms,orfourthousand,orsixthousand,ortenthousand,etc.,alltheseshotswouldtakeplaceintimesequaltoeachother,andeachequaltothetimetheballwouldtaketofallfromthecannon’smouthtoearth,whendropped,withoutanyotherimpulse,forasimpleverticalfall.Indeeditlooksreallywonderfulthatinthesameshorttimeoftheverticalfallfromaheight,forexample,ofonehundredarms,couldthesameballtraveleitherfourhundred,oronethousand,orfourthousand,orevententhousandarms,insuchawaythatinallthepoint-blank(horizontal)shotsitwouldbeintheairforequaltimes.

AlittlelaterGalileispecifiesthatthatwouldbetrue

…whentherewerenoaccidentalimpedimentsbytheair…

1.17 Scalars,Pseudoscalars,VectorsandPseudovectorsInSect.1.4wehavedefinedthevectorasanorderedtripleofrealnumbersthatunderrotationsofthereferenceframetransformsinthesamewayasthetripletrepresentingthepositionvector.

InSect.1.6wehavemetascalarquantity,thedotproductoftwovectors.Wehaveseenthatitisthesameintworeferenceframesdifferingforarotationoftheaxes.Indeed,ingeneral,aquantityis,bydefinition,ascalarifitisinvariantunderchangeofthereferenceframe.Forexample,thexcomponentofavectorisasinglenumberbutisnot,properlyspeaking,ascalar,becauseitisnotinvariantunderrotationsoftheaxes.

Hence,bothvectorandscalarpropertiesareexpressedintermsoftransformationsbetweenreferenceframes.Weshallnowconsiderthebehaviorsofthesequantitiesundertheinversionoftheaxes.Itiscalledparityoperation.Itleadsfroma,say,left-handedframetoaright-handedone.

Wenowconsiderthetransformationpropertiesofphysicalquantities.Aquantitycanbescalarorpseudoscalar.Bothareinvariantunderrotations

1.1.

1.2.

1.3.

1.4.

buttheformerisinvariantunderparityoperation,thelatterchangessign,whilekeepingitsabsolutevalue.

Thedotproductoftwovectorsisascalar;the“scalar”tripleproductisapseudoscalar.Thisisimmediatelyevidentconsideringthatunderinversionoftheaxesallthethreevectorfactorschangesign.

Aquantitycanbeavectororapseudovector(alsocalledanaxialvector).Bothtransforminthesamewayunderrotations,butthecomponentsoftheformerchangesignunderinversionoftheaxes,asthepositionvectordoes,whilethecomponentsofthelatterdonotchangesign.

Thecrossproductoftwovectorsisapseudovector,becauseboththevectorfactorschangesignandtheirproductdoesnot.Wemetbothtypesofphysicalquantities.Positionvector,velocityandaccelerationare(proper)vectors;angularvelocityandmomentofavectorarepseudovectors.

Thistypeofpropertiesofthephysicalquantitiesbelongtoaclassgenericallycalledsymmetryproperties.

1.18 ProblemsThevectorVvariesbyΔV,itsabsolutevaluevariesbyΔVinthetimeintervalΔt.(a)CanΔVbelargerthanthemagnitudeofthevariation,namely|ΔV|?Cantheybeequal?

ThevectorVchangesitsverse.ExpressΔV,ΔVand|ΔV|?

Attheinstantt1thevelocityofabodyis,withcertainunits,v1=(1,3,2),attimet2isv2=(5,3,5).Find:(a)ThevariationofthevelocityΔv,(b)themagnitudeofthevariationofthevelocity|Δv|and(c)thevariationofthemagnitudeofvelocityΔυ.

Aparticletravelsonacirclewithvelocityυconstantinmagnitude.Afteracompleteturn,(a)whichisthemeanvalueofυ?(b)whichisthemeanvelocity<v>?

1.5.

1.6.

1.7.

1.8.

1.9.

Aparticlemoveswithapositionvector,inthegivenframe,

.Find:(a)velocityandaccelerationasfunctionsof

time,(b)thevelocityatt=2s.

Apointmovesuniformlyonaplanecurvetrajectorywithvelocityυ.Themagnitudeofaccelerationonacertainpointofthetrajectoryisa.Whatisthecurvatureradiusinthatpoint?

Thepositionvectorofapointis .(a)Findthevelocityandaccelerationvectorsandtheirmagnitudes.(b)Expressthescalarproductofrandv.Whatdoestheresultmean?(c)Expressthescalarproductofranda.(d)Findthetrajectoryofthepoint.(e)Howwouldthemotionchangechangingthesignofy(t)?

Acyclisttravelsat10km/hheadingnorth.Windblowswithaspeed(relativetoground)of6km/hfromadirectionbetweenNandE.Tothecyclistthewindappearstocomefromthedirectionat15°fromNorthtoEast.(a)Findthespeedofthewindrelativetothecyclistandthedirectionofthewind,relativetoground.Whenthecyclistgoesback,whicharevelocityandapparentdirectionofthewind(winddidnotvary).

Weareonashiptravellingat10knheadingeast.Weseeanothership,whichweknowmovesat20kntoNorth,6milesdistantintheSouthdirection.Whatistheminimumdistancethetwoshipswillbe(withoutchangingtheircourses)?Afterhowmuchtime?RefertoFig.1.16.N.B.Ontheseadistancesaremeasuredinnauticalmilesandvelocitiesinknots(1kn=1mile/h).Assumeforthemiletheroundfigureof1800m.

1.10.

1.11.

1.12.

Consideraflatplatformrotatingwithangularvelocityω1=Kt2kwherekistheunitvectorofthez-axisdirectedverticallyupwards.Abodyontheplatformrotateswithangularvelocity,relativetoit,ω2=2Kt2i(thexaxisishorizontal).K=1rad/s3.(a)Findthedirectionofthebodyrelativetotheground.(b)Findtheangleϕofwhichthebodyhasrotatedrelativetogroundatt=3s.(c)Doesthemagnitudeoftheresultantangularvelocityvaryintime?Anditsdirection?

AnairplaneisflyingatconstantvelocityV,ofhorizontaldirectionandmagnitudeV=100m/sattheheighth=5000m.A(super)cannononearthshootsagainstitaballatthemomentinwhichtheplaneisjustabovetheweapon(Fig.1.23).Thevelocityoftheballisυ0=500m/s.Neglectingthepresenceofair,find:(a)theangleαatwhichwemustshoottohittheplane;(b)thetimeofthecollision(whichofthetwosolutionshouldbechosen?);(c)howmuchdidtheplanetraveluptothismoment.

Fig.1.23 Theplaneandthecannonofproblem11

ThewheelshowninFig.1.24rotateswithoutslipping.Itsaxismovesforwardatthevelocityv.Findthevelocities(namelytheircomponentsonthetwoco-ordinateaxes)ofthepointsA,B,C.

Fig.1.24 Thewheelofproblem12

(1)

©SpringerInternationalPublishingSwitzerland2016AlessandroBettini,ACourseinClassicalPhysics1—Mechanics,UndergraduateLectureNotesinPhysics,DOI10.1007/978-3-319-29257-1_2

2.DynamicsofaMaterialPoint

AlessandroBettini1

DipartimentodiFisicaeAstronomia,UniversitàdiPadova,Padova,Italy

AlessandroBettiniEmail:alessandro.bettini@pd.infn.it

Inthischapterwestudythedynamicsofamaterialpoint,namelythelawsgoverningmotionbyitscauses,whicharetheforces.Weshallthenstartbydefininganddiscussingtheconceptofforce.TheexperimentalmethodwasintroducedbyGalileoGalileiattheendoftheXVIcentury.Healsodiscoveredpartofthelawsofmechanics.ThecompletetheoryofmechanicswasbuiltbyIsaacNewton,whopublishedin1686the“PhilosophiaeNaturalisPrincipiaMathematica”,knowngenerallyassimply“Principia”.

ThelawofinertiawasdiscoveredbyGalileiandassumedbyNewtonasthefirstlawofmechanics.ItwillbestudiedinSect.2.3.Thelawstatesthatabodyinabsenceofforcesactingonitmovesnaturallywithconstantvelocityinastraightline,arectilinearuniformmotion.ThesecondlawwasalsodiscoveredbyGalileiandpreciselyformulatedbyNewton.Itstatesthattherateofchangeofthemomentum,avectorthatweshalldefine,namelyitstimederivative,isequaltotheforceactingonthebody.Inanequivalentmannertheaccelerationisproportionaltotheforce.ThisisthesubjectofSect.2.4.InthesamesectionweshalldiscussNewton’sthirdlaw,theaction-reactionlaw.

ThereareseveraltypesofforceinNature,asweshallseeinthenextchapter.Inthisone,however,inSect.2.5,weshalltalkofweight,theforceactingonallthebodiesnearthesurfaceoftheearth.AfewexampleswillbediscussedinSects.2.6and2.7.

InSect.2.8weintroducetwoofthefundamentalmechanicalquantities

(beyondmomentum,orquantityofmotion,alreadyintroducedinSect.2.4),theangularmomentumandthemomentofaforce.

InSect.2.9weshallstudyasimplebutveryimportantsystem,thependulumanditsharmonicmotion.Weshallalsoseehowtwoconceptsofmass,theinertialandthegravitationalmass,areinfactonlyone.

AfterhavingintroducedtheconceptofworkmadebyaforceandshownthetheoremofenergyconservationinSect.2.10,weshalldescribeaninterestingexperimentbyGalilei.Itestablishesthattheworkdoneonabodybytheweightforcedependsonlyonthedifferencebetweeninitialandfinalheights,notontheparticularpathfollowed.Inmodernlanguagetheexperimentestablishedthattheweightforceisconservative.ThisveryimportantconceptwillbedefinedinSect.2.13.Wethendemonstratetheenergyconservationtheorem.Energyconservationisafundamentallawofallphysics.Weshalldealinthisbookonlywithmechanicalenergy,initskineticandpotentialforms,butwewarnthereaderthatotherimportantformsofenergyexist,inparticularthermalenergy,asweshalldiscussinthesecondvolumeofthiscoursewhendealingwiththermodynamics.

Thehistoricalprocessleadingtoaprecisedefinitionoftheconceptofenergyandtotheestablishmentofthelawofenergyconservationtookmorethantwocenturies.StartingwithGalilei,itcametomaturityaroundmidXIXcentury,withtheexperimentsofMayerandJouleandenunciationoftheenergyconservationlawbyMayerandHelmholtz.WeshallgivesomehintsinSect.2.14.

InSect.2.15weshalldiscussaparticulartypeofforce,thecentralforces.Thegravitationalattractionofthesunonaplanetisanimportantexampleofthiscategory.

Inthelastparagraphweintroducetheconceptofpower,whichistheworkdonebyaforceperunitoftime.

2.1 Force,OperationalDefinitionTheprimitiveconceptofforceislinkedtomuscularstrain.Ifweliftaweight,pushanobject,wemustexertaforcewithourhandsandarmsandwefeelstrain.Sinceancienttimeshumansdevelopedsimplemechanicaldevicestoexertforcesoramplifythemusculareffect.Thestringofanarcher’sdrawnbowexertsaforceonthearrow,throwingitintheair;alevercanbeusedtoliftbigweights,etc.However,inphysicstheconceptmustbequantitative.Forthat,wemustdefineforceaccuratelyenoughtobeabletomeasureit.Thismeansthatwemustbeabletocomparetwoforcesandestablishwhentheyareequal,whenoneis

twicetheother,etc.Inotherwordswemustbeabletodeterminetheratiobetweentwodifferentforces.

Adirectmethodtocomparetwoforcesisbasedontheleverrule,whichwasdiscoveredbyArchimedesofSyracuse(287–212BC)morethentwothousandyearsago.Therulestatesthattwoequalforcesbalancewhenappliedatequaldistancesontwosidesofthepivot(Fig.2.1a)andthattwodifferentforcesF1andF2balancewhenappliedatdistancesfrompivot(l1andl2respectively)inverselyproportionaltotheforces(Fig.2.1b),i.e.suchas

Fig.2.1 Comparisonoftwoforces

(2.1)Thefirststatementcanbeprovensimplywithsymmetryarguments.Ifthe

twoforcesareequalandthetwoarmsareequal,thesystemissymmetric.Howcoulditchooseonwhichsidetobend?Thesecondstatementonthecontrary,namelythevalidityofEq.(2.1),mustbeexperimentallyverified.

Weknowthataspringexertsaforcewhencompressedorstretchedrelativetoitsnaturallength;wefeelthemuscularstrainwhenwecompressorpullit.Webuildacertainnumberofspringsasequalaspossibletoeachother.Wecanthenverifythattheyexertequalforceswhencompressed(orstretched)inthesamemeasurebyapplyingthoseforcesatequaldistancesfromthepivotofaleverasinFig.2.1a.Wecannowdefineasunitarytheforceexpandedofaspecificlength(N.B.:thisisnottheofficialdefinition).

Wecanthendefinethemultiplesoftheunitforce.Ifforexample,wewantaforceofthreeunits,weputthreeofourspringsinparallel.WecanexperimentallyverifytheleverruleEq.(2.1)asshowninFig.2.2bwithdifferentcombinationsofunitforces.Oncewehavestatedthat,wecanuseittomeasureforces.Asamatteroffactthemethodhasbeenusedinsteelyardssinceveryancienttimesandisstillusednowinfruitorothergoodsmarketstoweighawidevarietyofgoods.Theweighttobemeasurediscomparedwiththeweight

ofastandardobjectseekingforequilibriumbychangingthelengthoftheleverarmofthelatter.

Fig.2.2 Thedynamometer

Intheoperationaldefinitionoftheforcewehavejustchosen,wedidnotmakeanyhypothesisontherelationbetweentheforceexertedbythespringanditslength.However,thisdefinitionisnotsimpletouseinpractice.Ahandierdeviceisthedynamometer(fromtheGreekdynamiforforceandmetroformeasure).

Thedynamometer,shownschematicallyinFig.2.2,ismadeofaspringfixedatoneextremeonawood,orothermaterial,plateandwitharingatitsotherextreme.Theforcetobemeasuredisappliedtothering.Apointermovingonascalegivesameasurementofthedilationofthespring.Oncewehavebuiltthedevicewemustcalibrateit.Withtheabovedescribedprocedurewehavebuiltanumberofsprings,multipleandsubmultiplesoftheunit.Weapplyeachofthemtotheringandmarkthepositionofthepointeronthetable.Inthiswaywebuildascaleonwhichwewillreadthevaluesofunknownforces.Inpractice,wefindthatthescaleislinear,namelythestretchisproportionaltotheappliedforce,ifthestretchisnottoolarge.However,thispropertyiscomfortable,butnotnecessary.

Themethodwehavedescribedisusedinpractice,butdoesnotallowaprecisedefinitionofforce.IntheSItheunitofforceisaderivedone,Itistheforceimpartingtheunitacceleration(1m/s2)totheunitmass(1kg).Itiscallednewton(N).Tohaveanideaoftheorderofmagnitude,thinkthattheweightofoneliterofwater,1kg,isabout9.8N.InotherwordsoneNewtonisabouttheweightofthewaterfillingaglass.

2.2 ForceIsaVectorIngivingtheoperationaldefinitionofforceintheprevioussectionwehaveimplicitlyassumed,andwedidthatbydefinition,thattwoequalandoppositeforceswhenappliedtoapointdonotcauseacceleration.Namely,thetwoforcesareinequilibrium.Clearly,aforcenotonlyhasamagnitudebutalsoadirection.Wecanexertaforceonabodyapplyingoneofourspringsandpullingin

differentdirections.Weareledtothinkthatforceisavectorquantity.However,theconclusioncannotbereachedbylogic,ratheritneedsexperimentalverification.Tobeavector,aquantitynotonlyshouldhaveamagnitudeandadirection,butalsosatisfytheruleofadditionofvectors.

TheexperiencewiththreeforceswasoriginallydevisedbyPierreVarignon(1654–1722),acontemporaryofNewton.ItsdeviceisshowninFig.2.3.Intheplaneofthefigure,whichisvertical,threepulleysarefixed.Thethreeweightsofmassesm1,m2andm3,actbymeansofwires,drawninthefigure,joinedinthepointO.Theforcesexertedbythewireshavemagnitudesproportionaltotheweightsandthedirectionsofthewires.OncewehavejoinedthethreewiresinOandletthesystemalone,thesystemmovesuntilitreachesitsequilibriumconfiguration,theonerepresentedinthefigure.Weknowthevaluesoftheweights,sayF1,F2andF3,andmeasuretheanglesθ1andθ2.Wefindthatthefollowingrelationsaresatisfied:

or

Fig.2.3 Varignonexperimentshowingthecompositionofforces

TheVarignonexperimentandsimilaronesmadeafterwardsverifythevector

characteroftheforce.Themostprecisetests,however,areindirectandcomefromtheagreementoftheexperimentaldatawiththepredictionsmadeunderthishypothesisinthemostdifferentconditions.

Oncewehaveestablishedthatforcesaddasvectors,wedefineastheresultantofthesetofforcesF1,F2,F3,…andtheirvectorsum

(2.2)Letusnowthinkofsomeforcesthatweknowfromoureveryday

experience.Wecandistinguishtwotypes.Thejustconsideredforcesexertedbyaspring,theforceatableexertsonanobjectitsupports,theforceweexertwithourhandpushinganobject,areeachexertedbycontact.Abody,thespring,theplaneofthetableandthehandeachapplyforcetotheobjecttouchingit.Theeverydayexampleofthesecondtypeofforceisweight.Weightistheforcewithwhichearthgravitationallyattractsallbodies.Itisdirectedverticallydown,towardsthecenterofearth.Thisforceisexertedatadistancei.e.,itdoesnotneedcontact.

2.3 TheLawofInertiaOneofthemostrevolutionarydiscoveriesofGalileiwastheestablishmentofthebehaviorofabodynotsubjecttoforces.Theproblemliesinthefactthatinpracticeitisimpossibletoeliminatealltheforces.Weightisalwayspresentonearth.Itcannotbeeliminated,butitcanbebalanced.Ifweputabodyonahorizontalplane,thelatterwillexertonthebodyaforceequalandoppositetoitsweight.However,whenthebodymoves,frictionalforcesduebothtothecontactbetweenthesurfacesoftheplaneandthebodyandtheairarepresent.Theeffectofthese“passive”forcesismuchmoredifficulttocontrolandwasnotknownbeforeGalilei.

Considerthefollowingexperiment.Weputabronzesphereonahorizontalplane.Wethengiveitapush.Thatis,weapplyaforceforabrieftimeinterval,givingitacertaininitialvelocityontheplane.Weobservethesphere’smotionandseethatitsvelocitygraduallydecreasesandfinallystops.Tohavethespheremovingatconstantvelocityweneedtoapplyaforcecontinuously.Theconclusionseemstobethat,whennotactedonbyforces,abodystandsstill.Ifitmovesatconstantvelocityitisacteduponbyaforceproportionaltoitsvelocity.Wenowknowthattheconclusion,thoughttobetrueforcenturies,isactuallyfalse.

Galilei’sargumentcanbesummarizedasfollows.Thefactthat,whenweapplyaforcetoabodyandthenweceasetoapplyit,thebodyslowsdownand

finallystopsisobviouslytrue.Butthecauseisnottheabsenceofactingforces.Onthecontrary,thecauseisthepresenceofforcesthatwedonotapply,wedonotsee,yetexist(theyarecalledpassive)andweareunabletoavoid,likefrictionandairdrag,

Galileicouldnotprovehisstatementexperimentallybyeliminatingallthepassiveresistiveforces.Heobservedhoweverthat,whenlaunchingasolidpolishedsphereofbrassorivoryonahorizontalguide,thedistancetravelledbythespherebeforecomingtorestwaslongerandlongerwhenthesurfacesoftheguideandthespheresweresmootherandsmoother.Mentallygoingtothelimitofinfinitesmoothness,heconcludedthatinthoseconditionsthespherewouldneverstop,butwouldcontinuetomoveforeverwiththesamevelocity.

Theconclusionisthelawofinertia.InthewordsofNewton

Everybodypreservsinitsstateofrest,orofuniformmotioninarightline,unlessitiscompelledtochangethatstatebyimpressedforces.

Thelawofinertiaisnothowevervalidinjustanycircumstance.Whetheritisvalidornotdependsonthereferenceframe.Uptonowwehavemadeexperimentsinareferencefixedtoearth.Wenowsupposethatwewanttobuildalaboratoryonacarriagemovingonstraightrailsatconstantvelocity,relativetoearth.Inourlaboratorywehaveasmoothhorizontalplane.Welayabronzesphereonthetableandobservethat,asexpected,itremainsstill.However,suddenlythespheremoves,acceleratesandmovesquicklyforward,withoutanyvisibleforceactingonit.Whatdidhappen?Ithappenedthatthecarriagesuddenlystartedtoslowdowntillcomingtorest.Evenifourlaboratoryisclosedwithnowindowtolookout,weknowthatthecarriagedeceleratesbecausewealsoexperienceamysteriousforcepushingusforwards.

Anobserveronearth,namelyintheframewehadbeenconsideringabove,easilyinterpretsthephenomenon.Thesphereisfreetomovehorizontally,thetablebeingsmooth.Aforceacteduponbybrakesonitsreelshasslowedthecarriagedown.Thisforce,however,doesnotactonthesphere,becausethesupportplaneissmooth.Theresultantoftheforcesonthesphereisnull.Forthelawofinertiaitwillcontinueinitsmotionwithconstantvelocity.Thisisrelativetotheground.Buttheobserveronthecarriage,whichslowsdownrelativetotheground,seesthesphereacceleratingtoreachthevelocitythatthecarriagehadbeforebraking.

Areferenceframeinwhichthelawofinertiaisvalidiscalledaninertialframe.Weshallseethatinertialframeshaveaprivilegedroleinmechanics,andmoregenerallyinphysics.

Moreprecisely,thelawofinertiacanbestatedas:Referenceframesdoexistinwhicheverybodynotsubjecttoforceindefinitelyremainsinitsstateofrestoruniformrectilinearmotion.

Onemightthinkthatthelawofinertiaisaconsequenceofourdefinitionofinertialframe,inotherwordsthattheargumentiscircular.Butthisisnottrue.Indeed,wecangivearbitrarilyanydefinitionwelike,butwecanneverestablishbydefinitionalawofnature,namelyhowshebehaves.Theexistenceofinertialframesisalawofnaturenotadefinitionbymen.

Wefurtherobservethatwehaveconsideredinertialanyreferencestationaryonearth.Theconclusioncomesfromthefactthat,whiledoingexperimentsinsuchlaboratories,weneverobserveobjectssuddenlymovingwhennoforceactsonthem,nordowefeelasthoughwearebeingpushedinonedirectionoranother.However,theconclusionisvalidonlyinafirstapproximation.Accuratemeasurementsshowthatframesthatarestationaryoneartharenotexactlyinertial.Thisisduetothefactthatearthmovesaroundthesunandrotatesonitsaxis.WeshallcomebacktothatinChap.4.Forthemomentitwillbeenoughtoknowthatstationaryreferenceframesoneartharecloseenoughtobeinertialforthevastmajorityofmeasurementscarriedoutinlaboratoriesand,ontheotherhand,proceduresexisttodefineinertialreferencesystemswithalltherequestedprecisionincasethisisneeded.

2.4 TheNewtonLawsofMotionInthePrincipia,Newtonbeginsbystating,asaxiomsinducedfromtheexperiments,thethreefundamentallawsfromwhichthedescriptionofallthemechanicalphenomena,bothonearthandintheUniversecanbededuced.Thefirstlawisthelawofinertiawealreadydiscussed.Thecausesofanychangeofthestateofrestorrectilinearuniformmotionofabodyaretobesearchedforinthebodiesaroundit.Forexampletheracketthathitsitchangesthestateofmotionofatennisball,thestateofthecompassneedleischangedbythepresenceofamagnet,etc.Thesamehitimpartedwitharackettoatennisorping-pongballproducesdifferentaccelerationsinthetwobodies.Bytheterminertialmasswemeanthecharacteristicofagivenobjectthatmakesitmoreorlessresistanttochangingitsstateofmotionundertheactionofagivenforce.Galileihadalreadyprovenwithhisexperimentsthatabodyundertheactionofaconstantforce,itsweightoracomponentofitsweight,moveswithaconstantaccelerationinthedirectionoftheforce.

Letusstudythephenomenonquantitatively.Wehavealreadybuiltspringsproducingforcesofdifferentmagnitudes.Wehaveperformedananalogous

1.

2.

procedureformass.Wehavebuiltanumberofblocksofthesamematerialmakingthemasequalaspossibletoeachother.Wecansaythatoneblockhasunitinertialmass,twoblocksinertialmassequaltotwo,etc.

Wehavealsopreparedahorizontalplane,thefunctionofwhichistoequilibratetheweightsofourblocks.Inourexperimentsweshallputtheblocksinmotionslidingontheplaneandwewanttoreduceasmuchaspossiblethefrictionforcesbetweentheplaneandtheblocks.Wepreparethesurfaceoftheplaneassmoothaspossible.Wecanalsoplaythefollowingtrick.Wecanbuildtheblockswithacavityinsideandaseriesofholesbetweenthecavityandthelowerface.Wefillthecavitywithdryice(frozenCO2),whichwillsublimatepushingCO2gasthroughtheholes.Thethinlayerofgasbetweentheblockandplanesurfacesreducesfrictiontonegligiblevalues.

Weattachoneofourspringstooneblock,wegiveitacertaindeformation,stretchorcompression(Fig.2.4a).Weobservethatthebodymoveswithconstantacceleration,saya0,inthedirectionoftheforce,aslongaswekeepconstanttheforce(i.e.thedeformation)

Fig.2.4 Simpleexperimentstostudytherelationbetweenforce,accelerationandinertialmass

Weattachtwosprings(Fig.2.4b)totheblockandgivethemthesamedeformationasinthefirstexperiment.Weobservethebodymovingagainwithconstantaccelerationinthedirectionoftheforce.Theaccelerationistwiceaslarge,2a0.

3.

Wefixtwoblocksoneontopoftheotherandattachonespringtowhichwegiveoncemorethesamedeformation.Theaccelerationisnowonehalfasinthefirstexperiment,a0/2(Fig.2.4c).

Continuingwithsimilarexperimentschangingtheforceonabodyorthe

inertialmass,wecometotheconclusionthatitsaccelerationaisproportionaltotheforceFandinverselyproportionaltoitsinertialmassmiandwewrite

Wecandobetter,becausewehavefoundthataccelerationandforce,whicharetwovectors,havethesamedirection.Thesecondlawstatesthat

(2.3)Thisistheformthatismoreoftenexpressed.However,Newtonstateditas

Achangeofmotionisproportionaltothemotiveforceimpressed,andtakesplaceinthedirectionoftherightlineinwhichtheforceisimpressed.

ThequantitycalledbyNewton“motion”isafundamentalvectorquantity,p,nowcalledquantityofmotion,ormomentum(sometimeslinearmomentum).Itisthevelocitytimestheinertialmass

(2.4)Twobodiesofdifferentmassescanhavethesamequantityofmotioniftheir

velocitiesareintheinverseproportionofthemasses.ThesecondNewtonlawis

(2.5)Inwords,therateofchangeofthemomentumofamaterialpointisequalto

theforceactingonit.Consideringthatmiisaconstant,andusingEq.(2.4)wehave

(2.6)Asforthelawofinertia,thesecondlawisnotvalidineveryreference

frame.Recalltheexampleofthesphereinalaboratoryonacarriagethatstarts

suddenlytoacceleratewithoutanyforcebeingacting.Likethefirstlaw,thesecondNewtonlawisvalidonlyininertialframes.

Equation(2.3)saysthataccelerationhasthesamedirectionastherelevantforce.Thismayappeartobeobviousbutitisnottrueineverycircumstance.Theequationalsosaysthattheaccelerationduetoagivenforceactingonagivenbodyisindependentofthevelocityofthebody.Experimentsshowthatbothofthese,whiletrueatcommonexperiencevelocities,arenotsoforvelocitiesclosetothespeedoflight.Intheseconditions,calledrelativistic,Eq.(2.3)fails.However,eveninthesehighvelocitiesregimes,Eq.(2.5)remainsvalid,namely,asNewtonstated,theforceandthetimederivativeofmomentumareequal.Whatneedstobechangedistherelationbetweenmomentumandvelocity.

WeshallstudyrelativisticmechanicsinChap.6;weanticipatethatinarelativisticregime,theconceptofinertialmassremainsexactlythesame.Massisaconstant,independentofvelocity,characteristicofthebody.Theconceptofmomentumhowevermustbemademoregeneral.Itsexpressionis

(2.7)whereγ(υ)isafunctionofvelocity,calledtheLorentzfactor,afterHendrikLorentz(1853–1928),oneofthefathersofrelativisticmechanics.Itsvalueisverycloseto1uptovelocitiesclosetothatoflight,c≈3×108m/s,butincreasesveryrapidlywhenυapproachesc.1

Forcomparison,thespeedoftheearthrelativetothesunisabout3×104m/s,10−4ofthespeedoflight,thespeedsofthestarsrelativetotheirgalaxies,includingoursun,areanorderofmagnitudelarger,butstill10−3ofthespeedoflight.Forthelatter,theLorentzfactordiffersfrom1onlyby0.5×10−6.

AsecondlimitofvalidityoftheNewtonlawsisatverysmalldimensions.Indeed,classicalphysicsceasestobevalidandmustbemodifiedinquantumphysics,atatomicscales.Thesehoweverareverysmallcomparedtotheobjectsofeverydayexperience,e.g.,atomicradiusesaretypically30–300pm.

TheNewtonlawgivestheaccelerationoncetheforcesareknown.Consequently,intheanalysisofanymotionwedealwiththepositionvector,thevelocity,whichisitsfirsttimederivative,andtheacceleration,itssecondtimederivative.Wedonotneedhigherderivatives.Forthesereasonswedidnotgobeyondthesecondderivativeofthepositionvectorwhenwestudiedkinematics.Werecallonpurposethattoknowthemotionofaparticleweneedtoknownotonlytheactingforces,butalsotheinitialpositionandvelocity.

Letusnowlookatanotheraspect.Thesecondlawcanbeusedinthreemainways:

1.

2.

3.

Ifweknowtheinertialmassofabodyandalltheforcesactingonit,andtheinitialconditions,wecancalculateitsmotion

Ifweknowthemotionofabodyanditsinertialmass,wecaninfertheforcesactingonit.

Distinguishingthetwopointsofviewisnotastrivialasitmaylook.Thefirstpointofviewisdeductive.Thelawsofmechanicsareusedtocalculatethemotionofbodiesinallpossiblecircumstances.Inthiswayphysicistsandengineersdesignmechanicaldevicesandengines.Thesecondpointofviewisinductiveandisthepointofviewtakentomakeprogressinphysics.Thechallengeofthephysicsresearchistounderstandfromthestudyofmotionthefundamentalnatureoftheforcesthatcauseit.ThisisthewayfollowedbyNewtontodiscoveruniversalgravitationfromstudyofthemotionsofheavenlybodies.ThisisthewayinwhichErnestRutherford(1871–1937)discoveredtheatomicnucleusin1911whenstudyingthescatteringofenergeticalphaparticlebyathingoldsheet.Thisisthewayfollowedtodaytostudythepropertiesofatomicnucleiandelementaryparticles.

WecanstatethatthesuccessoftheNewtonlawisjustasfollows.Itsubstantiallytellsus:ifyouseeabodythatdoesnotmoveinauniformrectilinearmotion,aforceshouldact.Searchforitandsearchforthephysicalagenttowhichitisdue.Youwillfindaforce,themathematicalexpressionofwhichwillbesimpleand,asaconsequence,youwillbeabletolaydownasimpletheory.FromthispointofviewtheNewtonlawisaresearchprogram.WeshallseeinChap.3that,indeed,thevariousforcesofnaturehavesimpleexpressionsintermsoftheco-ordinatesandcharacteristicsofthesystem.Theprogramissuccessful.

Athirdpossibilityisthat,ifweknowbothforcesandmotionwecandeducetheinertialmassofthebody.Toknowthemassoftheprotonforexample,wecanmeasurehowitsmomentumandenergyvaryundertheactionofaknownforce.

Thelawofcompositionofforces.Ifmorethanoneforceactatthesametime

onthematerialpointwearediscussing,theireffectisthesameasifonlyone

forcewereacting,equaltotheresultantofthoseforces.ConsiderforexamplethattwoforcesareappliedasinFig.2.5.ThefirstspringexertstheforceF1inthexdirection.WhenactingaloneitproducestheaccelerationF1/mialongx.ThesecondspringexertstheforceF2intheydirection.WhenactingaloneitproducestheaccelerationF2/mialongy.Toknowwhathappensifthetwoforcesactcontemporarilyissomethingthatcannotbefoundbylogic,ratherithastobefoundexperimentally.Indeed,whatexperimentsshowisthattheaccelerationisjustwhatonecalculatesassumingthatonlyoneforcewereacting,equaltotheresultantFofF1andF2.Inotherwords,theobservedaccelerationisa=F/mi.

Fig.2.5 Twoforcesactingatthesametime

ThethirdNewtonlawisthelawofaction-reaction.Ifabodyexertsaforce(anaction)onasecondbody,thesecondalways

exertsonthefirstaforce(areaction)thatisequalandoppositeonthesamelineofaction.

Givenitsimportance,wereproducehowitisstated,inanequivalentmanner,byNewton.

Toeveryactionthereisalwaysopposedanequalreaction:or,themutualactionsoftwobodiesuponeachotherarealwaysequal,anddirectedtocontraryparts.

Newtongivesthenafewexamples.

Whateverdrawsorpressesanotherisasmuchdrawnorpressedbythatother.Ifyoupressastonewithyourfinger,thefingerisalsopressedbythestone.Ifahorsedrawsastonetiedtoarope,thehorse(ifImaysosay)willbeequallydrawnbacktowardsthestone;forthedistendedrope,bythe

sameendeavortorelaxorunbenditselfwilldrawthehorseasmuchtowardsthestoneasitdoesthestonetowardsthehorseandwillobstructtheprogressoftheoneasmuchasitadvancesthatoftheother.

Wenoticethat,differentlyfromthefirsttwo,thethirdlawdealswithtwo,ratherthanone,bodies.Ittellsusthatisolatedforces(actions)donotexist,onlyinteractionsdoexist.

Payattentiontothefactthatactionandreactionsareappliedindifferentpoints,oneononebody,theotherontheotherbody.Ifwepushastonewithafinger,theactionofthefingerisappliedinapointofthestone;thereactionofthestoneisonthetipofourfinger.Theforceexertedbythehorsedrawingthestoneisexertedonthestonethroughtherope,thereactionacts,againthroughtherope,inthepointofthehorseattheendoftherope.Everyobjectwhetheritisfallingorlayingonasupport,weighs,meaningthattheweightforceisappliedonit.Weightistheforcewithwhichtheearthattractsallbodies.Asareaction,eachbodyattractstheearthwithanequalandoppositeforce.Thereactionisappliedtoapointoftheearth,itscenter.

Theaction-reactionprinciple,asallphysicallaws,mustbeexperimentallyverified.Directverificationsarebasedonthefactthatinacollisionbetweentwobodiesthetotalquantityofmotion,namelythevectorsumofthetwo,isconserved,meaningthatitsvaluesbeforeandafterthecollisionareequal(whileeachofthetwovary).

Thevectorswehavemetsofar,positionvector,velocityandaccelerationdepend,aswehaveseen,onthereferenceframe.Onthecontrary,forcedoesnot.

2.5 WeightWeknowfromeverydayexperiencethatallthebodiesoneartharesubjecttoaforce,verticallydirecteddownwards,calledtheweight.Wecanmeasuretheweightofabody,forexample,attachingittoadynamometerverticallypositionedandreadingonitsscalethepositionofthepointer,namelythestretchofthespring.Ifwerepeatthemeasurementindifferentpointsofourlaboratorywefindthatitdoesnotvary.However,ifwerepeatthemeasurementatmuchlargerdistances,forexampleattheEquatorandat45°latitude,oratdifferentaltitudes,forexampleatthesealevelandat2000maltitude,wenoticesmalldifferences(oftheorderofafewpermille)betweenthem.AsweshalldiscussinSect.5.7,thesesmallvariationsareduetotherotationoftheErath.Apartfromthesesmallcorrections,theweightisthegravitationalattractionexertedbytheearthonthebody.Thisisuniversal;itisthesameforcewithwhichtheearth

attractsthemoon.WeshalldiscussthisfundamentalforceinChap.4.Weanticipatethatthegravitationalattractiondecreasesasthereciprocalofthedistancesquared.Thisisoneofthereasons(theotheristherotationmotionofearth)whytheweightofanobjectisabitsmalleronamountainthanatthesealevel.

Differentobjects,inthesameplace,mayhavedifferentweights.Thismeansthattheforcewithwhichearthattractsabodydependsonacharacteristicofthebody.Westatethatthegravitationalforceonabodyisproportionaltoitsgravitationalmass,whichwedenotewithmg.Thisissimilartotheelectricattraction.AchargedbodyAatacertaindistancefromanotherbodythatisalsocharged,issubjecttoanelectricalforce.IfintheplaceofAweputabodyBwithtwicethecharge,theforceonitisdouble.Hence,theelectricforceonabodyisproportionaltoitselectriccharge.Inasimilarwaytwomassivebodies,forexampletwospheres,atacertaindistanceattractwiththegravitationalforcethatisproportionaltothegravitationalmassofeachofthem.Thisforce,ifbetweentwoobjectsofeverydaylifeisquitesmall,butcanbemeasuredwithverydelicateexperiments,asweshallseeinSect.4.7,butislargebetweenHeavenlybodies.Consideringthatthegravitationalmassisforthegravitationalforcetheanalogousoftheelectricchargefortheelectricforces,wemightcallitgravitationalcharge,butweshallsoonseethereasonwhywecallitmass.

TheweightforceFWactingonabodyofgravitationalmassmgisthen

(2.8)Thevectorquantitygdoesdependonthelocation,butinagivensiteitis

equalforallbodies.Ifristhepositionvector,thevectorg(r)isthegravitationalforceatrperunitgravitationalmass.Itiscalledgravityacceleration.Weshallseesoonthereasonforthename.Wenoticethatthegravitationalmassbeingacharacteristicofabodyisthesameinanypoint,differentlyfromitsweight.Ifwemeasuretheweightsoftwobodiesindifferentpointsontheearthwefindthateachofthemvariesabit,asalreadymentioned,buttheratioofthetworemainsrigorouslyequal.Evenifweshoulddothisexperimentonthemoon.

Operationally,thegravitationalmassisthephysicalquantitymeasuredbyabalance.Abalance,seeFig.2.6,consistsofaleverwithpivotinOandtwopans,whichweshallconsider,tomakeitsimple,exactlyatthesamedistanceonthetwosidesofO.Thebalancecomparestheweightsofthetwoobjectsonitspans.Iftheyareequalthebalanceisinequilibrium.Wehaveseenthat,bydefinition,theweightsofdifferentobjectsinthesameplaceareproportionaltotheirgravitationalmass.Wecanthenthestatethattwoobjectshavethesamegravitationalmasswhen,putonthepansofthebalance,theyareinequilibrium.

Fig.2.6 Comparingtheweightsoftwoequalmasses

Wenowneedabodyhavingunitmassbydefinition.Weputitonapan.Anotherbodyhasgravitationalmassequaltoonewhen,putontheotherpanitisinequilibrium.Abodyhasgravitationalmassequalto2,ifputonapanisinequilibriumwithtwooftheunitmassesontheother,etc.

Gravitationalmassandinertialmassaretwodifferentpropertiesofeverybody.Theformerisameasureofthestrengthofthegravitationalattractiontowhichitissubject,thelatterofhowdifficultitistomodifyitsquantityofmotion.However,weknowfromeverydayexperience,thatheavierbodiesarealsomoredifficulttoacceleratebecausetheyaremoreinert.Tosearchforamathematicalrelation,supposetoobservethefreefalloftwodifferentbodies.Theirinertialmassesarem1iandm2iandtheirgravitationalmassesm1gandm2g.Theweightofthefirstis,F1w=m1gg,theweightofthesecondF2w=m2gg.Callinga1anda2thetwoaccelerations,wehave:

whichcanbewrittenas

(2.9)Weseethatthefreefallaccelerationsofdifferentbodiesinthesameplace

areproportionaltotheratiosoftheirgravitationalandinertialmass.Consequently,ifthisratioisequalforallthebodies,lightorheavy,allofthemfallwiththesameacceleration.ThisfundamentalpropertywasexperimentallyshowntobetruebyG.Galilei.

ItisoftentoldthatGalileidroppedcontemporarilytwoballs,onemadeoflead,oneofwood,fromthePisatowerandthatheobservedthemreachinggroundatthesameinstant,showinginthiswaythattheyfallwiththesameacceleration.Theexperimentwasabsolutelysuccessandspectacularlycarriedoutin1971bytheNASAApollo15astronautD.Scottdroppingahammerandafeatheronthemoon.AsamatteroffactGalileinevermentionshavingmadehisfundamentalexperimentsinsuchaway.Henewverywellthatitcouldnotwork,

bothfortheperturbingeffectsoftheatmosphereandduetothesmallnessofthefalltimes,afactthatdidnotallowhimprecisemeasurements.Hisverypreciseexperimentsweredonewithreduced,tosayso,weightforces,withspheresoninclinedplanesandwithpendulums.WeshalldiscussthisinSect.2.9.

Wecanconcludethatthefreefallaccelerationsofallbodiesinagivenplaceareequal,actionoftheatmosphereapart.Theratiobetweengravitationalandinertialmassisauniversalconstant,thesameforallbodies.Thevalueoftheconstantisarbitrary,becausedependsonthechoiceofthetwounits.Clearly,themostconvenientchoiceistohavetheratioequaltoone.Withthischoicegravitationalandinertialmassarenotonlyproportional,theyareequal.Theunitofbothisthekilogram.Fromnowonweshallindicatewiththesamesymbol,forexamplemwithoutanysubscript,bothquantities.

2.6 ExamplesInthissectionwestudyanumberofexamplesofapplicationoftheNewtonlaws.Agoodwaytoproceedisthefollowing.

Thefirststepistoidentifyallthebodiespresentintheproblem.Nextweidentifyforeachofthemalltheforcesactingonit.Todothatitisconvenienttowrapit,ideallyinanenvelope,inordertoidentifyalltheforcesactingonthebodyfromitsexterior.Tothisaimitisoftenusefultodraweachobjectseparately,initsidealenvelope,andtheactingforcesandwritedownforeachofthemitstypeanditsagent(forexample:weightduetoearth,normalforceduetotheconstraint,frictionduetothesupportingsurface).Iftheproblemcontainsmorethanonebody,wemustidentifytheactionandreactionpairs,andthebodiesonwhichtheyact.Oncealltheforcesareidentifiedwemustcalculatetheresultantsoneachofthebodies.Todothatwechooseareferenceframe.Thechoiceshouldbeguidedbyanysymmetrytheproblemmighthave.WemustthencalculatetheCartesiancomponentsoftheresultantbysummingthecorrespondentcomponentsofalltheforces.Thecomponentsdividedbythemassofthebodyarethethreecomponentsoftheaccelerationofthebody.FromtheaccelerationwefindthelawofmotionwiththeprocedureswestudiedinSects.1.15and1.16.

ExampleE2.1.Placeablockonahorizontalfrictionlesssurfacehorizontallydrawnbyarope.

Frictionlessmeansaphysicalsurfacethatdoesnotexertforcesparalleltoit.Itisanidealization.Frictionalwaysexists,butwecanreduceit,forexamplewiththedryicetrickofSect.2.4.Weattacharopetotheblockanddrawit

horizontallywiththeforceFr.ThesituationisshowninFig.2.7.

Fig.2.7 Nnormalconstraintforce,Frforceexertedbytherope,Fwweight,duetoearth

KnowingFrandthemassmoftheblockwewanttoknowitsmotion,consideringitasapoint.Wedrawthebodyinitsidealenvelope.Weidentifytheforcesactingthroughthesurface:(1)theweightoftheblockFw,duetoearth,verticallydirecteddownwards,(2)theconstraintforceexertedbytheplane.Aswehaveassumedittobefrictionlesstheforceisnormaltothesurface,upwardsandwecallitN,(3)theforce(tension)exertedbytherope,Fr.WehavedrawnallofthatinFig.2.7b.Asweareconsideringtheblockasamaterialpoint,alltheforcesareappliedinthesamepoint.Oneoftheforces,N,isnotgiven.Thisisalwaysthecaseofconstraintforces.Thebodycannotpenetratethesupportplanebecausethemoleculesofthebodyandtheplanerepeleachother.Weknowthatthebodyhasnoverticalacceleration.Weinferthatthesupportdevelopstheforcethatisexactlywhatisneededtokeepitsteady.Wewillfinditbysolvingtheequations.

Alltheforcesoftheproblemlayinthesameverticalplane.Itisthenconvenienttochooseareferenceframewithoneaxis,sayz,verticalupwardsandasecondone,sayx,horizontaltotherightinthefigure.Wedonotneedthethirdaxisbecausethereareneitherforcesnormotioninthatdirection.WenowwritethesecondNewtonlawanditstwocomponents

Weconcludethatthenormalforceexertedbythesupportplanehasmagnitudeequaltotheweight.Bothforcesareverticalandhaveoppositedirection;hencetheirresultantiszero.Theresultantoftheforcesisthetensionoftherope,whichcausesauniformlyacceleratedmotioninthexdirection.

ExampleE2.2Ablockmovingonahorizontalfrictionlesssurfacedrawnbyaropeatananglewiththehorizontal.

Thesituationisthesameasinthepreviousexample,butfortheropenowpullingatanangleθwiththehorizontal(seeFig.2.8a).However,westillassumethatthemotionisontheplane,namelythatthereisnoverticalacceleration.Theforcesarethesame,butFrhasdifferentcomponents.Wehave

Fig.2.8 Nnormalconstraintforce,Frforceexertedbytherope,Fwweight,duetoearth

Theequationforthezcomponentsgivesagainthenormalconstraintforce,N=Fw–Frsinθ.Ifθ>0asinthefigure,Nissmallerthaninthepreviousexamplebecausetheropehelpsinsustainingtheblock,theoppositeifθ<0.Thesecondequationgiveshorizontalacceleration.

Noticethataphysicallimitationofthisanalysisexists.Thenormalforcecannotbenegative,becausethesupportplanecannotattractthebody(thereisnoglue).Hence,ifFrsinθ>Fw,theassumedconditionscannotbesatisfied.Clearly,inthissituationtheblockisliftedupanditsaccelerationhasaverticalcomponent.

ExampleE2.3Blockonaninclinedfrictionlesssurface.

Therearetwoforcesactingonthebody(Fig.2.9),theweightFwandtheconstraintforceNperpendiculartothesupportplane,whichisnowinclined.Theconvenientchoiceoftheaxesistotakezperpendiculartotheplaneandxalongtheplane,downwards.Clearly,thebodywillslideacceleratingdownwards,namelyinthexdirectionwehavechosen.

Fig.2.9 Ablockonafrictionlessincline

TheNewtonequationanditscomponentsare

ThezcomponentgivesusthenormalforceN=Fwcosα.Thexcomponentgivestheacceleration(a=ax).RecallingthatFw=mg,wehavethatthenotiononaninclinedfrictionlessplaneisuniformlyacceleratedwithacceleration

(2.10)Weseethatthemotiononaninclineiscompletelysimilartothemotionof

freefall,aslongaswecanneglecttheresistiveforces.Thedifferenceisthattheaccelerationissmallerontheinclinebyafactorsinα.Wecanreduceaccelerationbyreducingtheslopeoftheplane.Ifthemotionstartsfromrestfromtheorigin,thelawofmotionisobtainedbyintegratingtwiceEq.(2.10),obtaining

(2.11)Inwords:thedistancestravelledareproportionaltothesquaresofthetimes

takentotravelthem.Theinclineallowsustoslowdownthefreefallmotionandtostudyitslaws

overlongertimes,whichcanbemeasuredwithbetterprecision.AsmentionedinSect.2.5thisisoneofthegreatdiscoveriesofGalilei.He

didnothaveamodernchronometer,butinventedaningeniouswaterchronometer,withwhichhewasabletomeasurethetimesofthemotion,afewsecondslong,withaprecisionbetterthan0.1s.Hedescribeshisexperimentsinthebook“Dialoguesandmathematicaldemonstrationsconcerningtwonewsciences”or“Twonewsciences”publishedin1638.Hewrites:

Apieceofwoodenmoldingorscantling,about12cubitslong,halfacubitwide,andthreefinger-breadthsthick,wastaken;onitsedgewascutachannelalittlemorethanonefingerinbreadth;havingmadethisgroove

verystraight,smooth,andpolished,andhavinglineditwithparchment,alsoassmoothandpolishedaspossible,werolledalongitahard,smooth,andveryroundbronzeball.Havingplacedthisboardinaslopingposition,byliftingoneendsomeoneortwocubitsabovetheother,werolledtheball,asIwasjustsaying,alongthechannel,noting,inamannerpresentlytobedescribed,thetimerequiredtomakethedescent.Werepeatedthisexperimentmorethanonceinordertomeasurethetimewithanaccuracysuchthatthedeviationbetweentwoobservationsneverexceededone-tenthofapulse-beat.Havingperformedthisoperationandhavingassuredourselvesofitsreliability,wenowrolledtheballonlyone-quarterthelengthofthechannel;andhavingmeasuredthetimeofitsdescent,wefounditpreciselyone-halfoftheformer.Nextwetriedotherdistances,comparingthetimeforthewholelengthwiththatforthehalf,orwiththatfortwo-thirds,orthree-fourths,orindeedforanyfraction;insuchexperiments,repeatedafullhundredtimes,wealwaysfoundthatthespacestraversedweretoeachotherasthesquaresofthetimes,andthiswastrueforallinclinationsoftheplane,i.e.,ofthechannel,alongwhichwerolledtheball.Wealsoobservedthatthetimesofdescent,forvariousinclinationsoftheplane,boretooneanotherpreciselythatratiowhich,asweshallseelater,theAuthorhadpredictedanddemonstratedforthem.

Forthemeasurementoftime,weemployedalargevesselofwaterplacedinanelevatedposition;tothebottomofthisvesselwassolderedapipeofsmalldiametergivingathinjetofwater,whichwecollectedinasmallglassduringthetimeofeachdescent,whetherforthewholelengthofthechannelorforapartofitslength;thewaterthuscollectedwasweighed,aftereachdescent,onaveryaccuratebalance;thedifferencesandratiosoftheseweightsgaveusthedifferencesandratiosofthetimes,andthiswithsuchaccuracythatalthoughtheoperationwasrepeatedmany,manytimes,therewasnoappreciablediscrepancyintheresults.

ExampleE2.4Ablockatrestinalift.

Ablockofmassmliesinaliftonahorizontalpanofabalance,oneofthose,forexample,thatareusedtoweighpeople.Whatistheapparentweightoftheblockwhentheliftacceleratesupordown?

Asusualweimaginetheblockinanidealenvelope(Fig.2.10).Twoforcesactonit,theweightFwverticaldown,andthenormalconstraintofthepanNupwards.ThebalancemeasuresthereactiontoN,namelytheforceonit,whichis–N.Hence,Nistheapparentweightoftheblock.

Fig.2.10 Ablockinanacceleratinglift

Iftheliftmoveswithaccelerationaupward,theunknownNisgivenbytheNewtonlaw Hence,theapparentweightis ,whichislargerthanthetrueweight.Iftheliftacceleratesdownwards,theapparentweightis ,smallerthantherealone.Noticethatiftheaccelerationdownwardsisgtheapparentweightisnull.Indeed,theblockisfallingwiththesameaccelerationofthelift.

Iftheliftmovesuniformlybothupwardsanddownwardstheapparentweightisequaltotherealone,asifitwerestanding.Wefeelanincreaseofourweighteitheriftheliftacceleratesgoinguporifitdeceleratesgoingdown.Inbothcasesitsaccelerationisupwards.Similarlywefeeladecreaseofourweightwhentheliftslowsdowngoinguporacceleratesgoingdown.

Tensionoftheropesandwires.Insomeoftheexampleswemadewehaveusedastretchedropeorwiretoapplyaforceinapointofabody.Thisforceisequaltothetensionofthewire.Wegenerallyassumethewiretobeinextensible,meaningthatitslengthdoesnotvarywhicheverthetensionmaybe,andperfectlyflexible,meaningthatthetensionisalwaysparalleltothewire,andofnegligiblemass.Oncemore,theseareidealizations.

Letusclarifytheconceptoftension.Considerawire,stretchedandsteadyasinFig.2.11a.Wementallyisolateasmallsegment,enlargedinFig.2.11b.Twoforcesactonthesegment(neglectingtheweight),appliedtoitsextremesandduetothecontiguouselementsofthewire.Thesearethetensionforces.Asthewireisatrest,thetwoforcesareequalandopposite.Consequently,thetensionisthesameineverysectionofthewire.

Fig.2.11 aThetensionforcesonawireand,bonasegment

Eachoftheextremesofthewireisnotincontactwithanotherelement.Asitdoesnotaccelerate,aforcemustactonitfromoutsideequalinmagnitudetothetensionanddirectedoutwards,asinFig,2.11a.Theforcesontheextremesareequalandoppositeandhavethemagnitudeofthetension.

Considernowthecaseinwhichthewiremoves.Asanexample,supposethatoneextremeisfixedtoablockofmassMlyingonahorizontalplaneofnegligiblefriction.WedrawtheblockapplyingtothefreeextremeofthewireaforceF1obtaininganaccelerationa,asshowninFig,2.12a.Wewanttounderstandunderwhichconditionswereallycanneglectthemassofthewire.Todothat,letusstartassumingthemassofthewiretobem.

Fig.2.12 aAcceleratedmotionofablockdrawnbyarope,bNnormalconstraintforce,Mgweightduetoearth,F2forceduetothewire,cT2forceonthewireduetotheblock,F1forcepullingthewire

Wearenowdealingwithtwobodies,theblockandthewire.Weideallyisolateeachofthemanddrawtheforcediagramsoneachofthem,inFig.2.12b,c.

Wenextidentifytheactionreactionpairs.Thereisonesuchpair,consistingoftheforcesF2appliedtotheblockandT2appliedtotheleftextremeofthewire.Theyareequalandopposite.TheforceF1appliedtotherightextremeofthewireisitstensionandwecancallitT1.TheNewtonequationsforthetwobodiesare

hence,forthemagnitudes, and .Weseethatthetensionsatthetwoextremesaredifferent.IndeedT1>T2becauseT1mustacceleratewireandblock,T2onlytheblock.Letusconsidertheirratio

whichbecomesunityform/M→0.Wecanthenstatethatthetensionsattheextremescanbeconsideredequalifthemassofthewireisnegligiblecomparedtothemassoftheblock.Whenwespeakofmasslessropesorwireswemeanof

negligiblemasscomparedtothemassesoftheotherobjects.Noticethatwecanarrangeastretchedwire,orrope,tohaveforcesatits

extremesofequalmagnitudebutdifferentdirections,byusingpulleys.WedidsoalreadydiscussingtheVarignonexperiment(Fig.2.3).Noticethatinthesecases,ifthemotionisaccelerated,themagnitudesofthetensionsattheextremescanbeconsideredequalonlyifalsothemassofthepulleyisnegligibleandifitcanrotatewithnegligiblefrictiononthepivot(Fig.2.13).

Fig.2.13 Withapulley,thedirectionoftheforceexertedbyawirecanbechanged

ExampleE2.5Twoblockslinkedbyaropeofnegligiblemass.

Figure2.14ashowstwoblocksofmassesm1andm2lyingonahorizontalfrictionlessplane,connectedbyaninextensiblewireofnegligiblemass.Tothesecondblock,attheright,ahorizontalforceFisapplied.Themotionisonthesupportplane.Toknowit,wedonotneedtoanalyzetheverticalforces,whichhavezeroresultants(Fig.2.14a,b,c,d).

Fig.2.14 aTwoblocksconnectedbyawire,bforceonm1,cforcesonthewire,dforcesonm2

Westartbyconsideringthewholesystem,thinkingofitasauniqueidealenvelope.TheonlyforceactingonthissurfaceisF.Hencewehave

.whichgivesanaccelerationaequalforthetwobodies.Wenowisolateeachofthebodies.Theblockontheleft(Fig.2.14b)is

attachedtoanextremeofthewire.Thisexertsontheblockthehorizontalforce

F1.Fortheaction-reactionlawtheblockexertsontheextremeofthewireanequalandoppositeforce,whichisthetensionofthewireatthatextreme(F1=–T1).Twootherforcesactontheblock,theexternalforceFandtheforceF2duetotherightextremeofthewire(Fig.2.14d).Again,fortheaction-reactionlawtheblockexerts,ontherightextremeofthewire,aforceequalandoppositetoF2thatisthetensionT2atthatextreme(F2=–T2).Aswehavediscussedabove,themagnitudeofthetensionisthesameinallpointsofthewire.TakingintoaccountthedirectionswehaveT1=–T2(Fig.2.14c).CallingTthemagnitudeofthetensionwecanwritetheNewtonequationsas

.Thesumofthetwoequationsgivestheaccelerationofthesystema=F/(m

1+m2).Ifwewantthevaluefortension,wesubstituteainthefirstequationobtaining

Wesee,inparticular,thatT<F,namelythetensionissmallerthantheforcewithwhichwepull.

2.7 CurvilinearMotionIntheprevioussectionwehavestudiedafewexamplesinwhichtheforceswereknown,apartoftheconstraintones,andthemotionthathadtobefound.Inthisactionweshallconsidertheinverseproblem,namely,themotionofamaterialpointbeingknown,findtheresultantoftheforces.Thesingularforces,incasemorethanoneispresent,cannotbefound,becausesystemsofforceswiththesameresultantproducethesamemotioninthecaseofmaterialpoints.

CircularuniformmotionConsiderthemotionofamaterialpointPwithmassmconstrainedtomove

onacircumferenceofradiusR.Supposethemotiontobeuniform,namelythemagnitudeofitsspeedυtobeconstant,asinFig.2.15.Themotionishoweveraccelerated,becausethedirectionofthevelocityvaries.Aswealreadyfound,theaccelerationhasaconstantmagnitude(Eq.1.57) andisineverypointdirectedtothecenter(centripetalacceleration).Thisaccelerationmustbegivenbyaforceofmagnitude

Fig.2.15 Circularmotion,auniform,bincreasingvelocity,cdecreasingvelocity

(2.12)Thecorrespondingforcehasthesamedirectionastheaccelerationandis

calledcentripetalforce.Theadjective“centripetal,fromtheLatin“petere”for“pointtowards”,recallsonlyitsdirectionbutdoesnotspecifyatallitsnature.Itmaybethetensionofawire,thenormalforceofacircularguide,thegravitationalforceoftheearthonthemoon,etc.WeshalldiscussafewexamplesinSect.3.4.

Variablespeedmotion.Ifthemagnitudeofthevelocityofaparticlemovingonacirclevaries,its

accelerationhastwocomponents.Onecomponent,an,isperpendiculartothetrajectory,or,thelatterbeingcircular,directedtothecenter.Itisagainthevariationofthedirectionofthevelocity,namelythejustdiscussedcentripetalaccelerationofvalueυ2/Rwhereυ,wemustnowspecify,istheinstantaneousvelocity.Thesecondcomponent,at,isinthedirectionofthemotion,i.e.tangenttothetrajectoryandexpressesthevariationintimeofthemagnitudeofthevelocity.Wehave

(2.13)Theaccelerationvector,andtheforce,isdirectedatananglewiththeradius

thatisforwardifthevelocityisincreasing(Fig.2.15b),backwardifitis

decreasing(Fig.2.15c).Themagnitudeoftheforceis

Asanexample,considerablocklyingontheplatformofamerrygoround,whichisinitiallystill.Whentheplatformstartsmoving,graduallyincreasingitsangularvelocity,theaccelerationoftheblockhastwocomponents,onecentripetalandonetangential.Thecorrespondingforce,equaltothemassoftheballtimesthisacceleration,isgivenbythefrictionontheplatform.Ifthelatterisnotenough,theblockslidestowardstheperipheryoftheplatform.

Asasecondexampleconsiderthelaunchofthehammer.Theathleteactingontheropeheholdsinhishandsputsthehammerinrotationwithincreasingspeed.Theforceonthehammermustbeadequatetokeepitonacircularorbit(componentmυ2/Rtowardsthecenter)andmakesitsspeedincrease(acomponentinthedirectionofthemotion).Theropemustthenbedirectedforward,asinFig.2.15b

Generalplanemotion.Weconsidernowamaterialpointofmassmmovingonaplanetrajectoryof

arbitraryshapewithvelocitynotnecessarilyconstantinmagnitude.WehavealreadystudiedthekinematicsoftheprobleminSect.1.14.Eveninthiscase,theaccelerationhastwocomponents,atangentialandanormalone,asinEq.(1.62).TheyaregivenbyEq.(2.13).

TheonlydifferencefromthecircularcaseisthatnowRisthelocalcurvatureradius,whichisnotfixedbutvariesalongthetrajectory.ThesecondNewtonlawtellsusthattheresultantoftheforcesactingonthepointmustbeitsaccelerationtimesitsmass.

Ifweknowonlythetrajectory,butnothingofthevelocity,wecanstillsaythatineverypointofthetrajectoryinwhichthecurvatureisnotzero,theresultantoftheforcesmustbedirectedonthesideofthecurvaturecenter,pointingforwardfromit(Fig.2.16a)orbackwards(Fig.2.16b)dependingonwhetherthemotionisacceleratedordelayedrespectively.

Fig.2.16 Generalplanemotion.aIncreasingspeed,bdecreasingspeed

2.8 AngularMomentumandMomentofaForceConsideramaterialpointPmovinginaninertialframeasshowninFig.2.17.Letp=mvbeitsmomentumandritspositionvector.ConsideragenericpointΩ,whichmaybeatrestormovingrelativetotheframe.WeshallnowintroducetheconceptsofangularmomentumandmomentofaforceaboutthepoleΩ.

Fig.2.17 Thevectorsrelevantforangularmomentum

WehavealreadydefinedthemomentofaboundvectorinSect.1.8.Theangularmomentumisthemomentofthelinearmomentum,consideringit,forthispurpose,asappliedtothematerialpoint,asshowninFig.2.17.

Hence,theangularmomentumofthepointPaboutthepoleΩisthevectorproductofthevectorfromΩtoPanditsquantityofmotion(ormomentum).

(2.14)ConsidertheforceFappliedtoP.ThemomentoftheforceaboutthepoleΩ

isthevectorproductofthevectorfromΩtoPandF

(2.15)Rememberthattheorderofthefactorsmattersincrossproducts.Noticealso

thatthemomentschangeifthereferenceframechanges.Letusnowseehowtheangularmomentumchangesintime.Forthat,we

takethetimederivativeofEq.(2.14)usingtheruleofthederivativeofproducts,payingattentiontotheorderofthefactors

(2.16)Tofindthederivativeofthevector wenoticethatitisthedifferenceof

twovectors,bothvaryingwithtime, .Derivingwehave.

Themeaningofthisexpressionisclear:thederivativeofavectorjoiningtwomovingpointsistherelativevelocityofthosepoints.Wesubstitutethis

expressioninEq.(2.16)andalsonoticethatthederivativeofthemomentumisequaltotheresultantFoftheforcesactingonP,becausetheframeisinertial.Weget

Thefirstterminthesecondmemberiszero,beingthecrossproductoftwoparallelvectors;thelasttermisthemomentoftheresultantaboutthepoleτΩ.Inconclusion

(2.17)Thisisaveryimportantequationthatweshalluseofteninthefollowing.It

becomesparticularlysimpleifwechooseastationarypoleinthereferenceframe.Theequationbecomes

(2.18)Inwordstheequationiscalledtheangularmomentumtheoremforamaterial

point:thetimederivativeoftheangularmomentumofamaterialpointaboutapolefixedinaninertialreferenceframeisequaltothemomentoftheresultantoftheforcesactingonitaboutthesamepole.

Noticethatifthebodyisextended,asweshalldiscussinthefollowingchapter,thedifferentforcesactingonit,sayf1,f2,…,maybeappliedindifferentpointsandthemomentoftheirresultantF=f1+f2+···,isingeneraldifferentfromthevectorsumoftheirmoments.Inthecaseunderstudyhowever,alltheforcesareappliedinPand

Theresultantofthemomentsisequaltothemomentoftheresultantoftheforces.Westressthatthisistrueonlyifalltheforcesareappliedatthesamepoint.

2.9 TheSimplePendulumThependulumisamaterialpointconstrainedtomoveonanarcofacircumference.Itcanbesimplymadebyfixingathinwiretoasmallsphereonanextremeandtoafixedpointontheother,whichwecallΩ.Thelengthlofthewire,orbetterthedistancebetweenthefixedpointandthecenterofthesphere,iscalledthelengthofthependulum.IfwetakethependulumawayfromitsequilibriumpositionOandabandonitwithzerovelocity,thebodymoves

towardsOundertheactionoftwoforces,theweight,directedverticallydown,andthetensionofthewire(T),directedasthewire.Theaccelerationhasthedirectionoftheresultantofthesetwoforces.Consequentlyitisalwaysintheplanedefinedbythewireandthevertical.Iftheinitialvelocityiszero,themotionisontheplane.AsthedistancefromΩiskeptfixedbythewire,whichweassumeinextensible,thetrajectoryisanarcofacircleofradiusl.

AsshowninFig.2.18,wetakeareferencesystemwiththeoriginOintherestpositionofthependulum,they-axisverticalupwards,thex-axishorizontallyintheplaneofmotionandzsuchastocompletethetriplet.Thez-axisisnormaltothefiguretowardstheobserver.Wecallθtheanglebetweenthewireandthevertical,takingitpositiveifseenanticlockwisebytheobserver.

Fig.2.18 Thesimplependulum

Historically,aswehavealreadymentioned,thestudyofthemotionofpendulums,withtheirperiodicmotion,madeafundamentalcontributiontothedevelopmentofmechanics.Galileidiscoveredtwoimportantproperties.Thefirstoneistheisochronismofsmalloscillations;iftheamplitudeisnottoolarge(weshallbemorepreciseinthefollowing),theoscillationperiodisindependentoftheamplitude.Thispropertyallowedbuildingofpreciseclocks.Thesecondpropertyisevenmoreimportant;theoscillationperiodsofpendulumsofthesamelengthsanddifferentmassesareidentical.Thisproves,asweshallnowsee,thatgravitationalmassandinertialmassareequal.Thepropertywaslatercalledequivalenceprincipleandisatthebasisofgeneralrelativity.

Inourdemonstration,westartbyassumingthatthetwomassesmightbedifferent.Wecallmitheinertialmassofthependulum,namelythe

proportionalityconstantbetweenaccelerationandforce,andmgitsgravitationalmass,theconstantthatappearsintheweight,whichisthenmgg.

Thetensionisaconstraintforce,duetothewire,whichweassumetobeperfectlyflexibleandinextensible.Theconstraintdevelopsaforce,ingeneralunknownapriori,automaticallyadjustedtomakethemotionhappen,inourcase,atafixeddistancefromΩ.WedonotknowtheintensityofthewiretensionT,butweknowitsdirection,whichisalongthewire.

Inourstudyofthemotionweshallusetheangularmomentumtheorem.WechoosethepoleinthesuspensionpointΩ,forreasonsthatwillbecomeclearsoon.WeuseEq.(2.18)with

Wenowseethereasonforourchoiceofpole.Thefirsttermisalwayszero,beingthevectorproductoftwoparallelvectors.Consequentlywedonotneedtoknowtheintensityofthetension.Wehave

(2.19)Theangularmomentumaboutthesamepoleis

(2.20)wherethemassistheinertialone.Equation(2.18)gives

(2.21)Allthevectorsintheseequations,inanypositionofthependulum,belongto

theplanexy.Bothvectorproductsareconsequentlyinzdirection.Theequationhasonlythezcomponent.Thezcomponentof is .Thevelocityisalwaysperpendicularto .Asaconsequencethezcomponentof

issimply ,where .So,wehave

Andfinallywecanwrite

(2.22)Thisisadifferentialequation,whoseunknownisafunctionoftimeθ(t).

Onceitissolved,weknowthemotionofthependulum,becauseifweknowθ,weknowitsposition.Equation(2.22)cannotbesolvedanalytically.However,iftheoscillationsare“small”,wecanapproximatethesinewithitsargumentandtheequationbecomes

(2.23)Thisisawell-knowndifferentiallinearequationwithconstantcoefficients,

whichweshallmeetseveraltimes.Weleaveitsstudytocalculuscoursesanddirectlygivethegeneralsolution,whichis

(2.24)where

(2.25)

iscalledproperangularfrequency.Asonesees,itdependsonlyonthecharacteristicsofthependulum,includingitsweight.

Thereadercaneasilyverify,withtwoderivatives,thatthisexpressionindeedsatisfiesEq.(2.23),forwhatevervaluesoftheconstantsθ0andϕ.Theseconstantsdonotdependonthecharacteristicsofthependulumbutonhowthemotionhasstarted.Theyshouldbefoundineachcaseonthebasisoftwoinitialconditions.Wecanusethepositionandvelocityatthestartingtimethatweshalltakeast=0.Weimmediatelyseethat

Theinitialvelocitybeingzero,thesecondequationgivesϕ=0(θ0=0isalsoasolutionofEq.(2.24)butidenticallynull).Thefirstconditionsaysthatθ0isjusttheinitialangle,theangleatwhichwehaveletthependulumgo.Inconclusionthemotionofthependulumisdescribedbytheequation

(2.26)Themotionisperiodic,meaningthat,foranyinstantoftimetwecan

consider,boththepositionandthevelocitybecomethesameafteracertaintimeintervalT,calledtheperiod,namelyattheinstantt+T.FromEq.(2.26)weimmediatelyseethattheperiodis

(2.27)

whereweusedEq.(2.25).ThemotionisrepresentedinFig.2.19.Thisisthemostcommonperiodic

motioninNature.Itiscalledharmonicmotion.Inthenextchapterweshallstudyitindepth.

Fig.2.19 Angularharmonicmotion

Wenowmakeanimportantobservationontheexpressionsofproperangularfrequencyandperiod.Angularfrequencyandperioddependonthelengthofthependulum,butnotontheoscillationamplitude:twopendulumsofthesamelength(andinthesameplace,henceatthesameg)areisochronous.Ontheotherhand,angularfrequencyandperioddependontheratiobetweengravitationalandinertialmasses.Ifthisratioisthesameforallbodies,independentlyofthesubstancetheyaremadeofandoftheirposition,thatratioisaconstantandangularfrequencywillbeindependentofthemassofthependulum.Ifwewanttoexperimentallytestifgravitationalandinertialmassesareproportionalornot,wecantestwhetherpendulumsofthesamelength,anddifferentmassesormadeofdifferentsubstances,dooscillateornotwiththesameperiod.

Galileinoticedthatthismethodismuchmoreaccuratethanothersheknew.Inprinciple,onecouldthinktodroptwospheres,e.g.oneofwoodandoneoflead,fromthetopofatowerandcheckiftheythereachthegroundsimultaneously.However,Galileinevermentionshavingdonesuchanexperiment,fromtheleaningtowerofPisa.Thisisalegendwithoutanyhistoricalsupport.Indeed,Galileiobserved,andwrote,thatthatmethodisnotaccurateenough,becauseistoofastandathighspeedstheairresistancenoticeablyperturbsobservations.Galileiusedinclinedslopes,aswehavediscussed,toslowdownthemotion,reducetheairdrag,andincreasetherelativemeasurementaccuracyduetothelongertimestobemeasured.Theuseofpendulumsallowsevenbetteraccuracy.Heusedpairsofpendulumsmadeofdifferentmaterialsandofexactlythesamelengths,tookthemoutofequilibriumandletthemgoatthesametime.Hefoundthattheykeeposcillatinginphaseforhundredsofperiods.Theairdragdidactmoreeffectivelyonthelighterpendulumgraduallyreducingitsamplitudemorethanthatoftheheavierone.Howeverthisdidnotmatterbecausetheperiodisindependentofamplitude.

G.Galileidescribesaccuratelyhisprogresstowardamorepreciseexperiment,graduallyeliminatingthespuriouseffectsandthesourcesoferrors

inthe“Dialogsconcerningtwonewsciences”(1638)(translationfromItalianbyHenryCrewandAlfonsodeSalvio).Heestablishedtheproportionalityofinertialandgravitationalmasswithanuncertaintyof2–3×10−3.

Theexperimentmadetoascertainwhethertwobodies,differinggreatlyinweightwillfallfromagivenheightwiththesamespeed,offerssomedifficulty;because,iftheheightisconsiderable,theretardingeffectofthemedium,…willbegreaterinthecaseofthesmallmomentumoftheverylightbodythaninthecaseofthegreatforceoftheheavybody;sothat,inalongdistance,thelightbodywillbeleftbehind;iftheheightbesmall,onemaywelldoubtwhetherthereisanydifference;andiftherebeadifferenceitwillbeinappreciable.ItoccurredtomethereforetorepeatmanytimesthefallthroughasmallheightinsuchawaythatImightaccumulateallthosesmallintervalsoftimethatelapsebetweenthearrivaloftheheavyandlightbodiesrespectivelyattheircommonterminus,sothatthissummakesanintervaloftimewhichisnotonlyobservable,buteasilyobservable.Inordertoemploytheslowestspeedspossibleandthusreducethechangewhichtheresistingmediumproducesuponthesimpleeffectofgravityitoccurredtometoallowthebodiestofallalongaplaneslightlyinclinedtothehorizontal.Forinsuchaplane,justaswellasinaverticalplane,onemaydiscoverhowbodiesofdifferentweightbehave:andbesidesthis,Ialsowishedtoridmyselfoftheresistancewhichmightarisefromcontactofthemovingbodywiththeaforesaidinclinedplane.AccordinglyItooktwoballs,oneofleadandoneofcork,theformermorethanahundredtimesheavierthanthelatter,andsuspendedthembymeansoftwoequalfinethreads,eachfourorfivecubitslong.Pullingeachballasidefromtheperpendicular,Iletthemgoatthesameinstant,andthey,fallingalongthecircumferencesofcircleshavingtheseequalstringsforsemi-diameters,passedbeyondtheperpendicularandreturnedalongthesamepath.Thisfreevibrationrepeatedahundredtimesshowedclearlythattheheavybodymaintainssonearlytheperiodofthelightbodythatneitherinahundredswingsnoreveninathousandwilltheformeranticipatethelatterbyasmuchasasinglemoment,soperfectlydotheykeepstep.Wecanalsoobservetheeffectofthemediumwhich,bytheresistancewhichitofferstomotion,diminishesthevibrationofthecorkmorethanthatofthelead,butwithoutalteringthefrequencyofeither.

Inconclusion,Galileiexperimentallydemonstratedtheequalityofinertialandgravitationalmasseswithanaccuracyofaboutonepermille,namelythat

(2.28)NewtonrepeatedthislaterontheGalileiexperiments.Hewritesinthe

“Principia”:

Ithasbeen,nowofalongtime,observedbyothers,thatallsortsofheavybodies(allowancebeingmadefortheinequalityofretardationwhichtheysufferfromasmallpowerofresistanceintheair)descendtotheearthfromequalheightsinequaltimes;andthatequalityoftimeswemaydistinguishtoagreataccuracy,bythehelpofpendulums.Itriedthethingingold,silver,lead,glass,sand,commonsalt,wood,water,andwheat.Iprovidedtwowoodenboxes,roundandequal:Ifilledtheonewithwood,andsuspendedanequalweightofgold(asexactlyasIcould)inthecenterofoscillationoftheother.

Heconcludedthat:

Bytheseexperiments,inbodiesofthesameweight,Icouldmanifestlyhavediscoveredadifferenceofmatter(i.e.inertialmass)lessthanthethousandthpartofthewhole,hadanysuchbeen.

HenceNewtonconfirmedwhatGalileihaddiscoveredwithasimilarprecisionof1×10−3.Afterhavingfoundanexpressionofthegravitationalforce,Newtondidalsoacheckoftheequivalenceprinciple,onasolarsystemscale.Hedidthat,inparticular,onthesystemofJupiteranditssatellites.WeshallseehisargumentinSect.4.4.Herewejustsaythattheprecisionwas,oncemore,ofonepermille.

Havingestablishedtheproportionalityofthetwotypesofmass,wecanmakethemequalbychoosingtheirunits.WiththischoiceEqs.(2.25)and(2.27)become

(2.29)

Tohaveafeelingoftheordersofmagnitude,wecaneasilycalculatethata1-mlongpendulumhasaperiodofabout2s.

Wenowrecallhavingapproximatedthesineoftheanglewiththeangle(inradiants)itself.Letusverifywhentheapproximationisgood.Forexample,ifθ=30°,or0.52rad,itssineissin30°=0.50.Therelativeerroris(0.52–0.50)/0.50=4%,whichisquitesmall.Evenforθ=60°,or1.05rad,theerroris

notenormous,butalreadynoticeable.Indeed,sin60°=0.87andthecorrespondingerroris20%.Thesearetherelativeerrorsmakingthesineequaltotheangle,butthecorrespondingonesontheperiodareevensmaller,aswenowshallsee.

TheexactEq.(2.22),aswesaid,cannotbesolvedanalytically.However,itcanbesolvedbysuccessiveapproximations.Infact,theapproximationwemadeisaseriesexpansionstoppedatthefirstterm(sinθ=θ);thenextapproximationwestopatthesecondterm(sinθ=θ−θ3/6).Theresultingexpressionfortheperiodwithamplitudeθ0,callingTotheperiodgivenbyEq.(2.28),is

which,asitisseen,dependsontheamplitudeθ0.Therelativeerrormade

usingtheusualexpressionoftheperiodis .Goingbacktotheabove

examples,wefindthattherelativeerrorforθ=30°is1.6%,theoneforθ=60°is6.3%.Theyarenotlarge.

Wemakealastobservation.Iftheoscillationsaresmall,thependulummovessubstantiallyonthehorizontal,namelyonthex-axisinFig.2.18.Nowx=ltanθ,thatwecanapproximatewithx=lθ.Wecanthenconcludethatthemotion,asrepresentedbythexcoordinate,hastheequation

(2.30)Asexpecteditisaharmonicmotion,ofamplitudex0.

2.10 TheWorkofaForce.TheKineticEnergyTheoremInthissectionweintroducetheconceptsofwork,donebyaforce,andkineticenergy,ofabody.Themeaningof“work”inphysicsisratherdifferentfromitsmeaningsineverydaylanguageandconsequentlyfromwhatintuitionmightsuggest.Forexample,holdinginonehandaheavyobjectevenifwedonotmoveitwestillneedtoapplyaforcewithourmusclesandmakesomeeffort.However,wedonotperformanywork,inthelanguageofphysics.Inphysics,aforcemakesworkonlyifitsapplicationpointmoves.Intheexample,theworkdonebytheforceweexertonthebodyispositiveifweraise,negativeifwelowerit,butzeroifwedonotmoveit.

ConsiderthematerialpointPmovinginareferenceframewithpositionvectorr,alongacertaintrajectory,thecurveΓ.AsshowninFig.2.20,considerthepositionvectorintheinstantst,r(t),andimmediatelyaftert+dt,r(t+dt).

ThedisplacementofPintheintervaldtistheinfinitesimalvector

Fig.2.20 TheelementstodefinetheworkofforceF

(2.31)IfFisaforceactingonthepoint,itsworkfortheinfinitesimaldisplacement

(2.31)isdefinedas

(2.32)Thefiniteworkhavingbeendonebytheforce,afinitedisplacementofthe

point,sayfromAtoBalongthetrajectoryΓ,isthelineintegralalongthecurveΓfromAtoB

(2.33)

whereF(r)istheforceinthepointofpositionr.Thelineintegralisthesumofalltheelementarydotproducts onalltheelementsofthecurve.Clearly,theintegraldoesnotdependonlyontheinitialandfinalpointsAandB,butalsoonthespecificpathtakentogofromtheformertothelatter.Indeed,ifthepathchanges,alsotheforceinthenewpointsmaychange.TomakethisexplicitinthenotationwehaveincludedbothAandBandΓinthesubscriptsofW.ThecaseinwhichtheintegraldependsontheoriginandtheendbutnotonthepathishoweverimportantandwillbestudiedinSect.2.13.

Noticethatmoreforces,callthemFi,mayactcontemporarilyonthepointP,forexampleweight,friction,airresistance,etc.Inthiscase,thetotalworkmadebyalltheforcesisequaltothesumoftheworkseachforcewoulddoif

actingseparately

(2.34)

Clearly,theelementary(meaning“infinitesimal”)displacementoftheapplicationpointsoftheforcesdsisthesameforallthem.Consideringthatthesumofintegralsisequaltotheintegralofthesum,whichisinourcasetheresultantoftheforces ,wehave

(2.35)

Namely,thetotalworkmadebytheactingforcesisequaltotheworkmadebytheirresultant.Notice,again,thatthisistrueonlyifallforcesareappliedinthesamepoint.

Thephysicaldimensionoftheworkisthoseofaforcetimesadisplacement.Itsunitisthejule,withsymbolJ,whichistheworkdonebytheunitforce,1N,whenitsapplicationpointmovesoneunitoflength,1m,inthedirectionoftheforce.Toappreciatetheorderofmagnitude,ajuleisroughlytheworkyoudowhenyouraiseaglassofwaterby1m.

Wenowprovethework-kineticenergytheorem.BeingaconsequenceofthesecondNewtonlawitisvalidininertialframes.ConsideramaterialpointandtheresultantRoftheforcesactingonit.TheNewtonlawsays

Wetakethescalarproductwiththeelementarydisplacementds=vdtofthetwomembers

Nowconsiderthedotproduct .Werecallthatthesquareofavectoristhedotproductofthevectorbyitself,inthiscase .Differentiatingthisexpressionwehave

hence

TheworkdonebyRwhenthepointmovesfromAtoBonthegiven

trajectoryisthen

(2.36)

Wethendefinethekineticenergyofthematerialpointofmassmandvelocityυas

(2.37)whichisindependentoftheposition.Thekineticenergyhasthesamephysicaldimensionastheworkandismeasuredinjule.WefinallycanwriteEq.(2.36)as

(2.38)whichisthework-kineticenergytheorem.Inwords:whenamaterialpointmovesonacertaintrajectoryfromAtoB,theworkdonebytheforcesactingonitisequaltothedifferencebetweenthekineticenergyofthepointhasinBandthatithadinA.

Itissometimesusefultoexpresskineticenergyintermsofmomentumratherthanvelocity,namely

(2.39)

2.11 CalculatingWorkInthissectionweshallseetwoexamplesofcalculationofworks,maderespectivelybyweightandfriction,whentheapplicationpointPmovesonitstrajectoryfromtheinitialpositionAtothefinaloneB.Weshallseethatintheformercasetheworkdependsonlyontheinitialandfinalposition,andnotonthepathtakenbetweenthem,inthelatteritdependsonthepathtoo.

Startingwithweight,Fig.2.21showsthereferenceframe(notnecessarilyinertial)wherewehavechosenthez-axistobevertical.PointPmovesonthetrajectoryfromthepositionA,withthepositionvectorrA=(xA,yA,zA)tothepositionB,withthepositionvectorrB=(xB,yB,zB).Thefigureshowsalsothepositionvectoratthegenericinstanttandintheimmediatelyfollowinginstantt+dt.Theforceactingonthepointisitsweightmg,whichisequalinallpoints.Theelementaryworkdonebytheweight,whichisverticallydirecteddownwardsis .Thetotalworkisgivenbytheintegral

Fig.2.21 Trajectoryofthematerialpointanditsweight

(2.40)

Weseethatinthisrelevantcasetheworkisindependentofthepath,dependingonlyonthefinalandinitialposition,evenbetter,ontheirheightsonly.ThisconclusionwasexperimentallyprovenbyGalileiwithasimpleexperimentthatweshalldescribeinthenextsection.

Thisisnotthecaseofthesecondexample,thefrictionforce,whichweshallstudyinSect.3.5.

Supposewehaveanobject,sayabookorabrick,lyingonatable.Inrealcases,theconstraintdoesnotapplytothebodyonlythenormalforce,butalsoafrictionthatistangenttothecontactsurface.IfwewanttomovethebodyonthetrajectoryΓinFig.2.22ataconstantspeed,asweknowfromeverydayexperience,weneedtopullit,applyaforce,paralleltotheplaneinthedirectionofthedisplacement.Thismeansthattheplaneexertsonthebodyaforceequalandoppositetoourpull,becausethevelocityisconstantinmagnitudeandthentheresultantoftheforcesinthedirectionofthemotionmustbezero.Indeed,asweshallseeinSect.3.5,thefrictionforce,Fa,isalwaysparallelandoppositetotheelementarydisplacementds.WenowcalculatetheworkofFa.

Fig.2.22 Calculatingtheworkofthefrictionforce

Theelementaryworkis whichisalwaysnegative.Thetotalworkisgivenbythelineintegralonthetrajectory

(2.41)

wheresAB(Γ)isthelengthofthetrajectoryΓbetweenAandB.Theworkisproportionaltothelengthofthepath,aquantityobviouslydependingonthepath.

WeconcludewithanobservationthatweshallgeneralizeinSect.2.13.WehaveseenthattheworkoftheweightforcefordisplacementAtoBisWAB=–mg(zB–zA).SupposenowthatthepointgoesbacktoA.TheworkofweightisWBA=–mg(zA–zB)=–WAB.Namelythetotalworkoftheweightonaclosedpathiszero.Ontheotherhand,theworkofthefrictionforcetogofromAtoBonthecurveΓis .Ifwenowgobackonanothercurve,sayΓ′intheFig.2.22,theworkofthefrictionis ,whichisagainnegative.Consequentlytheworkofthefrictiononaclosedpathisnotzero,itisnegative.

2.12 AnExperimentofGalileionEnergyConservationOneofthediscoveriesofG.Galileiwasthefact,aswehavementioned,thatthevelocityofbodydescendingundertheactionofisweightonly,startingfromrest,dependsonthedifferencebetweentheinitialandfinallevels,andnotonthefollowedpath.

Inthe“DialogueonTwonewsciences”hestatesthatthevelocitiesofbodiesdescendingoninclinesofdifferentslopesandthesameheightareequal.Inhiswords(translationsbytheauthor):

Allcontrastsandimpedimentsremoved…aheavyandperfectlyroundball,descendingthroughthelinesCA,CD,CBwouldreachthefinalpointsA,D,Cwiththesamemoments

withreferencetoFig.2.23areproducedfromthebook.Noticethat,atthetime,Galileiwassearchingforanddevelopingthelawsofmechanicsandthatseveralconceptshadnotyetbeencompletelydefined.Inparticular,impetus,momentum,kineticenergywerenotwell-separatedconcepts.

Fig.2.23 aBallfallingoninclinesofdifferentslopes;bthependulumandnailexperiment

However,accuratemeasurementsofthosevelocitieswereimpossibletodo.Toprovethestatement,heinventedasimpleandgenialexperiment,usingapendulumandanail.Figure2.23bisalsoreproducedfromhisbook.

Salviati,thepersonwhointheDialoguesrepresentsGalilei,startswiththedescription:

Supposethissheettobeaverticalwallandtohavealeadballofoneortwoounceshangingfromanailfixedinthewall,suspendedtoathinwireAB,twoorthreearmslong,perpendiculartothehorizon…andabouttwofingerfarfromthewall.

ThendrawtheverticallineABand,perpendiculartoitDC.MovethewirewiththeballinACandletitgo.Weshallseetheball

descendingfirstthroughthearcBCD,andgoingbeyondpointBasmuchas,slidingonthearcBD,almostreachingthedrawnhorizontalCD,failingtoreachitbyaverysmallgap,whichhasbeentakenawaybytheimpedimentsoftheairandthewire;fromwhichwecanlikelyconcludethatthemomentum(impetus)gainedbytheballinB,inthedescentonthearcCB,wassomuchtopullitbackthroughthesimilararcBDtothesame

height.

Hecontinueswiththerequesttorepeattheexperimentsseveraltimestochecktheresult.Then

Iwantwefixinthewall,grazingtheverticalAB,anail,likeinEorinF,whichshouldprotrudeoutfiveorsixfingers.

Asbefore,thewirewiththeballismovedtoACandletgo.TheballwillagainmoveonthearcCB.But,whenitisinB,thewirehitsthenail,forcingtheballtomoveonthearcBG,havingcenterinE.

Now,myLords,youwillseewithenjoymenttheballreachingthehorizontallineinthepointG,andthesametohappeniftheobstaclewouldbelower,asinF,wheretheballwouldgothroughthearcBI,alwaysfinishingitsascentonthelineCD.

Salviaticoncludesthatthemomentumacquiredbyabodydescendingfromacertainheightisjustwhatisneededtobringitbacktothesameheight,throughwhateverpath.Heobservesthatthemomentumacquiredinthedescentonagivenarcisequaltothemomentumneededtorisethroughthesamearc.Heconcludesthatthemomentum,andwecansayalsothevelocityandkineticenergyinB,isthesamewhetheritdescendsthroughCB,orGBorIBoranyarcbeginningonthehorizontalDCandhavingitslowestpointinB.Ontheotherhand,thefallalonganarccanbethoughtofasthefallonan“incline”ofvaryingslope,provingtheassumption.

Theimportanceoftheresultofthisexperimentbecameclearinthefollowingevolutionofmechanics.InhisexperimentthekineticenergyoftheballinBisthesamewhateverthepathstartingfromstillnessfromthesamelevel.Wenowknowthatthisenergyisequaltotheworkdonebytheweightforce.Weconcludethattheworkdonebytheweightdependsonlyonthedifferenceoflevelandnotontheparticularpathfollowed.Wehavealreadydiscussedthispropertyintheprevioussection.Indeed,itisafundamentalone;itshowsthatthereisaquantity,theenergy,whichisconserved,doesnotchangeinthemotionundertheactionofweight.Weightisaconservativeforce,asweshallnowsee.

2.13 ConservativeForcesIngeneraltheworkofaforceonapointdependsonthetrajectoryofthepoint.

However,wehaveseenacase,thecaseoftheweightforce,inwhichtheworkdependsonlyontheoriginAandendBandnotonthetrajectorybetweenthem.Forceshavingthesepropertiesaresaidtobeconservative.Intheoppositecase,asforthefriction,theyaresaidtobenon-conservativeordissipative.

LetrbethepositionvectorinthechosenreferenceframeandF(r)beaconservativeforce,afunctionoftheposition.Thedefinitionofconservativeforcestatesthat,forwhatevercurveΓwithorigininAandendinB,

(2.42)

wherefisafunctionoftheco-ordinatesofAandofB.Itiseasytoshowthatinthiscaseitisalwayspossibletofindafunctionoftheco-ordinates,whichweshallindicatewithUp(r),suchas

(2.43)Toshowthat,consideranarbitrarilychosenpointo,asinFig.2.24.The

workfromAtooonwhateverpathis

Fig.2.24 Differentpaths

(2.44)andsimilarlytheworkfromotoBis

(2.45)But,wecangofromotoBalsogoingfromotoAandthenfromAtoB.

ConsideringthatworkisanadditivequantitywecanwriteWoB=WoA+WAB.Hence

(2.46)BysubtractingEq.(2.43)fromthisexpressionwehave

(2.47)Wethenreachtheresultbyputting .ThefunctionUp(r)is

thepotentialenergyoftheforceF(r)andisafunctionoftheco-ordinatesonly.Inconclusionthepotentialenergy,orbetteritsdifference,isdefinedbytherelation

(2.48)

Inwords:thedifferenceofpotentialenergyoftheforceFinthepointBandinthepointAisequaltotheoppositeoftheworkdonebytheforcewhenitsapplicationpointmovesfromAtoB,followinganytrajectory.

Thereasonofthe—sign,ortheword“opposite”,isthefollowing.Tobeconcrete,considertheweight.IfwemoveabodyofmassmfromthelevelzAtothehigherlevelzB,thedisplacementisoppositetotheforceandthework–mg(zB–zA)isnegative.Thepotentialenergyofthebodyisthenlargerwhenitslevelishigher.Theworkdonebytheweightforceisequalandoppositetothegainofpotentialenergyofthebody.Thisenergycanbegivenbackasworkbythebody,takingitdowntotheoriginallevel.Thehigherthebody,thegreaterisitspotentialtoproducework.

Wecanconclude,andthisistrueincompletegenerality,bystatingthatthepotentialenergydifferencebetweentwostatesofabodyisequaltotheworkweneedtodoagainsttheforceactingonthebodytochangeitfromthefirsttothesecondstate.

Noticeagainthatapotentialenergycanbedefinedforaforceonlyifitsworkisindependentofthepath.Nopotentialenergyexists,forexample,forthefrictionforces.

Noticealsothatonlydifferencesofpotentialenergycanbedefined,notitsabsolutevalue.Inotherwords,potentialenergyisdefineduptoanarbitraryadditiveconstant.Inpractice,wefixtheconstantchoosingareferenceposition,sayo,inwhichwedefinethepotentialenergytobezero(Up(o)=0),ThepotentialenergyinthearbitrarypointPisthen

Forexamplefortheweight,wearbitrarilyfixareferencelevelatwhichthepotentialenergyiszerobydefinition.Thismaybethegroundlevelbutsomeotherleveltoo.Wetakethatlevelastheoriginoftheverticalupwarddirectedz-axisandthepotentialenergyis

(2.49)WehavestatedthataforceFisconservativeiftheworkitdoesonapoint

whenitmovesfrompositionAtoBisindependentofthepath.Therearetwoequivalentwaystostatethesame,whichmaybeusefulincertaincircumstances.

1.

2.

AforceisconservativeiftheworkitdoesmovingfromAtoBonanypathisequalandoppositetotheworkdonemovingfromBtoAonanypath(Fig.2.25).Thisfollowsimmediatelyfrom(2.48).

Fig.2.25 Thepathsdiscussedinthetext

Theworkofaconservativeforceonanyclosedpathiszero.

Insummarywecanbrieflysaythatthe(equivalent)propertiesofconservativeforcesare:(1)itsworkdoesnotdependonthepath,(2)admitsapotentialenergy,(3)theworksgoingandgoingbackareequalandopposite,(4)theworkonaclosedpathiszero.

2.14 EnergyConservationConsideramaterialpointPofmassmmovingfromthepositionAtothepositionBonthetrajectoryΓundertheactionofthe(only)forceF.Whethertheforceisconservativeornotitsworkisequaltothechangeofthekineticenergyofthepoint.DenotingwithUkthekineticenergy,wewrite

(2.50)If,andonlyif,Fisconservative,thesameworkisalsotheoppositeofthe

changeofpotentialenergyoftheforce

(2.51)Itimmediatelyfollowsthat

(2.52)ConsideringthatthepositionsAandBarearbitrary,weconcludethatthe

sumofthekineticandpotentialenergiesisthesame,i.e.,isconstant,ineverypositionofthemotion.Thesumisthetotalmechanicalenergy,sayUtotofthe

materialpoint.Theconclusionissoimportantthatitisoftencalleda“principle”.Theprinciple,orlaw,ofenergyconservationstatesthat

(2.53)IfmorethanoneforceisactingonthepointPandallofthemare

conservative,Eq.(2.53)isstillvalid,providedthatUpisthesumofthepotentialenergiesofalltheactingforces,or,inanequivalentmanner,ifitisthepotentialenergyoftheresultantofthoseforces.Noticehowever,thatthelawisnolongervalidevenifonlyoneoftheforcesisdissipative.

Inwords,thelawofenergyconservationstatesthatifapointmovesundertheactionofconservativeforcesonly,itstotalmechanicalenergyisconservedduringitsmotion.

Considernowthecase,whichiswhathappensinpractice,thatalsodissipativeforcesarepresent.Considerforexamplethemotiononaninclineundertheactionsofweightandfriction.Thekineticenergytheoremisstillvalid.TheworkdonebytheforcesforthedisplacementfromAtoBonthecurveΓ,canbewrittenasthesumofthework oftheconservativeforcesandthat

ofthedissipativeonesandwehave

but ,andinconclusion

(2.54)Weseethat,ifnon-conservativeforcesareactive,thetotalmechanical

energyvariesanditsvariationisequaltotheworkofthenon-conservativeforces.Theworkoftheseforcesisnegative,aswesawforfriction.Hencetheenergydiminishes.Thisisthereasonofthedissipativeterm.

Thephysicaldimensionofkinetic,potentialandtotalenergiesarethesameasofthework.Themeasurementunitisconsequentlythejule.

ExampleE2.1LetusgobacktothediscussionmadeinSect.2.12ontheexperimentsbyGalileioninclinedplanes.Figure2.26showsabodyofmassm,whichcanfall,startingfromrestfrompointC,oninclinesofdifferentslopesCAorCDorverticallyonCB.Takeaverticalupwardsaxisz,anddenotebyzCtheheightofC(thatistheheightoftheinclinedplane).

Fig.2.26 Falloninclinesorvertical

ConsiderthemotiononCA.Iffrictionisnegligibletheforceexertedbytheconstraintisnormalanddoesnotmakework.Theotheractingforceistheweightmg.

TheenergyconservationprincipleappliedtothedisplacementCAfromC,wherethevelocityiszero,toA,wherez=0,gives

(2.55)or

(2.56)Weseethatthefinalvelocitydependsonlyonthedifferenceinlevelnoton

theinclination.Ifthefrictionisnotnegligible,thefinalenergyislessthanwehavejust

calculated.WecanobtainitwithEq.(2.54)calculatingtheworkoffriction.Thelatterdoesdependontheinclinationfortworeasons:thelengthsofthepathsaredifferentandthebodypusheswithdifferentforcesontheplane.Todothecalculation,however,weneedtoknowsomethingmoreonfriction.Weshalldothatinthenextchapter.

Wefinallyobservethattheaboveargumentsarevalidifthebodycanbeconsideredamaterialpoint.Ifthebodyalsorotates,likeballsdo,thereisalsokineticenergyassociatedtothelatterthatshouldbeconsidered.WeshalldiscussthispointinSect.8.16.

Aswehavejustseen,inthepresenceofdissipativeforces,thetotalmechanicalenergy,namelythesumofkineticandpotentialenergy,isnotconserved.However,theseareonlytwoofmanyformsofenergy.Asamatteroffactthelawofenergyconservationisoneofthebasiclawsofphysics.Thelawisuniversallyvalid,withoutanyexception,providedalltheformsofenergyare

includedinthebalance.Otherformsofenergyarechemicalenergy,thermalenergy,electricenergy,nuclearenergy,etc.Everytimeenergyseemsnottobeconserved,itisbecausewehavefailedtoincludeoneofitsforms.Theissueisoneofthemainobjectsofthermodynamics,whichwillbediscussedinthesecondvolumeofthiscourse.Thehistoricprocessthatledtoclarificationoftheconceptofenergyandtotheestablishmentoftheuniversallawofenergyconservationwasverylong.Starting,aswehaveseen,alreadywithGalilei,theprocesscametomaturityonlyinthemiddleoftheXIXcentury.Itwasthenestablishedwiththefirstlawofthermodynamics,mainlybyJuliusvonMayer(1814–1878)andJamesPrescottJoule(1818–1889).Energyisconservedalsointhepresenceofdissipativeforcesifinternalthermalenergyisincludedinadditiontomacroscopicmechanicalenergy.

2.15 ATheoremConcerningCentralForcesAregionofspaceinwhichaforcethatisafunctiononlyofthepoint,andpossiblyoftime,iscalledaforcefield.Iftheforcedoesnotdependontime,thefieldissaidtobestationary;ifitdoesnotdependontheposition,itissaidtobeuniform.

Themostcommonexampleofauniformstationaryfieldisweight,whichisconstantintimeandspace(atleastwithinthelimitsofalaboratory).Onthecontrary,theviscousdrag,theresistanceofairtothemotion,say,ofacaroranairplane,isan(increasing)functionofspeedandconsequentlyisnotaforcefield.

AforcefieldissaidtobecentralifineverypointPtheforceisdirectedasthelinebetweenPandafixedpoint,calledthecenteroftheforces.ThesituationissketchedinFig.2.27,whereCisthecenteroftheforces.

Fig.2.27 Acentralfieldofforces

Itisclearlyconvenienttochoosethecenteroftheforcesastheoriginofthereferenceframe.Weshallemploypolarco-ordinatesinwhichr=(r,θ,ϕ)isthepositionvector.LetF(r)betheforceunderconsideration.SayingthattheforceiscentralmeansthatthetwovectorsFandrareeverywhereparallel.Theymayhavethesameoroppositedirections.Thecomponentoftheforceonthepositionvector,theradialcomponent,isitsmagnitudeintheformercase,theoppositeinthelatter.Thisquantitymaydependonthethreecoordinated,thetwoanglesandthedistancerfromthecenter.Iftheforcedependsonlyonr,thefieldissaidtohaveasphericalsymmetry.Ontheotherhand,acentralforcemaybeconservativeornot.Weshallnowprovethatthesetwopropertiesarecorrelated:ifafieldofcentralforceshassphericalsymmetry,theforceisconservativeand,viceversa,ifacentralfieldofforceisconservativeitpossessessphericalsymmetry.

Westartwiththefirststatement.Theradialcomponentoftheforce,sayFr(r),isbyhypothesisafunctionofthedistancefromthecenterronly.GivenanytwopointslikeAandBinFig.2.28,letuscalculatetheworkdonebytheforceonanarbitrarycurveΓ,havingAasoriginandBasend.Weshallproofthatitisindependentofthechosencurve.Weindicatewithdsthegenericelementofthecurve.Theworkcorrespondingtothiselementarydisplacementis

Fig.2.28 Workbyacentralforce

(2.57)whereαistheanglebetweenFandds,whichisalsotheanglebetweenthedirectionsofrofds.Hence,dscosαistheprojectionofdsonthedirectionofr,namelysimplydr,i.e.theelementaryvariationofthedistancefromcenter.N.B.Payattention!Thisnotationisuniversallyemployed,butisambiguous.Thedesignationdrmeansthevariationofthemagnitudeofthevectorr,namelyd|r|,notthemagnitudeofthevectorvariationofr,namely|dr|.

Anywaywehave

(2.58)NoticethatthiselementaryworkmaybepositiveornegativedependingonF

randdrhavingthesameoroppositesign.ThetotalworkonthecurveΓis

(2.59)

whichisindependentofthechosencurve,provingthattheforceisconservative.

Whatwehaveprovenisvalidforwhateverdependenceonr.AparticularlyimportantcaseisthegravitationalforceexertedbyamassM,whichweshallconsidertobepoint-like,onanothermassm,point-liketoo.WeshallstudythegravitationalforceinChap.5.WeanticipateherethatsuchaforceactingonmisinanypointdirectedtowardsthepositionofM;namelyitiscentral.Itsmagnitudeisproportionaltotheproductofthetwomassesandinverselytothesquareoftheirdistancer.IndicatingbyGNtheproportionalityconstant,theforceis

(2.60)wheretheminussignindicatesthattheforceisalwaysinthedirectionoppositetor,namelyisattractive.TheworkdoneonadisplacementfromAtoBis

(2.61)

Asexpected,itisindependentofpath.Wecanthendefinethepotentialenergyofthegravitationalforce.ThepotentialenergydifferencebetweenthepointofpositionvectorrBandthepointofpositionvectorrAistheoppositeoftheworkEq.(2.61),namely

(2.62)Asalways,thepotentialenergyisdefineduptoanarbitraryadditive

constant,namely

(2.63)Theconstantisfixedchoosingapointinwhichthepotentialenergyiszero

bydefinition.Inthiscaseitisobviouslyconvenient(butnotatallnecessary)tochoosethispointatinfinitedistance,obtaining

(2.64)Thisisthepotentialenergyofapoint-likemassm(theearthforexample)in

thegravitationalfieldofthepoint-likemassM(thesun).Noticethat,infact,thisistheenergyofthepairofmassesmandM(seeChap.7).

Wenowprovethesecondoftheabovestatedproperties.Weassumetheforcetobecentralandconservativeandshowthatitscomponent(magnitudewithsign)onthepositionvectorcannotdependonangles.

Letusconsiderforsimplicitydisplacementsonaplane.Consideraclosedpath,asinFig.2.29,composedoftwocirculararcscenteredonthecenterofforcesC,andtworadialsegmentsjoiningtheirextremes,attheanglesθ1andθ2respectively.TaketheradialsegmentsofaveryshortlengthΔs.AssumebycontradictionthatthemagnitudeoftheforceFwoulddependnotonlyonrbutalsoontheangleθ.UnderthishypothesisFrhasdifferentvaluesonthetworadialsidesthatareatdifferentangles,sayFr1andFr2.Letuscalculatetheworkoftheforceonthispath.Thecontributionsofthearcsarezerobecauseonthemtheforceisperpendiculartodisplacement.Thecontributionsoftheradialsegmentsare–Fr1ΔsandFr2Δs.Thetotalworkisthen ,incontradictionwiththehypothesisthattheforceisconservative.

Fig.2.29 Theclosedpathusedinthedemonstration

2.16 PowerInphysics,powerisdefinedastheworkdoneperunittime.Foragivendeliveredwork,thepowerislargerforshorterdeliverytimes.Thesimplestcaseistheworkdonebyaforce,sayF,onamaterialpoint,sayP.Considertheelementarydisplacementdsofthepoint,takingplacebetweentheinstantstand

2.1

2.2

2.3

t+dt.Theworkdonebytheforceis .Thepowerwgivenbytheforcetheworkdividedbythecorrespondingtimeinterval,thatis

(2.65)Inwords:thepowerdeliveredbytheforceFactingonamaterialpoint

movingatthevelocityvinagiveninstantisequaltothedotproductoftheforceandthevelocityofthepointinthatinstant.Iftheforceisafunctionoftheposition,itmustbeobviouslyevaluatedinthepositionofthepoint.

Thephysicaldimensionsofthepowerarethoseofaworkdividedbyatime.Itsunitisthewatt,afterJamesWatt(1736–1819)Onewattisthepowerdevelopedbyaforcedeliveringtheworkofonejouleinonesecond(1W=1J/1s).Tohaveanideaoftheorderofmagnitude,youdevelopabout1Wifyouraiseaglassofwaterby1minonesecond.

2.17 ProblemsApersonissittingonachairsupportedbyahorizontalground.Drawthediagramsoftheforcesfortheperson,thechair,andtheearth.Describeeachoftheforces,identifyingthebodythatproducesthemandthebodyonwhichtheyact.Identifytheactionreactionpairs.

Ablockhangsfromtheceilingthrougharope.Asecondropeisattachedtothebottomoftheblock.Ithangsverticallyandyoudrawitwithyourhandsdownwards.Drawthediagramsoftheforcesfortheblock,eachoftheropes,yourbody,theceilingandtheearth.Describeeachoftheforces,identifyingthebodythatproducethemandthebodyonwhichtheyact.Identifytheactionreactionpairs

Fig.2.30representstwoblocksofmassesm1andm2onfrictionlessplanes.Theplaneofthefirstblockishorizontal;theplaneofthesecondisatanangleθ.Thetwoblocksaretiedbyamasslessinextensiblewirethatcanslideoverapulleywithoutfriction.(a)mentallyinsulateeachblockanddrawtheforcediagrams;thenwritethreeequationsofmotion,(b)findthetensionofthewireandtheaccelerationofm2.

2.4

2.5

2.6

2.7

Fig.2.30 Thetwoblocksofproblem2.3

Abodyofmassm=1kgmovesinacircularuniformmotiononacircleofradiusR=0.1m.Whatisthevalueofthecentripetalforce?

ThesystemrepresentedinFig.2.31isinaverticalplane.M>m.Lettingitfree,Mgoesdownandmgoesup.Neglectingthefrictions,drawthediagramsoftheforcesanddeterminetheaccelerationsofMandofm.

Fig.2.31 Thetwoblocksofproblem2.5

Withahammerofmassm=0.1kgwebeatonanail,whichisalreadypartiallystuckinapieceofwood,withaspeedofυ=1m/s.Thenailadvancesadistanceofs=1cm.Findtheforceexertedbythehammer.

Twopeoplepullarope,eachononeend,eachwithaforceofmagnitudeF.Whatisthetension?For2F?Why?

2.8

2.9

2.10

2.11

2.12

Tworopeshangfromtheceiling.Twospheresofdifferentmasseshangatthetwoends.WithbothyourhandsyouapplytothetwospheresthesameforceF,whichisnotnecessarilyinthedirectionoftherope.Whataretheforcesoneachhand?

ThethreecurvesinFig.2.32representthreerigidguidesinaverticalplane.Threeringsofdifferentmassesslidewithoutfriction,oneoneachofthem.ThethreeringsstartfromAatthesametimewithnullvelocity.Stateforeachofthefollowingstatementsifitistrueorfalse.1.TheringsreachBcontemporarily.2.TheringsreachBwithvelocitiesequalinmagnitude.

Fig.2.32 Thethreeguidesofproblem2.9inaverticalplane

Amanofmassm=80kgjumpsfromaplatformattheeighth=0.5maboveground.Reachingthesoilheforgetstofoldhislegs.Fortunatelythegroundisquitesoftandstopsthemotioninadistances=2cm.Whatistheaverageforceexertedonhisbonesduringthestoppage?

Giveanapproximateevaluationoftheheightreachedbyapolevaulterathleteabletoreachinhisrunthespeedofυ=10m/s.

Fig.2.33showsthreeblocksofequalweightFp.Thepulleyisfrictionless.Ifwegraduallyincreasealltheweights,keepingthemequaltoeachother,whichropewillbreak?

2.13

2.14

2.15

1

Fig.2.33 Thesystemofproblem2.12

Twospheres,onewithmassdoublethatoftheother,arelaunchedupwardswiththesameinitialmomentump0.Iftheresistanceofaircanbeneglected,whatistheratiooftheheightstheyreach.

Aparticleofmassm=2kgoscillatesonthex-axis.Theequationofitsmotionis ,withxinmetersandtinseconds.(a)Whatisthemagnitudeoftheforceactingontheparticleattimet=0?Whatisthemaximumvalueoftheforce?

AtwineoflengthlcanholdamaximumtensionT.Itisemployedtorotateamassmonacircle.Findthemaximumvelocitythebodycanrotateif(a)therotationisinahorizontalplane,(b)inaverticalplane.Drawineachcasethediagramoftheforces.

FootnotesThereaderiswarnedthatonecanstillfindbooksandarticlescallingtheproductmiγ(υ)“relativisticmass”andmi“restmass”.Theformerinarelativisticregimeincreaseswithincreasingvelocity.Theseconceptswereintroducedinthelastyearsofthe19thcenturyandthefirstonesofthe20thwhen

relativitytheorywasbeingdevelopedandthingswerenotyetcompletelyclear.Theyaremisleadingconcepts(whatvarieswithvelocityistheLorentzfactor,notthemass,whichisinvariant)andshouldbeavoided.WeshalltreatrelativityinChap.6.

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©SpringerInternationalPublishingSwitzerland2016AlessandroBettini,ACourseinClassicalPhysics1—Mechanics,UndergraduateLectureNotesinPhysics,DOI10.1007/978-3-319-29257-1_3

3.TheForces

AlessandroBettini1

DipartimentodiFisicaeAstronomia,UniversitàdiPadova,Padova,Italy

AlessandroBettiniEmail:alessandro.bettini@pd.infn.it

In1686,inhisPrefacetotheFirstEditionofthePrincipia,Newtonwrote…thewholeburdenofphilosophyseemstoconsistinthis–fromthephenomenaofmotiontoinvestigatetheforcesofnature,andthenfromtheseforcestodemonstratetheotherphenomena.

Thesecondlawofmotionstatesthatthetimederivativeofthemomentumofabodyisequaltotheforceactingonit.Thelawisnotcompleteaslongastheforcesarenotknown.Asamatteroffact,theforcespresentinnaturehavesimpleexpressions.Therearefourfundamentalforces:thegravitationalforce,theelectromagneticforce,thestrongnuclearforceandtheweaknuclearforce.Thetwolatterforcesexplainhowmatterbehavesatafundamentallevel.Theyappearatnuclearandsubnuclearscales,atwhichquantumphysicsisvalid,anddonotdirectlyappearineverydaymacroscopicphenomena(evenif,forexample,theweakforceisresponsiblefornuclearfusionprocessesintheSunthatgiveuslightandenergy).Inthenextchapterweshallstudyinsomedetailthegravitationalforceandrelatedphenomena.Theelectromagneticforceistheobjectofthe3rdVolumeofthiscourse.

Wehaveexperienceofseveralotherforces.Apartfromweight,whichis(mainly)duetothegravitationalattractionofearth,alltheotherforcesaremacroscopiceffectsofelectromagneticnatureatmicroscopiclevel.Sucharetheelasticforce,thenormalforceofconstraints,frictionandviscousdraginafluid,bothgasandliquid.Theseforcesarenotfundamentalbutareextremely

importantforthestudyofeverydayphenomena.Weshallstudytheseforcesandthecorrespondingphenomenainthischapter.

ThegravitationalforcewillbetreatedinChap.4,fullydedicatedtoit.Furtherstudyoftheviscousdragwillbedoneinthesecondvolumeofthiscourse.

Theelasticforceismetinawidevarietyofcircumstances.Itgivesrisetothemostimportantperiodicmotion,theharmonicoscillations.Harmonicoscillationsandtheconnectedresonancephenomena,ofwhichthemechanicalonesareprototypes,arepresentwithverysimilarcharacteristicsinallthebranchesofphysics,electromagnetism,optics,atomicannuclearphysics.Also,thevastmajorityofthestronglyinteractingparticles,whicharecalledhadrons,areextremelyunstable,livingonlyafewyoctoseconds.Theyaredetectedasresonances.WeshallstudytheharmonicoscillatorinSects.3.8and3.9and,atadeeperlevel,inVolume4ofthiscourse.

Inthelasttwosectionsweshalldiscusstheinformationthatwecangatheronthemotionofthebodiesstartingfromthepotentialenergy,ratherthenfromtheforce,whichispossibleiftheforcesareconservative.WeshallintroduceenergydiagramsinSect.3.10andemploytheminthreeimportantcases,elasticforce,pendulumandmolecularforces,inSect.3.11.

3.1 ElasticForceThesolidobjectsofeverydayexperiencehaveadefiniteshape.However,ifstressedbyaforce,orasystemofforces,theydeform.ConsiderthegeometricallysimplesituationofFig.3.1,showingacylindricalmetalbarattachedononefacetoafixedsupport.Ifweapplyaforce,paralleltoitsaxis,ontheotherface,thebarshortensifwepushandlengthensifwepull.Iftheforceisperpendiculartotheaxis,thebarflexes.

Fig.3.1 AmetalbaranditsdeformationsunderanappliedforceFa

Asanotherexampleconsiderarubberband.Ifwepullonitatoneendwhilekeepingtheotherendfixed,thebandlengthensaccordingtotheforceexerted.Whentheforceisremoved,theribbonreturnstoitsnaturallength.

Manymoreexamplescanbecited.Theirstudyshowsthatifaforceisappliedtoabodyandthenremoved,thebodyresumestheshapeithadbeforebeingdeformed,providedthedeformationhasnotbeentoobig.Intheseconditionswespeakofelasticdeformation.

Forlargerdeformations,theoriginalsizeisnotcompletelyrecovered,rathersomedeformationisleftpermanently.Thisregimeiscalledofplasticdeformation.Thetransitionbetweenelasticandplasticregimesissmooth.Ittakesplaceatstressvaluesthatstronglydependonthechosenmaterial.ThebarinFig.3.1,forexample,willbepermanentlydeformedbymorefeebleforcesifitismadeofwaxratherthanofsteel.

LetusstudythephenomenonwithreferencetothemetalbarofFig.3.1.Whenitisnotstressedbyaforce,itslengthiscalledrestlengthornaturallengthish.Wechooseareferenceaxisparalleltothebardirectedoutsidewiththeorigininitsfreeend.WhenweapplyaforceFa(aforapplied)theextremeendmovesandanewequilibriumstateisreached.Thismeansthatthebarhasreactedbydevelopingaforce,sayF,equalandoppositetoFa.Thedeformation,namelythedifferencebetweentheactualandoriginalshapesis,inthis,case,achangeinthelengthofthebar.Withthechosenco-ordinate,theinfinitesimalchangedx,ispositiveforlengthening,negativeshortening.Thexcomponentoftheforcedevelopedbythebarisinthepositivexdirectionincaseofcompression,negativeincaseofstretching.Themagnitudeofthisforceincreaseswiththedeformationandisexperimentallyfoundtobe,fornottoolargedeformations,proportionaltothedeformationx,namely

(3.1)Theconstantkiscalledelasticconstantorspringconstant.Itisaproperty

ofthematerialcharacterizingitsstiffness.Itsphysicaldimensionsareaforcedividedbyalength,itsunitsarethenewtonpermeter(N/m).

TheproportionalitybetweenforceandlengtheningwasexperimentallydiscoveredbyRobertHooke(1635–1703)in1676andEq.(3.1)iscalledHooke’slaw.Hemadeitpublicinacuriousway.Initiallyhechallengedhiscolleagueswiththeanagram“ceiiinosssttuv”.Twoyearslater,consideringthatnobodyhadsolvedthequiz,hegavethesolution:“uttensio,sicvis”(asisthestretchsoistheforce).

TheHooke’slawisverysimpleandveryuseful.However,itisnotexact,butapproximate.Letusstudythephenomenonmoreprecisely.Weapplytotheextremeofthebaraforceofincreasingandknownvaluesofintensity.Atequilibrium,theseareequaltothemagnitudeoftheforcedevelopedbythebar.Foreachvaluewemeasurethedeformation,bothforextension(positivedeformation)andcompression(negativedeformation).PlottingtheresultsinadiagramweusuallyfindthebehaviorofFig.3.2.Forsmallenoughvaluesthedependenceoftheforceonthedeformationislinear,Hook’slawholds.Iftheforceistoolargehowever,thedeformation,inthecaseofmetalsweareconsidering,islargerthanforeseenbytheHooklawincompression,smallerinextension(weshallunderstandthereasoninSect.3.11).Thenon-linearitystartsatvaluessmallerthanthoseatwhichthedeformationispermanent.Theregimeinwhichthebodyreturnstoitsoriginalshapewhenthestressisremovediscalledelastic,bothifthedeformationislinearornot.Theformeriscalledalinearregime.

Fig.3.2 Forceversusdeformationintheelasticandnon-elasticregimes

Thetransitionbetweenlinearandnon-linearbehaviorissmoothandisfoundatdifferentvaluesfrommetaltometal.Itissmaller,forexample,forleadthanforsteel.

Supposethatwenowkeepincreasingtheforcefurther,forexampleincompression.ThedependenceofthedeformationontheformisshowninFig.3.3,curve(a).Letusnowsupposethat,havingreachedpointQ,westartdecreasingtheforce,alwaysmeasuringthedeformation.Wefindthattherepresentativepointinthediagramdoesnotgobackonthecurve(a)buton(b).Namely,forthesamevalueoftheforce,thedeformationislarger,inabsolutevalue,whenwestartfromadeformedstate.Inparticular,whentheexternalforce,andtheforceofthebarwithit,isbacktozero,thedeformationhasavalue,xr,differentfromzero.Itiscalledpermanentdeformation.Wehavedeformedthebarsomuchthatwewentoutoftheelasticregimeandenteredthe

plasticregime.

Fig.3.3 Theelastichysteresis

Figure3.3shows,foronevalueofthedeformation,twovaluesoftheforce.Infact,thevaluesarenotonlytwo,butafullrangebetweenaminimumandmaximum.Ifweperformthesameprocess,changingthepointQatwhichweinvertsomewhatfurtherorsomewhatsooner,thereturnbranchisnolonger(b),butasimilaronelowerorhigherinthediagram,butalwaysbelowthecurve(a).Inconclusion,theforcedoesnotdependonlyonthedeformationbutalsoonthepastelastichistoryofthebody.Thephenomenoniscalledelastichysteresis.

Foragivenmaterialwecandefinetheelasticlimit,whichweindicatewithL.Itisthemaximumvalueofthedeformingforce(andoftheforcedevelopedbythebody)dividedbythesectionofthebartoremainintheelasticregime.Itismeasuredinnewtonpersquaremeter(N/m2).

Asalltheforcesthatdependonlyondistance,astheelasticforce(withintheelasticlimit),areconservative.Withreferencetotheco-ordinateinFig.3.1,wenowexpresstheworkWofFwhentheextremeofthebarmovesfromx1tox2inthelinearregime.Theworkfortheelementarydisplacementdxis,inthisregime, ,hence

andwecandefinethepotentialenergyfunctionofxor,better,itsdifference

Asalways,todefineitsabsolutevalueweneedtochoosearbitrarilyapointinwhichthepotentialenergyiszerobydefinition.Inthiscaseitisquite

obvious,butnotnecessary,tochosethepointx=0(thatiszerodeformation).Withthischoicewehave

(3.2)Thisexpressionisvalidwithinthelinearregime.Intheelasticnon-linear

regimetheforceisstillconservativeandapotentialenergycouldbedefined,butwithamorecomplicatedexpression.Intheplasticregimetheforceisdissipativeandnopotentialenergycanbedefined.Indeed,tobeveryrigorous,smalldissipativeeffectsexistalsointheelasticregime,buttheycanbeneglectedformanypracticalpurposes.

Adeeperstudyoftheelasticforceshowsthatitistheresultantofanenormousnumberofmicroscopicforcesactingbetweenthemoleculesofthematerial.Theseareultimatelyelectromagneticforces.TheelasticforceandtheHook’slawareamacroscopicdescriptionofaverycomplexsituation,whichdependsonthespecificmicroscopicstructureofthematterofthebodyunderconsideration.

Letusgobacktothelinearregime.Ifthebodyhasasimplegeometry,acylinder,aparallelepiped,awireorabandwecandefineitslength,sayh,anditssection,sayS.Inthesecasesitisfoundwithagoodapproximationthattheelasticconstantofabodyisdirectlyproportionaltoitssectionandinverselytoitslength

(3.3)ThecoefficientE,whichdependsonthematerial(anditstemperature),is

calledYoungmodulusafterThomasYoung(1773–1829)UsingEq.(3.1)inabsolutevalue,wecanexpresstheYoungmodulusas

(3.4)Namelyitistheratioofthedeformingforceperunitsection,whichiscalled

stress,andthedeformationperunitinitiallength,calledthestrain.Thestressisapurenumber,thestrainandtheYoungmoduleareforcesperunitareaandaremeasuredinN/m2.

Itisusefultoappreciatetheordersofmagnitude.TheYoungmodulusvaluesofthemetalsrangeintheorderof1011N/m2(E=2×1011N/m2forsteels,E=1011N/m2forCu,etc.).Theelasticlimitsarearound108N/m2(L=3×108N/m2forsteel,L=108N/m2forCu).Athirdquantityisthefracturestrengthσf,whichisthestressunderwhichthebarbreaks.Forthemetalsthevaluesaretwo

orthreetimeslargerthantheelasticlimits.Oncetheplasticregimeisentered,thefractureisnearing.Theissueoftheresistanceunderstressisveryimportantforengineeringandthedefinitionofsafetylimitsisamuchmorecomplexissuethanthedefinitionwehavegiven.

TypicalvaluesofthethreequantitiesforsomesubstancesaregiveninTable3.1.

Table3.1 ElasticcharacteristicsofsomematerialsinN/m2

Youngmodule Elasticlimit Fracturestrength

10−10×E 10−7×L 10−7×σfIron 20 20 35Steel 22 30 50–200Copper 10 10 20–40Lead 1.5 1 1Glass 6 2.5 3–9Rubber 10−4 10−4 3×10−4

Goingbacktotheordersofmagnitude,considerasteelwireS=1mm2insectionandh=1minlength.WefixitatoneextremeandpulltheotherwiththeforceFa.Itsstretchis

Themaximumforceintheelasticregimeis(Forexampleifthewiresupportsa30kgweight).Thecorrespondingelongationis andthestressisquitesmall,around1.5permille.Astressafewtimeslargerwouldbreakthewire.

Muchlargerstresswithoutfracturecanbeobtainedwithothermaterials,likerubbers.TypicalvaluesarearoundE=106N/m2,L=106N/m2andfracturestrengthof3×106N/m2.Considerarubberwireofthesamegeometryofthesteeloneweconsideredabove,namelywithS=1mm2andh=1m.UndertheactionoftheforceFa,theelongationis .The

maximumforceintheelasticregimeis ,whichisquitesmall,asexpectedforrubber.

Thecorrespondingelongationis .Hence,arubberbandcanbestretchedtotwiceitsoriginallengthwithoutreachingtheelastic

limit.Thereasonforsuchdifferentbehaviorofthemetalsandtherubbersis

explainedbythemolecularstructureofthematerials.Themetalsaremadeofmicroscopiccrystals.Ineachofthesemicrocrystalstheatomsarearrangedatthenodesofaregularlattice.Thedistancesbetweentheatomsaresuchthattheintermolecularforcesareinequilibrium.Whenwetrytodeformacrystalweareattemptingtochangethosedistancesbyactingagainstintermolecularforcesthatarequitestrong.Consequentlythesystemisstiff,difficulttodeform.Ontheotherhand,rubberiscomposedofverylong,spaghetti-like,molecules.Thesemolecularspaghettiformasortoftangledskein.Themoleculesinteractamongstoneanotherlikeapastathathasbeencookedtoomuchandbecamesticky.Whenwepulltherubberwemakethemoleculesstraighter,butwedonotchangetheirlength.Consequently,theprocessismuchsofterthanforacrystallatticeandisreversiblewithinmuchwiderlimits.Wementionthatwhenheatedametalexpands,apieceofrubbercontracts.Withincreasingtemperaturetheequilibriumintermoleculardistancesinacrystalincrease,whileintherubbertheincreasedrateofcollisionsbetweenmoleculesincreasestheirentanglement.

Insummary,themetalwirescanbeloadedwithratherlargestressandtheirstrainissmall.Therubberbandscanhavelargestrains,butdonotbearlargeloads.Ifweneedtoworkbothwithratherintenseforcesandrelevantelongationswecanuseasteelhelicalspring,asinFig.3.4.Whenwepullthespring,itsturnsflexbutthewiredoesnotchangeappreciablyitslength.Theelasticforceisproportionaltothetiltangle,hencetotheelongation(orcontraction)andtheHook’slawholds.

Fig.3.4 Helicalsteelspring

3.2 HarmonicMotionWehavealreadydiscussedthemotion,i.e.theoscillations,ofthesimplependulum.Thistypeofmotionisimportantineverybranchofphysicsandinphysicsbasedtechnologies.Weshallnowstudythemotioninitsdetails.

Tobeconcrete,letusconsiderthesystemshowninFig.3.5.Theblock,of

massm,liesonahorizontalplane,whichweassumetobefrictionless(wecanusethetrickexplainedinSect.2.4).Intheseconditionstheresultantoftheverticalforcesiszero,theweightbeingequilibratedbythenormalforceoftheconstraint.Aspringisconnectedtotheblockatoneendandtoafixedpointontheother.Wetakeaco-ordinateaxis,x,horizontallyinthedirectionofthespringelongationsandwiththeoriginatthepointinwhichthespringisattachedtotheblockwhenitisatrest.Inthisway,xwillmeasurethedeformationofthespring.WeassumeweareintherangeofvalidityoftheHooklaw.Theforceactingontheblockisthen

Fig.3.5 Amechanicaloscillator

(3.5)whichisarestoringforceproportionaltothedisplacement.

Theequationofmotionis

whichwewriteinthecanonicalform

(3.6)Wenowintroducethepositivequantity

(3.7)Thishasaveryimportantdynamicalmeaning. istherestoringforceper

unitdisplacementandperunitmass.Itdependsonthecharacteristicsofthesystem.WecanthenwriteEq.(3.6)as

(3.8)Wehavealreadymetit(withadifferentexpression,Eq.(2.29)forω0)when

discussingthependulum.Thisveryimportantdifferentialequationdescribesthe

motionofmanysystems,includingpendulums,neartheirstableequilibriumposition,whensubjectedtoareturnforceproportionaltothedisplacement.Thegeneralsolution,aslearnedbycalculus,is

(3.9)wheretheconstantsaandbmustbedeterminedfromtheinitialconditionsofthemotion.Theyaretwoinnumberbecausethedifferentialequationisofthesecondorder.

Thegeneralsolutioncanalsobeexpressedinthe,oftenmoreconvenient,form

(3.10)wherenowtheconstantstobedeterminedfromtheinitialconditionsareAandϕ.

Tofindtherelationsbetweentwopairsofconstants,westartfrom

Hence

(3.11)andreciprocally

(3.12)Wenowintroducethetermsusedwhendealingwiththistypeofmotion.To

dothatinageneralway,considertheexpression(withagenericω)

(3.13)Themotionisnotonlyperiodic,butitstimedependenceisgivenbya

circularfunction.Suchmotionsaresaidtobeharmonic.Aiscalledtheoscillationamplitude,theargumentofthecosine, ,iscalledthephase(orinstantaneousphaseincaseofambiguity)andtheconstantϕiscalledtheinitialphase(indeed,itisthevalueofthephaseatt=0).Thequantityω,whichhasthephysicaldimensionsoftheinverseoftime,iscalledangularfrequencyandalsopulsation.Itskinematicphysicalmeaningistobetherateofthevariationofthephasewithtimeand,notice,isindependentoftheinitialconditionsofthemotion.Inthespecificcasewehaveconsideredabove,theharmonicmotionisthespontaneousmotionofthesystem(inSects.3.8and3.9weshallstudymotionsundertheactionofexternalforces)andtheangularfrequency,ωo,asinEq.(3.10),iscalledproperangularfrequency.

Themotionisperiodicwithperiod

(3.14)Thenumberofoscillationsperunittimeiscalledthefrequency,ν.

Obviouslyitislinkedtotheperiodandtotheangularfrequencyby

(3.15)Theperiodismeasuredinseconds,thefrequencyinhertz(1Hz=1s−1),the

angularfrequencyinrads−1orsimplyins−1.TheunitisnamedafterHeinrichRudolfHertz(1857–1899).

Theharmonicmotioncanbeviewedfromanotherpointofview.Consideracirculardiscandasmallballattachedtoapointofitsrim.Thedisccanrotateinahorizontalplanearoundaverticalaxisinitscenter.Supposethediscisrotatingwithaconstantangularvelocityω.Ifwelookattheballfromabove,inthedirectionoftheaxis,weseeacircularmotion,butifwelookhorizontally,withoureyeintheplaneoftherotation,weseetheballoscillatingbackandforthperiodically.Indeed,themotionisnotonlyperiodic,itisharmonic,aswenowshow.

Figure3.6showsthematerialpointPmovingonacircumferenceofradiusAwithconstantangularvelocityω.Wecallϕtheanglebetweenthepositionvectoratt=0andthex-axis.Theco-ordinatesofPatthegenerictimetare

Fig.3.6 ApointPmovingofcircularuniformmotion

Theprojectionofthemotionontheaxes,inparticularonx,isharmonic.Theconclusionleadstothesimplegraphicalrepresentationoftheharmonic

phenomenashowninFig.3.7.TorepresentanharmonicmotionofamplitudeA,angularfrequencyωandinitialphaseϕ,wetakeafixedreferenceaxisxandavectorA,ofmagnitudeA,rotatingarounditsoriginintheplaneofthefigureattheconstantangularfrequencyωandformingwiththexaxistheangleϕatt=0.

TheprojectionofAonthereferencexaxisisourharmonicmotion.

Fig.3.7 Vectordiagramfortheharmonicmotion

Wecanusethisrepresentationalsoforvelocityandaccelerationoftheharmonicmotion.ThederivativeofEq.(3.13)gives

(3.16)Aswritteninthelastside,thevelocityisseentovaryinaharmonicwaytoo,

withaphasethatisforwardofπ/2radianstothedisplacement.ThisisshowninFig.3.8a.

Fig.3.8 Vectordiagramforharmonicmotionavelocity,bacceleration

Differentiatingoncemorewehavetheacceleration

(3.17)Theaccelerationisproportionaltothedisplacementwiththenegative

proportionalityconstant–ω2,or,asseeninthelastside,itsphaseisatπradianstothedisplacement,orinphaseoppositionwithit.

Wenowgobacktotheoscillatorinthelinearregimeandconsideritspotential,kineticandtotalenergies.Theformeroneisthepotentialenergyoftheelasticforce,whichwehavealreadyexpressedinEq.(3.2).WecanusenowEq.(3.7)andwritedirectlyforthetotalenergy

wherex(t)isgivenbyEq.(3.10),andweobtain

(3.18)Weseethatneitherthekineticnorthepotentialenergyareconstantintime,

rather,theyvaryas and respectively,buttheirsum,thetotalenergyis,asweexpected,constant.Noticealsothatkinetic,potentialandtotalenergiesareallproportionaltothesquareoftheamplitudeandtothesquareoftheangularfrequency.

Themeanvalueofaquantityinagiventimeintervalistheintegralofthatquantityonthatinterval,dividedbytheinterval.Itisimmediatetocalculatethatthemeanvaluesofbothfunctionscos2andsin2overaperiodareequalto½(theperiodofthesquareofacircularfunctionishalftheperiodofthatfunction).Consequentlythemeanvaluesofbothpotentialandkineticenergyoveraperiodareonehalfofthetotalenergy.

(3.19)

3.3 IntermolecularForcesAllbodiesarecomposedofverysmallparticlesthatwecall“molecules”.Thesemoleculescombinetoformgases,liquidsandsolids.Moleculesarecomposedofatoms,adifferenttypeforeachchemicalelement.Atomsarealsocompositeobjects.Eachonehasapositivelychargedcentralnucleuscomposedofprotonsandneutrons,whileelectronsformwhatmaybethoughtofasacloudsurroundingthenucleus.Electronsandprotonsareequalinnumber,theatomicnumber,sothateachatomisgloballyneutral.Differentelementshavedifferentatomicnumbers.Quantummechanics,notclassicalmechanics,correctlydescribesthemolecularandatomicphenomena.Wecanhowevergivehereafewsemi-quantitativepiecesofinformation,withaclassicallanguage,thatareconsistentwiththepredictionofquantummechanics.

Anelectroninsideanatomcannotbethoughtofasmovingonawell-definedtrajectorysimilartotheorbitofaplanet(aswasassumedintheearlystagesofthedevelopmentofthetheory,i.e.,theBohrmodel).Wemustinsteadconsidertheprobabilityoffindinganelectronataparticularlocationaroundthenucleus.Thisprobabilityisaknownfunctionofpositiondifferentforeachdifferentatom.It,inparticular,vanishesatacertaindistancefromthenucleus.Itisthe“cloud”wehavementionedabove.Inorderofmagnitude,theradiusesaretenthsofnanometers(or10−10m).Nucleiaremuchsmaller,1–10fm(10−15−10−14m).Ifwemagnifiedanucleustothesizeofthedotonan“i”ofthispage,thediameter

oftheatomwouldbeoftheorderofmeters.Theelementaryconstituentsofchemicalsubstancesaremolecules.For

example,awatermoleculeismadeoftwohydrogenandoneoxygenatoms,nitrogenoneoftwonitrogenatoms,etc.Atomsareboundinamoleculebyelectricforcesthataredescribedbyquantummechanics.Asamatteroffact,electricforcesareverystronginsidemolecules,but,themoleculesbeinggloballyneutral,arealmostnulloutsidethe“cloud”oftheirelectrons.Notcompletelyhowever,astwomoleculeswhentheyarecloseenoughdointeractwithaforce,muchweakerthanthoseinsidethecloud,calledvanderWaalsforce,afterJohannesDiderikvanderWaals(1837–1923)

ThevanderWaalsforcebetweentwomoleculesasafunctionofthedistancebetweentheircentersrisshowninFig.3.9.Itisrepulsiveatsmalldistances,attractiveatlargerones.Inthediagramweadoptedtheconventionofhavingpositiverepulsiveforces.

Fig.3.9 Theforcebetweentwomoleculesasafunctionofthedistancebetweentheircenters

Whenthecentersofthemoleculesareatthedistancer0,atwhichthevanderWaalsforceiszero,theyareinequilibrium.Atsmallerdistancestheforceisrepulsiveandbecomesquicklyenormous.Inaveryroughapproximationwecanconsiderthemasrigidspheresofradiusr0.ThedottedlineinFig.3.9isforanidealizedrigidbody,whichwouldbenon-deformable.Theforcewouldberepulsiveandinfinitewhentryingtosqueezeitandnullatdistanceslargerthanr0whereitisnottouched.

3.4 ContactForces.ConstraintForcesIfweputaheavybody,forexampleabrick,onahorizontalplane,itdoesnotaccelerate,itisinequilibrium.Thisimpliesthattheplane,ingeneraltheconstraint,hasdevelopedaforce,callednormalbecauseitisperpendiculartothe

plane,whichisexactlyequalandoppositetotheforcethatthebody,thebrick,exertsontheplane.Thelattermaybetheweight,asintheexample,ornot.Ifwepushwithourhandonthebrick,themagnitudeofthenormalforceisequaltothesumoftheweightandourpush.Similarly,ifwepushawallwithahand,itdoesnotmove.Thenormalforcemadebythewallisequalandoppositetothepush.

Thenormalforceisacontactforce.Ifweraisethebrickortakebackourhandfromthewall,evenatverysmalldistances,theforcedisappears.Contactforceistheresultantoftheforcesbetweenthemoleculesoftheconstraintandthemoleculesofthebody.Whenthetwoareincontact,moleculesonthetwosurfacesareatdistancesbetweentheircentersequaltomoleculardiameters.Theappliedforcetendstobringthemoleculesofthebodyandoftheconstraintnearertoeachother,namelytoreducetheirradii.ThisisopposedbythevanderWaalsforce,which,aswehaveseen,quicklybecomesenormous.Forthisreasontwosolidbodiescannotpenetrateintoeachother.Wehavealsoseenthattheintermolecularforcegoesquicklytozeroatdistanceslargerthantheequilibriumposition.Thisexplainswhytheforcedisappearsifweseparatethesurfacesevenbyverysmalldistances.Alreadyatafewmoleculardiametersthesurfacesnolongerinteract.

Contactforcesareusedinpracticewhenwewanttoconstrainabodytomoveonacertaintrajectory.Forexample,wehaverepeatedlyusedahorizontalplanetoforceablocktomoveinthatplane;therailsforcethetraintomoveonacertainpath,etc.Thephysicalsystemsusedforthispurpose,thesupportplane,therails,etc.,arecalledmechanicalconstraints,becausetheyconstrainthemotion.Theconstraintsmayinhibitmotionononesideonlyorboth,beingnamedrespectivelyunilateralandbilateral.Asupportplaneisunilateralbecauseitdoesnotinhibitabodyfromrisingaboveitssurface.Therailsofatrainareunilateralbutthoseofarollercoasterarebilateral,thecoastercannotdetachfromtherail.

Usually,forcesproducedbythemechanicalconstraintsarenotknownapriori.Theydependonthemotionofthebody,henceonotherforcesactingonit.Forexampletheforceexertedbytherailonthewheelofatraininagivencurvedependsonthecurvature,butalsoonthespeedofthetrainandonthemassofthewagon.Indeed,theraildevelopsaforcethatisexactlythecentripetalforceneededtohavethewagonmovingatthatspeedonthatcurvaturewithitsmass.Theforcesexertedbytheconstraintsaresaidtobepassive,theotherones,whichareusuallyundercontrol,arecalledactive.

Wecanhowever,calculatethepassiveforcesifweknowthemotionofthebodyandalltheactiveforcesactingonit.Letuslookattwoexamples.

ExampleE4.1WehavealreadystudiedthependuluminSect.2.9.Werecallthatthesimplependulumisamaterialbody,ofmassm,constrainedtomoveonacirculararcofradiusl.TheeasiestwaytoimplementamechanicalconstraintislikeinFig.3.10a,withaninextensiblewirefixedinΩthatexertsthetensionTonthematerialpoint.Clearly,theconstraintisunilateral,becausethewirecanfold.Wecouldmakeitbilateralbyusingalightbarinsteadofthewire.InFig.3.10btheconstraintisimplementedwithawoodenorplasticguideshapedasanarcofacircleofradiusl,inwhichthebodycanslide.Assumingfrictiontobenegligible,theguidewilldevelopanormalforce.Werepresentitwiththesamesymbolasthetensionofthewire,namelyT.

Fig.3.10 Twodifferentmechanicalconstraintsforthesamemotion.asimplependulum,bsolidguide

Inbothcases,thesecondlawgives: .WealreadyknowthemotionandareinterestedintheconstraintforceT.WeobservethatinbothcasesTisdirectedalwaystowardsthecenterΩ.Theradialcomponentoftheresultantoftheforcesmustbethecentripetalone,correspondingtothevelocityυofthebody,namely ,whereυisthevelocityattheconsideredinstantandtheminussignmeansthattheforceistowardsthecenter.Theradialcomponentoftheresultantis andwehave

(3.20)Clearly,Tisnotaconstant,ratheritdependsonthepositionofthe

pendulum,whichisdefinedbytheangleθ.WecoulddothatusingtheequationofmotionwehavefoundinSect.2.9.However,itiseasiertoemployenergyconservation.Thereasonisthetermmυ2inthelastexpression,whichistwicethekineticenergy.Ifthependulumisabandonedfromtheinitialpositionθ0,

correspondingtotheheighty0,theenergyconservationequationis.

Hence .But, and ,hence

,andwecanwrite andfinally,substitutinginEq.(3.20), .

ExampleE4.2Consider,inaverticalplane,aninclinedguideconnectedatitslowerextremewithacircularguide,asshowninFig.3.11.Wewanttostudythemotionofamaterialpoint,asmallrigidballforexample,onthecircularrail,whichisunilateral,ofradiusr.Weusetheinclinetolaunchtheballwithacertaininitialvelocityonthatrail.Moreprecisely,wewanttofindtheminimuminitialvelocityinorderthattheballwouldtravelthroughtheentirecirclewithoutdetachingfromtherail.

Fig.3.11 aTheforcesonaballmovingonaverticalcircularrail,bmotionoftheballincaseofdetachment

Twoforcesactontheball,itsweightmgandtheforceoftheconstraint,whichwesupposetobenormal,N.Thelatterisdirectedastheradius,towardsthecenter.Thenormalforcecannotbedirectedoutwards.

Again,theradialcomponentoftheresultantoftheforcesmustbethecentripetalforcerequestedbythemotion.ThiscomponentisthesumofNandoftheradialcomponentoftheweight.Thelatterisamaximumatthehighestpointoftheguide.

Tobesurethattheballdoesnotdetach,itisthensufficienttoverifythatinthispoint.Here,theweightandtheconstraintnormalforcearebothdirectedverticallydownwards.Theconditionofnon-detachmentisthen .

SolvingfortheunknownNwehave .

Theconditionofnon-detachmentisN>0,hencethetermυ2>gr.Ifthevelocityissmaller,theballdetachesfollowingatrajectoryasinFig.3.11b,whichgivesasequenceofimagesoftheballinitsmotion.Wecanthinkthatinthissituationtheweightisprovidingacentripetalforcetoolargefortheradiusofcurvatureoftheguide,atthatvelocity.Themotionmustfollowatrajectorywithasmallerradius,andtheballdetaches.

3.5 FrictionWehavealreadyseenseveraltimesthataphysicalrigidplane,whenpushedbyabodyincontactwithit,reactswithanormalforcewhichisequalandoppositetotheactiveforce.IntheexampledrawninFig.3.12theplaneishorizontalandtheactiveforce,whichisvertical,issimplytheweightFwoftheblocklyingontheplane.ThenormalreactionNisverticalupwards.

Fig.3.12 aActiveandconstraintforcesonablock,bfrictionforceversusappliedtangentialforce

WenowapplytotheblockaforceFparalleltothecontactsurface(horizontalinthisparticularcase),byattachingawiretotheblockandpulling.Supposethatwegraduallyincreasethetangentialforcestartingfromzero.Weobservethatinitially,whenFisnotverystrong,theblockdoesnotmove,itisstillinequilibrium.Thisimpliesthattheresultantoftheforcesmuststillbezero,notonlyinthedirectionnormaltotheplane,wherenothingischanged,butalsointhetangentone,wherenowthereisaforce.Theconstraintmusthavedevelopedalsoaforceparalleltothecontactsurface,FtequalandoppositetoF,namely

Theforcedevelopedbytheconstraintparalleltothecontactsurface,whenthereisnomotion,iscalledstaticfriction.

IfwecontinuetoincreasethetangentialforceontheblockF,thetangentialforcebytheconstraintincreasestoo,aslongastheblockdoesnotmove.This

happensatacertainvalueoftheactiveforce,meaningthatthefrictionforcecannotbelargerthanamaximumvaluethatwecallFt,max.

Thisbehaviorisfollowedinallcasesinwhichtwodrysurfacesareincontact.Intheseconditions,itisexperimentallyfoundthatthemaximumvalueofthestaticfrictionisproportionaltothenormalforce,namelythat

(3.21)Theproportionalityconstantµsiscalledthecoefficientofstaticfriction,

whichisclearlyadimensionlessquantity.Wenowstudythemotionoftheblockwhenthetangentialappliedforceis

largerthanFt,max.Bymeasuringitsacceleration,weinferthatatangentialcontactforceFtispresent,whichisingeneralsomewhatsmallerthanFt,maxasshowninFig.3.12b.Alsointhecaseofrelativemovementsofthetwocontactsurfaces,itisexperimentallyfoundthatthetangentialforcebytheconstraintisproportionaltothenormalone.Itsdirectionisalwaysparallelandopposedtothevelocity,namely

(3.22)where istheunitvectorofthevelocity.Thedimensionlessconstantµdiscalledcoefficientofkineticfriction.

Figure3.12bshowsschematicallythetangentialforceoftheconstraintversustheappliedtangentialforce.WeseethatFtgrowstobeequaltotheappliedforceuptoFt,max.Then,whenthemotionisstarted,itdiminishessomewhat,aswehavealreadynoticed,andthenremainsapproximately,butnotexactly,constant.Noticethatinthemajorityofthecasesµd<µsbuttherearealsooppositecases.

Asamatteroffact,thestaticanddynamicfrictionforcesareduetotheinteractionsbetweenthemoleculesonthesurfacesofthetwobodies.Consequently,Eqs.(3.21)and(3.22)areamacroscopicdescriptionofacomplexmicroscopicsituation.Weobservethatfrictioncoefficientsdependcriticallyonthestatusofthesurfacesincontact,onhowtheyhavebeenmachined,ontheircleanliness,etc.Noticecarefullythatthemoleculesonthesurfaceofabodymadeofacertainsubstance,forexamplecopperorsteel,arenotonlyofthatsubstance.Waterisalmostalwayspresent,oxidationtoo.Onecanfindmentionedvaluesofthefrictioncoefficientsbetween,say,copperandcopper,copperandsteel,etc.But,thereisnosinglecopperoncopper,etc.frictioncoefficient,forthejustmentionedreasons.

Asamatteroffact,forexampleinthecaseofapieceofcopper,itispossibletoobtainsurfacespopulatedbycoppermoleculesonly.Thepiecemustbe

processedwithadhocproceduresunderavacuum,becauseinthepresenceofair,copperwilloxidizeandwatermoleculeswillbedepositedonthesurfaceimmediately.Nowsupposewehaveproducedtwosuchblocksinavacuumandputtheirsurfaceincontact.Theyimmediatelystickoneontotheotherandyouwillnotbeabletoseparatethem.Theybecameauniquecopperbock.Howaremoleculessupposedtoknowtowhichblocktheybelong?

ThefirstastronautstolandontheMoonobservedthisphenomenon.Puttingtwostonesgatheredfromthesoilintouch,theyfoundthemstickingtogetheranddifficulttoseparate,eveniftheirsurfaceswereobviouslyirregular.

Thereisnouniversalmechanismattheoriginofthefrictionbetweentwocontactsurfaces.Considertheimportantcaseoftwometalsurfaces.Metallicsurfacescanbeworkedtobeextremelysmooth.Evenintheseconditions,surfacesarenotsmoothiflookedatnanometerscales.Figure3.13triestoshowthesurfacesasseenatalargemagnification.Theirregularpatternshaveatypicalscaleof10=100nm.

Fig.3.13 Pictorialviewofthecontactsurfacesbetweentwometals,atnanometerscale

Whentwosurfacesare,wethink,incontact,thecontactisindeedonlybetweenthe“crests”onthetwosides.Consequentlythesurfacereallyincontact,sayScismuchsmallerthanthenominalsurfaceS(typicalvaluesofSc/Sarebetween10−4and10−5).However,thelargeristhenormalforceNpushingthetwosurfacesoneagainsttheother,thelargeristhenumberofcreststouchingeachother.WecanthenunderstandwhythefrictionforceisproportionaltoN.Wecanalsounderstandwhyitisindependentoftheareaofcontact.SupposewekeepNconstantanddoublethecontactmacroscopicsurfaceS.Theactionofthenormalforcewilldistributeonadoubledareaanditseffectonthecrestsperunitareawillhalve.Thenumberofcontactsperunitsurfacewillhalvetoo,buttheywillcoveratwiceaslargearea.Thetotalnumberofcontacthasnotvaried.In

conclusion,ScisproportionaltoNandindependentofS.Inthecontactpointsthemoleculesofthetwobodiesinteractstrongly

attractingeachotherandbecoming,sotosay,welded.Tohaveonesurfaceslidingontheother,thesemicroweldingpointsmustbebroken.AgainthenecessaryforceisproportionaltoScandconsequentlytoNandindependentofS.

Whatwehavejustdescribedisrelativetodrysurfacesbetweensolidbodiesandhasnothingtodowiththefrictionbetweenlubricatedsurfaces.Inthiscase,afilmofliquidispresentbetweensolidsurfaces,themoleculesofwhicharefarenoughawayfromeachothertohaveaninteraction.Inthiscasethefrictionisduetotheviscosityofthelubricant(seeSect.3.6).

Therollingresistanceorrollingfrictionistheforceresistingthemotiondevelopedbytheconstraint,forexamplethesupportsurface,whenacylindricalorsphericalbody,suchasareeloraball,rollsonthesurface.Figure3.14representsincrosssectionsuchacylinder,sayareel,ofradiusr.WeapplyaforceFtotheaxisofthereelparalleltothesupportplaneandnormaltotheaxis.Weassumethatthereeldoesnotslideontheplaneduetothestaticfrictionforce.Thistypeofmotioniscalledpurerolling.Whenthereelrolls,itdoesthataboutaninstantaneousaxisthatisthecontactgeneratorintheconsideredinstant.Themomentoftheappliedforceabouttheinstantaneousrotationaxisisτ=rF.ThemomentτnecessarytohavetherollingataconstantangularvelocityisexperimentallyfoundtobeproportionaltothemagnitudeofthenormalforceN,namely

Fig.3.14 Schematicsoftherollingresistance

(3.23)whereγistherollingresistancecoefficient.Itsphysicaldimensionisalength,andismeasuredinmeters.Theappliedmomentisequalandoppositetothemomentdevelopedbytheconstraint.

Therollingresistanceforceisgenerallysmallerthanthedynamicfriction.Asamatteroffactitisduetoquitecomplicatedphenomenaintheregionofcontactbetweenthereelandthesupportplane.InFig.3.14thisregionisshownasaflatareaoflongitudinalwithδ.Thisisanidealization,becauseactuallyboththecylinderandtheplanedeformintoshapesthatarenotforward-backwardssymmetrical.Weareheresimplifyingalot.Wecansaythatonthecontactareaanumberoftheaboveconsidered“crests”ofbothbodiesareincontact.Thedifferenceisthatnow,tohavemovement,themicroweldsarebrokenactinginadirectionnormal,ratherthanparallel,tothesurface.Thisrequires,caeterisparibus,asmallerforce.

ExampleE4.3ConsiderFig.3.15.Abrickliesonaninclinedsurface,theinclinationofwhich,α,canbevaried.Giventhecoefficientofstaticfrictionµs,whatisthemaximumvalueofαatwhichthebrickremainsstill?

Fig.3.15 Abrickonaslideandtheforcesactingonit

Theforcesonthebrickareitsweightmgandtheforceexertedbytheconstraint.Thelattercanbedecomposedinanormal,N,andatangential,Ft,component,whichisthefriction.Forequilibriumthecomponentsoftheresultantmustbezero.Namely, and .Hence

.But,thestaticfrictionforcecannotbelargerthanµsN,andtheno-slideconditionis .

Themaximumangle,say iscalledthefrictionangle.Forexample,theslopesofthepilesofsandorofthescreesinthemountainsnaturallysettleonthecorrespondingfrictionangle.

WehaveseeninSect.2.11thatfrictionforcesaredissipative,andthattheirworkisnegativewhentheirapplicationpointmoves,becausetheyarealwaysin

adirectionoppositetothemotion,seeEq.(2.41).Indeed,thefrictionforcesarealwayssuchastoopposetherelativemotionofthetwobodies.Thisdoesnotimplythatthefrictionactingonabodywouldalwaysacttoslowitdown,onthecontraryitcanalsoaccelerateit.

Asanexample,letusconsiderourbrick,ofmassm,lingonthehorizontalplatformofacart.Thelattermovesstraightforwardwithconstantaccelerationa(seeFig.3.16)inthedirectionofitsvelocityv.Iftheaccelerationofthecartisnottoolarge,theblockremainsstillrelativetotheplatform;itsmotionisacceleratedwiththesameaccelerationaasthecart.Itmustbeacteduponbyaforceequaltoma.ButtheonlyhorizontalforceactingonitisthefrictionFt.Hence,Ft=ma.Thefrictionacceleratesthebrick.WeknowthatFtcanbeatmostequaltoµsN=µsmg.Consequentlythemaximumaccelerationofthecartatwhichthebrickdoesnotslideisµsg.

Fig.3.16 Abrickonanacceleratingplatformandtheforcesactingonit

Noticethatinthiscasethefrictionhasthedirectionofthevelocity,namelyofthedisplacement.Consequentlyitsworkispositive.Inthesameway,whenwestartrunningweareacceleratedbythefrictionforcebetweenourshoesolesandtheground,whenacaracceleratestheacceleratingforceisthefrictionbetweenitsreelandtheroad.Noticehowever,thatinthesecasestheworkofthefrictionforceiszero,becausethepointofapplicationdoesnotmove.

3.6 ViscousDragAsolidmovingrelativetoafluid,aliquidoragas,issubjecttoaforce,differentfromfriction,butasfrictionopposingtherelativemotionofthebodyandthemedium.Itiscalledviscousdragorviscousresistance.Differentlyfromfriction,thereisnodragwhentherelativevelocityiszero,andanincreasingfunctionoftherelativevelocity.Thedirectionofthedragforceisalwaysequalandoppositetotherelativevelocity.

Themagnitudeofaforcedependsonthemagnitudeoftherelativevelocity,ontheshapeofthebodyandonthefluid.Movingrelativetothefluid,thebody

inducesanumberofeffectsthatmayperturbsubstantiallyitsflow.Thinkforexamplesofvortices.Consequently,thedependenceofthedragforceonvelocityiscomplicated.Weshallstudyitinthesecondvolumeofthiscourse,togetherwithfluiddynamics.Hereweanticipateonlyafewelementsthatareneededinourstudyofthemotionsofbodies.

Theforcedependsontheshapeofthebody,forexampleitisdifferentforacylinderorasphere,onitsorientation,forexamplethecaseofadiscisdifferentforitsorientationparallelorperpendiculartotheflow,and,foragivengeometricalshape,onitssize.Weshalllimitthediscussionheretoasphericalbody,ofradiusa.

Theforcedependsontwocharacteristicsofthefluid,itsdensityρ(massperunitvolume)andtheviscosityη.Thelatterwillbediscussedinthesecondvolume.Itsufficestoknowherethatitcharacterizesthedifficultywithwhichthefluidflows,so,forexample,oilhaslargerviscosity,ismoreviscous,thanwater,butislessviscousthanhoney.Foragivenfluid,theviscositydependsonthetemperature.

Thephysicalunitsofviscosityare

(3.24)where,inthethirdmemberwehavetakenintoaccountthatthedimensionsoftheforceare[F]=[MLT−2].Pressurehasthedimensionsofaforceperunitsurface(FL−2)anditsunitisthepascal(Pa),fromBlaisePascal(1623–1662).Theunitforviscosityisthenthepascalsecond(Pas).Forexample,forsomeeverydayfluidsatambienttemperature,theirviscositiesareforoilsη≈0.5–1.5Pas,forwaterη≈10−3Pas,andforairη≈1.8×10−5Pas.

TheReynoldsnumberisaparameterthatgivesrelevantinformationontheregimeofthemotion,namedafterOsborneReynolds(1842–1912).Itisdimensionless,namelyapurenumber.Thefourquantitiesoftheproblemhavethephysicaldimensions , , and .

Theycanbearrangedinadimensionlessquantityas

(3.25)whichistheReynoldsnumberforasphere.Itsexpressionsforothershapesaresimilar.

Figure3.17showsschematicallyhowthedragforceonabodycanbemeasured.Thebodyisfixedtoathinbarandtothepointerofadynamometerfixedonasupportandisimmersedinthefluidunderstudy,whichismovingataknownvelocityυ,thatwecanvaryinaknownmanner.Experimentsofthistypeshowthatatsmallvelocitiesthedragforcecanbewrittenasthesumofaterm

proportionaltothevelocityandoneproportionaltoitssquare

Fig.3.17 Measuringthedragforce

(3.26)wherethecoefficientsAandBdependonthebodyandthefluidbut,fornottoolargevelocities,areindependentofvelocity.Astheratiobetweenthesecondandthefirsttermisproportionaltothevelocity,thefirsttermdominatesatsmallvelocities,thesecondatlargerones.Wedefineascriticalvelocityυcthevelocityatwhichthetwotermsareequal.ItcorrespondstoaquitesmallvalueoftheReynoldsnumber

(3.27)Considernowthespheremovinginair,aspendulumsorfreefallingbodies,

atnormaltemperatureandpressureconditions.Theairdensityintheseconditionsisρ=1.2kg/m3.Withthevalueforviscosityalreadygiven,theReynoldsnumberis

(3.28)andthecriticalvelocity,inaroundnumber

(3.29)Ifforexamplea=1cm,thecriticalvelocityisυc=4cm/s.Thetimetaken

toreachitbyabodyfreely(inavacuum)fallingfromrestist=υ/g=4ms,whichisveryshortindeed.Inthistimeitwouldtravelinvacuumd=gt2/2=80µm.Forlargerdimensionsbodiesmovingintheairthecriticalvelocitiesareevensmaller.

Weconcludethatonlyforverysmallvelocities,smallerthanυc,istheviscousdragproportionaltothevelocity.However,itbecomesproportionaltothesquarevelocityverygradually,reachingthatregimeonlyatReynoldsnumberstwoordersofmagnitudelargerthaninEq.(3.27),correspondingtovelocitiesofafewmeterspersecondforasphereof1cmradius.

Asasecondexampleconsiderthesamespheremovinginwater.Withρ=103kg/m3andtheviscositygivenabove,η/ρ=10−6kg/m3,whichisavalue,notice,smallerthanforair.TheReynoldsnumberatvelocityυfora=1cmisRe=104υ.Thecriticalvelocityisonlyυc~2.5mm/s.

Intheelementarystudyoffreefall,ofthemotiononaninclineandofthependulum,theviscousdragofairisusuallyneglected.Isthisagoodapproximation?Letuscontrolonafewtypicalcases.Considerabronze(densityρ=8×103kg/m3)ballofa=2cmradiusandthreecases:freefallfromah=20mtalltower,descentofaninclineofelevationh=1mandoscillationofapendulumabandonedattheheightfromthepositionatresth=0.5m.TheweightoftheballisFp=2.7N.Neglectingthepresenceoftheair,andtheenergyoftherotationinthesecondcase,thevelocitiesattheendofthefallwouldbeinanycase ,henceυ1=20m/s,υ2=4.5m/s,υ3=2m/sin

thethreecasesrespectively.Inpresenceofairallvelocitieswouldbesomewhatsmaller,butlargerthanthecriticalvelocity.Thedragforceisapproximatelyproportionaltothesquarevelocity,butisnotverylarge.ForthejustmentionedvelocitiesitsvaluesareapproximatelyR1=2.4×10−2N,R2=1.2×10−2N,R

1=2.4×10−3N,whichareinanycasesmallcomparedtotheweight.Neglectingthedraginthesecasesisnotabadapproximation.However,theeffectwillbenoticeableonmuchlongertimes.

Finallynoticethat,whateveritsexpression,theviscousdragisadissipativeforce.Asitisalwaysdirectedoppositetovelocity,itsworkisnegativeforanydisplacementoftheapplicationpoint.

3.7 AirDragandIndependenceofMotionsInthestudyofthemotionofabodyundertheactionofone(ormore)force,itisoftenconvenienttodecomposethemotioninitscomponentsontheCartesianaxes.Thecomponentofthemotiononanaxisisduetothecomponentoftheforce(orforces)onthataxis.Thecomponentmotionsareindependentofeachother.ThislawoftheindependenceofmotionswasdiscoveredbyGalilei,andassumedbyNewtonasacorollaryofthesecondlaw.Wehavealreadyquotedin

Sect.1.16thefollowingexamplefromGalilei.Supposeweshootaballwithagunatthetopofatower,aiminghorizontally.Simultaneouslywedropaballwithzerovelocity.Theballleavestheriflebarrelwithaveryhighhorizontalspeedand,undertheactionofitsweight,describesaparabolafinallytouchinggroundatahorizontaldistancefaraway.Thesecondballfallsvertically.Galileiestablishedthatbothballstouchgroundatthesameinstant,providedthattheactionofairisneglected,ashespecifies.

Weshallnowanalyzethemotioninpresenceoftheairandweshallseethatthelawofindependenceofmotionsisnotalwaysvalid.

Wereferthemotionoftheball,ofmassm,toaframehavingthey-axisverticalandxhorizontalintheplaneofthemotionasinFig.3.18.Twoforcesareactingontheball,itsweightFp=mgverticaldownwardsandtheviscousdrag

Fig.3.18 Theforcesonaballmovinginair

(3.30)whereυisthevelocityanduυisitsunitaryvector.TheNewtonlawgives

(3.31)Thecomponentsontheaxesoftheequation,ifθistheangleofvwiththe

horizontal,are

(3.32)Thisisasystemoftwonon-lineardifferentialequations,whichcannotbe

easilysolved.However,weareonlyinterestedhereinknowingifandwhenthetwomotionsareindependent.Tobeso,onlyxandycomponentsshouldappearinthefirstandsecondequationrespectively.Thisisindeedthecaseforlowvelocities,whenthetermBcanbeneglected.Intheseconditions,consideringthat and ,Eq.(3.32)becomes

(3.33)Thetwomotionsareindependent.However,if,asitisoftenthecase,the

dragisproportionaltothesquarevelocity,Eq.(3.32)become

(3.34)Themotionsarenotindependent.Thisisanobviousconsequenceofthe

proportionalityofthedragforcetothesquareofthevelocity,whichdependsonbothcomponents.IntheexampleofGalilei,theairresistanceislargerforthegunballthanfortheverticallyfallingone,becausethevelocityoftheformerislarger.Thegunballtouchesgroundlaterthantheballfallingfromthetoweriftheeffectsoftheairarenotneglected.

3.8 DampedOscillatorInSect.3.2wediscussedthemotionoftheharmonicoscillator.Wethenneglectedthedissipativeforces,whichhowever,arealwayspresent.Asweknowthesearebasicallyoftwotypes,frictionandviscousdragoftheair.Weshallnowincludetheviscousdragoftheair,whichweshallassumetobeproportionaltothevelocity.

Fig.3.19 Amechanicaldampedoscillator

Tobeconcrete,considerthesysteminFig.3.19,whichissimilartothatinFig.3.5,withtheadditionofanelementprovidingtheviscousforce.Wecanthinkintermsofanabsorber,likeapistonmovinginafluid,buttheelementismeanttorepresentalltheviscousforces,includingthatduetotheair.Theviscousdragisproportionaltovelocityinmagnitudeandoppositetoitindirection,namely

(3.35)

whereβisaconstant.Weshallneglectthefrictionbetweenthesupportplaneandtheblock.Theforce(3.35)tendstoslowdownordampthemotion.Hencetheoscillatorissaidtobedamped.Thesecondlawgives

(3.36)whichwewrite,dividingbymandtakingallthetermstothefirstmember,inthe“canonical”form

(3.37)Inthisform,theequationisvalidforallharmonicdampedoscillators.The

twoparametersdependonhowtheoscillatorisbuilt,thestrengthofthespring,theviscosity,etc.Wehavealreadymetthefirstonewhilediscussingtheharmonicoscillator.Itistherestoringforceperunitdisplacementandperunitmass

(3.38)Thesecond,seeEq.(3.35)istheresistanceforceperunitvelocityandunit

mass

(3.39)Noticethatbothconstantshavethedimensionoftheinverseoftime.We

alreadyknowthatω0istheangularfrequencyoftheoscillatorinabsenceofdissipativeforces.Theinverseofthesecond

(3.40)isthetimethatcharacterizesthedamping,asweshallnowsee.

ThesolutionofthedifferentialEq.(3.37)isgivenbycalculus.Theruletofinditisasfollows.Firstwewritethealgebraicequationobtainedbysubstitutinginthedifferentialequationpowersofthevariableequaltothedegreeofthederivative.Inourcaseitis

(3.41)Thenwesolveit.Thetworootsare

(3.42)

Thegeneralsolutionofthedifferentialequationis

(3.43)whereC1andC2areintegrationconstantsthatmustbedeterminedfromthe

initialconditions.Letusdiscussthemotionwehavefound.Weobservethattheeffectofthe

dissipativeforce,whichistodampthemotion,islargerforlargervaluesofγ.Consideringthetworootsr1andr2ofthealgebraicEq.(3.41),threecasesshouldbedistinguishedcalledrespectively:under-dampingifγ/2<ω0,thetworootsarerealanddifferent,over-dampingifγ/2>ω0,thetworootsarecomplexconjugate,andcriticaldampingifγ/2=ω0,thetworootsarerealandcoincident.Letusanalyzethethreecases.

Over-damping.Thetwosolutions,whicharereal,arebothnegative.Themotionisthesumoftwoexponentialsdecreasingintime

(3.44)Thedampingissolargethatthesystemisnotabletoperformevenasingle

oscillation.Thedisplacementfromtheequilibriumpositiondecreasesmonotonically.Mathematicallyspeaking,Eq.(3.44)saysthatthetimetoreachthatisinfinite.Inpractice,aftersometimebothaddendaaresosmall,andsoisthevelocity,thatotherresistiveforcesthatarealwayspresent,asthefriction,stopthemotionintheequilibriumposition(x=0).Thishappensinatimeintervalofafewtimes1/|r2|(whichislargerthan1/|r1|).

Criticaldamping.Thetworootscoincide,r=–γ/2=–2/τ.Inthisparticularcase,Eq.(3.42)isnotthesolution.Thisis

(3.45)Inthiscasetoothesystemdoesnotoscillate.Thedisplacementreducesto

zero,inpractice,inatimeintervalofafewtimes2τ.Itcanbeshownthatinthecriticaldampingthetimetoreachequilibriumisminimum.

Under-damping.Wecanwritetheequationofmotionintheform

where

(3.46)

Wecannowchoosetwodifferentintegrationconstantsasa=C1+C2andb=i(C1–C2)andhavethesolutionintheform

(3.47)Fordampingtendingtozero( )theequationofmotionbecomes

Eq.(3.9),asweexpectsincetheoscillatorisun-dampedintheseconditions.The

solutioncanbewritteninaformanalogoustoEq.(3.10)

(3.48)wherenowtheintegrationconstantsareAandϕ.Themotionisanoscillationsimilartotheharmonicmotionwithanamplitude, ,whichisnotconstantbutdecreasesexponentiallyintimewithadecaytime2τ.Theoscillationsaredamped.Aweaklydamped,namelywithγ ω1,motionisshowninFig.3.20.Theoscillationamplitudesdiminishgraduallyinatimelongcomparedtotheperiod.Asamatteroffact,rigorouslyspeaking,themotionisnotperiodic,becausethedisplacementaftereveryoscillationissomewhatsmallerthanbeforeit.However,ifthedampingissmall,γ ω1,wecanstillidentifyaperiod

Fig.3.20 Weaklydampedoscillations

(3.49)Theweakdampingconditionγ ω1canbewrittenasτ T,inwords,the

decaytimeismuchlongerthantheperiod.Noticethattheproperangularfrequencyω1issmallerthantheproper

angularfrequencyofthefreeoscillatorω0,butthatforγ ω1thedifferencebecomesinfinitesimalofthesecondordercomparedtoγ/ω0.

WehaveseeninSect.3.2thatthetotal,kineticpluspotential,mechanicalenergyoftheharmonicoscillatorisconstantintime.Thedifferencenow,eveninthecaseofweakdamping,isthatadissipativeforceispresent.Weexpectthatenergydecreases.Withoutlosinggenerality,wecanassumetheinitialphasetobezero.TheinitialamplitudeofoscillationisA.Ateveryoscillation,thedisplacementreachesitsmaximumat,say,timet.Thedisplacementisthen

(3.50)Inthatinstantthevelocityiszeroandthetotalenergyisequaltothe

potentialenergy,whichisproportionaltothesquareoftheamplitude

(3.51)Thetotalenergydecreasesexponentiallyintime,reducingtoavalue1/eof

theinitialvalueinatimeτ,whichisonehalfofthetimeinwhichtheamplitudereducesofthesamefactor.τiscalleddecaytimeoftheoscillator.

Anobservationontheexponentialfunction.TheamplitudeofadampedoscillationintheEq.(3.48)andtheenergyofthedampedoscillator,Eq.(3.50)areexamplesofphysicalquantitiesdecreasingexponentiallyintime.Thisbehaviorisoftenmetinphysics.Wemakehereasimplebutimportantobservation.Considerthefunction

andtheratiobetweenitstwovaluesintwodifferentinstantst1andt2(t1<t2).Weimmediatelyseethatthisratiodependsonlyontheintervalt2−t1andnotseparatelyonthetwotimesortheconstant(theinitialvalue)f0.Indeed

(3.52)Inparticular,thefunctiondiminishesbyafactor1/eineverytimeinterval

andnotonlyintheinitialone.Inparticular,wecanreformulatetheabovestatementin:“τisthetime

intervalinwhichenergyreducesofafactor1/e”.

3.9 ForcedOscillator.ResonanceConsideragainthedampedoscillatoroftheprevioussectionandapplytothebodyaforceinthedirectionofthex-axisthatoscillatesasacircularfunctionoftimewithangularfrequencyωandamplitudeF0.Thecomponentoftheforceonthexaxis(itsmagnitudeoritsoppositedependingonthedirectionrelativetox)isgivenby

(3.53)wherewehavechosentheoriginoftimesandtheinstantinwhichtheforceiszero;itsinitialphaseisthennull.Thesecondlawis

(3.54)whichwewriteintheform

(3.55)Theleft-handsideofthisequationisthatoftheequationofthedamped

oscillation(3.37).Buttheright-handside,whichiszeroforthelatter,isnowproportionaltotheexternalforce.Equation(3.55)isanon-homogeneousdifferentialequationandEq.(3.37)isitsassociatedhomogeneousdifferentialequation.Amathematicaltheoremstatesthatthegeneralsolutionoftheformeristhesumofthegeneralsolutionoftheassociatedhomogeneousequationandofanyparticularsolutionofthenon-homogeneousone.

Weshalllimitourdiscussiontothecaseofweakdamping,asinFig.3.20.Wecanguessthatapossiblemotionmightbeaharmonicoscillationattheangularfrequencyoftheforce;namelyaparticularsolutionmightbe

(3.56)withsomeamplitudeBandinitialphase–δtobedetermined.Letuscheckifourguessiscorrect.Theeasiestwaytodosoistoconsideranequationexactlysimilarto(3.55)ofthecomplexvariablez(t)=x(t)+iy(t).Theimaginaryparty(t)issomefunctionthatisirrelevantinourarguments.Wethensearchforasolutionofthedifferentialequation,ofwhich(3.55)istherealpart

(3.57)Consideringthattheequationsarelinear,therealpartsofthesolutionsof

Eq.(3.57)aresolutionsof(3.55).Thefunctioncorrespondingtoourguessedsolutionis

(3.58)Letustryitin(3.57)

whichmustbesatisfiedineveryinstantoftime.Andsoitis,becauseallthetermsdependontimebythesamefactor.Hence,Eq.(3.55)isasolutionprovidedthat

(3.59)whichisanalgebraicequation.Theunknown,theparameterwemustfindtohavethesolution,isthecomplexquantityz0.Thisisimmediatelyfoundtobe

(3.60)

Weseethatthesolutioniscompletelydeterminedbythecharacteristicsoftheoscillator,ω0andγandoftheappliedforce,F0andω.Itdoesnotdependontheinitialconditions.

TheparticularsolutionofEq.(3.57)isthen

(3.61)TohaveaparticularsolutionofEq.(3.55)wemustnowtaketherealpartof

thisexpression.Todothatitisconvenienttowritez0intermsofitsmodulusBanditsargument–δ(weshallsoonseethereasonforthenegativesign)

(3.62)Equation(3.60)givesz0asaratio.Themodulusofaratioistheratioofthe

modulusofthenominatorandthemodulusofthedenominator

(3.63)

Theargumentoftheratioisthedifferencebetweentheargumentofthenominator,whichisnull,andtheargumentofthedenominator,anditsoppositeis

(3.64)TheparticularsolutionofEq.(3.57)isthen

(3.65)and,takingtherealpart,theparticularsolutionofEq.(3.55)is

(3.66)Finally,thegeneralsolutionofEq.(3.55)is

(3.67)Letusnowdiscussthemotionwehavefound.Itisthesumoftwoterms.

Thefirstonerepresentsadampedoscillationattheangularfrequencyω1thatisproperfortheoscillator.TheconstantsAandϕ,dependingontheconditionsfromwhichthemotionstarted,appearinthefirstterm.Thesecondtermdependsontheappliedforce.Themotionisundertheseconditionsquitecomplicated.However,theamplitudeofthefirsttermdecreasesintimethefasterthegreaterisγ.Itdiminishesbyafactorofeineverytimeinterval2/γ.Afterafewofsuchintervals,thefirsttermhaspracticallydisappeared.Oncethistransientregimehasgone,theregimeofthemotionisstationary.ThestationaryoscillationorforcedoscillationisdescribedbyourparticularsolutionEq.(3.66),whichis

calledastationarysolution.Wewriteitas

(3.68)Werepeatthatthestationarymotionisaharmonicoscillationattheangular

frequencyoftheforce,notattheproperfrequencyoftheoscillator.However,boththeamplitudeBandthequantityδ,whichisnottheinitialphasebutthephasedelayofthedisplacementxrelativetotheinstantaneousphaseoftheforce,dodependonthecharacteristicsofboththeoscillatorandtheforceasinEqs.(3.63)and(3.64).Animportantphenomenon,theresonance,happenswhentheangularfrequencyoftheforceisnearorequaltotheproperangularfrequencyoftheoscillator:theamplitudeisverylargeandthephasedelayvariesveryrapidly.

Figure3.21representstheamplitudeoftheforcedoscillationasafunctionoftheangularfrequencyBoftheforce.Ithasamaximumattheresonancefrequency

Fig.3.21 Dependenceontheappliedforceangularfrequencyofforcedoscillationsaamplitude,bphasedelay,γincreasesfromcontinuoustodashedcurve

(3.69)asoneobtainswiththeusualmethodsfindingthederivativeofEq.(3.63).NoticethatωRisclosebutnotexactlyequalbothtotheangularfrequencyofthedampedoscillationsω1andtheproperangularfrequencyofthefreeoscillatorω0.Howeverforsmalldamping,namelyforγ/ω0 1,allofthembecomealmostequal.

Asimplewaytoobservetheresonancephenomenon,andtounderstandthereasonforthenoun,isusingtwotuningforks.Thetuningforkisanacousticharmonicoscillatorthatvibratesataspecificfrequencywhensetvibratingbystrikingit.Itismade,likeatwo-prongedfork,withU-shapedprongs,calledtines,andastemofametal,usuallysteel.Theinstrumentisusedtohaveadefinitepitch,typicallyanAat440Hz,totunethemusicinstruments.

Westrikeatineofoneoftheforkstohaveitvibrating,andwehearthesound,withtheotheroneafewmetersfar.Wethenbringthelatternearbyandstopthefirstforkbytouchingitstines.Andwestillhearthepitch.Thesecondfork,thathasthesameproperfrequency,resonated.Thefirstforkhadexcitedsoundwavesintheair,namelypressureoscillationsatthefrequencyofitsvibrations(thesoundwehear).Thesepressureoscillationsactasaperiodicforceonthesecondforkatitsresonantfrequency.Wecandoublecheckthatthisistrueasfollows.Wefix,withalockingscrewnearthetopofoneofthetinesofthesecondfork,asmallweightandrepeattheexperiment.Thistimewedonothearthesecondforksound.Itsproperfrequencyisnowdifferentanditisnolongerinresonancewiththefirstone.

GoingbacktoFig.3.21a,weobservethatcalculationsshowthatthefullwidthoftheresonancecurveathalfmaximum(FWHM)isequaltoγandthatthemaximumisinverselyproportionaltoγ.Asamatteroffact,Eq.(3.63)immediatelyshowsthattheamplitudeisinfiniteintheidealcaseofγ=0.

Wediscussnowthebehaviorofδ,thephasedelayofthedisplacementrelativetotheforce,givenbyEq.(3.64)andshowninFig.3.21b.Whenthefrequencyoftheforceissmallrelativetotheproperone,ω ω0,thenδ≈0,namelyforceanddisplacementareinphase.Onthecontrary,ifthefrequencyoftheforceismuchlargerthantheproperfrequency,ω ω0,thenδ≈π.Wecaneasilyunderstandthephysicalreasonsforthat,consideringtherelativeimportanceofthedifferenttermsinEq.(3.53).Atlowfrequenciestheaccelerationsarequitesmallandtheappliedforceactsmainlyagainsttheelasticforce–kxandisconsequentlyinphasewithx.Athighfrequencies,aswehavejustseen,forceanddisplacementareinphaseopposition;whenthemassisontheright,theforcepushestotheleftandviceversa.Accelerationsarenowverylargeandthedominanttermis ,namelytheinertia.Forceactsmainly

againstaccelerationandisinphasewithit,whichweknowtobeinphaseoppositionwithdisplacement.

Wealsonoticethatourcalculationshowsthetransitiontobebetweenthetwojustdescribedregimesandtakesplaceinaangularfrequencyintervaloftheorderofγ.Thelessisthedampingthemoresuddenisthetransition.Inresonance,asimmediatelyseeninEq.(3.64)δ=π/2,namelythedisplacementisinquadraturewiththeforce,henceitisinphasewiththevelocity.Thepowerexertedbytheforcethatistheproductoftheforceandthevelocityisamaximum.

Theresonancephenomenonisverycommoninnatureandintechnology,notonlyinmechanicsbutalsoinelectromagnetism,optics,atomicphysics,nuclear

andparticlephysics.Infact,allthesystemsoscillateharmonicallywhendisplacedclosetoastableequilibriumconfiguration.WeshalldiscussthisinSect.3.11.Theseoscillationstakeplaceatdefinitefrequenciescharacteristicofthesystem.Engines,forexample,havealwaysarotatingpart.Irregularitiesintheirstructures,evenifsmall,mayproduceperiodicstressesofanaxisandofthesupportstructuresatthefrequencyofenginerotation.Whenthisisvariedandreachesoneoftheresonancefrequenciesofthesystem(theremaybemorethanone)theamplitudeofthevibrationmaybecomeverylargeand,ifthedampingissmall,evendestroytheengine,ifitisnotproperlydesigned.

Wenowconsidertheenergystoredintheoscillatorwheninitsstationarymotion,Eq.(3.68).Itisthesumofthekineticandpotentialenergies

(3.70)

Theexpressionissimilartowhatwefoundforthefreeoscillator.However,thetwotermsarenowproportionaloneto andonetoω2whileforthefree

oscillatorbothwereproportionalto andtheenergywasconstantintime.Nowthetotalenergyvariesperiodically.Thisisbecausethepowerdeliveredbytheforceisnotequalatasingleinstanttothepowerdissipatedbytheviscousforce,whiletheiraveragesonaperiodareequal.Theinstantaneousbalanceexists,however,inresonance,when andthetotalenergy

(3.71)isconstant.

3.10 EnergyDiagramsinOneDimensionInourpreviousdiscussion,theroleoftheforcehasbeentheprincipalone,whilethatofthepotentialenergywassomewhatsecondary.However,wheninamoreadvancedstudyofmechanicsandinotherfieldsofphysics,thepotentialenergyhasacentralrole.Wehavestudiedtheproblem:givenaconservativeforce,finditspotentialenergy.Wenowconsidertheinverseproblem:knowingthepotentialenergy,findanexpressionoftheforce.

Forsimplicityweconsiderthemotioninonedimensiononly.ThepointPmovesonaline,whichwetakeasthex-axis.SupposewehaveonlyoneforceactingonthepointandcallFxitsxcomponent.Supposethepointmovesfromx1tox2.Ingeneral,theknowledgeofx1andx2isnotsufficienttoknowthe

workdone,butwealsoneedtoknowthepathtaken.ThepointPmighthavegonedirectlyfromx1tox2,orhavemovedintheoppositedirectionandafterawhilehavecomeback,etc.Forexample,iftheforceisfriction,itsworkisproportionaltothetotallengthofthepath.However,iftheforceisconservative,asweshallassume,itsworkdependsonlyonx1andx2bydefinition.Forexampletheforcemightbeanelasticforce,ortheweightofthepoint.Inthiscasewehave

(3.72)

Wecanfixthearbitraryadditiveconstantbychoosingapositionx0inwhichthepotentialenergyisnullbydefinition andwrite

(3.73)

WenowwanttoinvertEq.(3.73).Todothatwetakethederivativeofbothitsmembers,immediatelyobtaining

(3.74)Inonedimension,theforceistheoppositeofthederivativeofitspotential

energywithrespecttotheposition.Forexample,thepotentialenergyoftheweight(xisverticalupwards)is andthecorrespondingforce,byderivation,istheoneweknow ,theelasticpotentialenergyis

and,byderivation,theforceis .Equation(3.74)canbewrittenas

(3.75)whichshowsthattheelementaryworkofaconservativeforceisthedifferentialofafunction,theoppositeofthepotentialenergy.

SupposenowthatthepotentialenergyUp(x)oftheforceFx(x)actingonourpointP(inonedimension)tobethefunctionshowninFig.3.22.Thestudyofthistypeofdiagram,calledenergydiagrams,isoftenusefultounderstand,evenifinasemi-quantitativeway,thepossibletypesofmotionofthesystem.

1.

2.

3.

Fig.3.22 Theenergydiagramexamplediscussedinthetext

WestartfromtheequilibriumpositionsofpointP.Aposition,moregenerallyastate,issaidtobeofequilibriumwhen,ifthesystemwasabandonedinthatpositionwithnullvelocity,itremainsthereindefinitely.Thismeansthatinthesepositionstheforceiszero.Wecanrecognizeimmediatelythesepositionsonthediagramasthoseinwhichthederivative,i.e.theslopeofthecurve,iszero,namelywherethecurvehasamaximum,aminimumoraflex,x4,x7,x9,x11inthefigure.However,inpracticewecanneverpositionthebodyexactlyinapositionandifwetrytodothatinamaximumorinaflex,thebodywillrunaway.Asamatteroffact,therearethreetypesofequilibriumstates.Tobegeneral(forthematerialpoint)wedefinetheminthreedimensions.

Stableequilibrium.Apositionofamaterialpointisofstableequilibriumif,whenitisremovedinwhateverdirectionbyaninfinitesimaldistance,theresultantoftheforcestendstobringitbacktowardstheequilibriumposition(restoringforce).

Unstableequilibrium.Apositionofamaterialpointisofunstableequilibriumifatleastadirectionexistssuchthat,whenthepointismovedinthatdirectionbyaninfinitesimaldistance,theresultantoftheforcestendstobringitfurtherawayfromtheequilibriumposition.

Indifferentorneutralequilibrium.Ifthepointisremovedbyaninfinitesimaldistanceinanydirection,thepointremainsthere.Inotherwordstheequilibriumpositionissurroundedbyotherequilibriumpositions.

GoingbacktoFig.3.10,inonedimension,thepositionx4,wherethe

potentialenergyhasarelativeminimum,isofstableequilibrium.Indeed,ifwemoveasmalldistanceontheleft,theforce–dUp/dxispositive,henceinthedirectiontowardsx4.Onthecontrary,ifwemovetotherighttheforce,–dUp/dx,isnegative,hencedirectedtotheleft.

Thepositionatamaximum,likex7inthefigure,isofunstableequilibrium.Ifwemoveabitonthelefttheforceistotheleft(–dUp/dx<0),ifwemovetotheright,theforceistotheright.Inbothcasestheforcetendstopullthepointfartherfromequilibrium.

Infact,oneoftheseconditionsisenoughtomaketheequilibriumunstable.Thishappeninthepositionsoftheflexes,likex9inthefigure.Movingthepointtotheleft,theforceisrestoring,butmovingittotherighttheforceisofremoval.

Considerfinallythepositionx11.Itisonanon-nullsegmentonwhichdUp/dx=0,namelyitissurroundedbyotherequilibriumpositionsandtheequilibriumisneutral.

WewarnthereaderthatthecurverepresentingUpsuggestsaballmovingonhillsandvalleys,namelytothinkoftheordinateaxisastheheight.Thisisindeedthecaseforweight,butnotforotherforces.Intuitionshouldbecontrolled.

Threeotherpiecesofinformationcanbeextractedfromthediagram.Foreveryvalueofpotentialenergy,thematerialpointmayhavedifferentvaluesofkineticenergy.Thesumofthetwo

(3.76)isconstantduringitsmotion.InFig.3.22wehavedrawn,asexamples,fourdifferentvaluesofthetotalenergy.Inanycase,thekineticenergyisthedifferencebetweentotalenergyandpotentialenergy,thedistancefromthelineandthecurve.

Wenowconsiderthat,whiletotalandpotentialenergiesmaybepositiveornegative(orzero),thekineticenergycannotbenegative.Consequently,ifthetotalenergyistoolow,asisUtot1inthefigure,forwhichineverypointthekineticenergywouldbenegative,itisnotpossibleforoursystem.Thetotalenergycannotbelessthantheabsoluteminimum(thedeepestinthefigure)ofthepotentialenergy.

Ifthetotalenergyissomewhatlarger,asisUtot2inthefigure,themotionofthepointcanhappenonlyintworegions,betweenx3andx5inandbetweenx10andx12.Inthisexample,thetworegionsareseparatedbyanon-reachableinterval.IfapointstartsmovingwithtotalenergyUtot2inoneofthetworegions,itcannotleaveit.Onemightthinkthatthepointmightjumpfromoneallowedintervaltoanother,becausetotalenergywouldremainthesame.Butthiscannothappenbecausebetweenthetworegionsthereisaforbiddenone,inwhichthetotalenergywouldbedifferentorkineticenergywouldbenegative.However,thistypeofphenomenonhappensinquantummechanics,inatomicnucleiforexample,whichiscalled“tunnelling”evenifnotunnelexists,becauseitlooksasthoughthesystemwouldcrossunderthebarrierinatunnel.

WecanlearnsomethingmorefromthediagraminFig.3.22.SupposeourmaterialpointwithtotalenergyUtot2tobeatacertaininstantinx3.InthispointthelineofUtot2intersectsthecurveofthepotentialenergy.Alltheenergyofthepointispotential,itskineticenergyiszero.Thepointhaszerovelocity.However,thepointdoesnotremainstill,becauseitisnotinanequilibriumposition.Theforce–dUp/dxispositiveandacceleratesthepointinthedirectionofincreasingx,withincreasingkineticenergy(thedistanceonthediagramfromthelinetothecurveincreases).Theforce,andconsequentlytheacceleration,slowsdownastheslopeofthepotentialenergycurvediminishes.Theybecomezerowhenthepointreachestheminimuminx4.Themotiondoesnotstopthere,itcontinuesnowdecelerated(theslopeispositive,hencetheforceisnegative,oppositetox).Wereadfromthediagramthatthekineticenergyisnowdiminishing.Itdoessouptozerowhenthepointisinx5.Thispointisreachedwithzerovelocity.

Whatdoeshappenafterwards?Theforce–dUp/dxactingonthepointisnownegative,namelyinthedirectionoppositetox.Thepointrestartsitsmotiongoingback.Inconclusion,themotionisanoscillationbackandforthbetweenx3andx5.Themotionisperiodic,butgenerallynotharmonic.Weshallseeinthenextsectionunderwhatconditionsitisso.ClearlyalsothemotionwithtotalenergyUtot2betweenx10andx12isperiodic.

Considernowalargervalueofthetotalenergy,Utot3inthefigure.Therearetwopossiblemotions.Thefirstoneisbounded,aperiodicoscillationbetweenx2andx6similartothatwehavejustdiscussed.Thesecondmotionis,forexample,themotionofapointapproachingfrominfinitedistanceontheright.Initiallyitaccelerates,then,oncex11ispassed,decelerateduptostopinx8.

Hereitbunchesbackgoingthroughininvertedorderallthephasesmovingfartherandnevertocomebackagain.Themotionisunbounded.

Atstillhighervaluesofthetotalenergy,asUtot4,noperiodicmotionispossible,butonlyunboundmotions.Aparticlecomingfromfaraway,oncereachedx1stopsandbunchesbacktoinfinity.

3.11 EnergyDiagramsforRelevantForcesInthissectionweshallusethemethodsdescribedintheprevioussectiontorelevanttypesofmotion:theoscillationundertheactionofarealelasticforce,theoscillationofapendulumandtheoscillationsofadiatomicmolecule.

Letusstartwithaperfectlyelasticspringonthex-axis,whichhasitsoriginintherestpositionofthespring.WeknowthattheforceitexertsonapointinthegenericpositionxisFx=–kxandthepotentialenergyisUp=–kx2/2.Inpractice,aswesawinSect.3.1,nospringisperfectlyelastic.Forlargedeformationsthedependenceoftheforceonthedeformationisnolongerlinear(seeFig.3.2)or,inotherwords,thecurveofthepotentialenergyisnotaparabola,butasshowninFig.3.23.

Fig.3.23 Thepotentialenergyversusdeformationforanideal(dashedcurve)andreal(continuouscurve)spring

Theequilibriumpositionx=0correspondstotheminimumofthepotentialenergy.Ifthematerialpointisabandonedoutsidetheequilibriumposition,itsoscillationsareperiodic.Theyarealsoharmonicifthedisplacementisnottoobig,saywithintheverticaldashedlinesthatmarktheregioninwhichthepotentialenergycurveisatagoodapproximationaparabola.

Considernowasimplependulum.Weattachasmallsphereofmassmtoawireoflengthlandnegligiblemasswiththeotherextremefixed.Ifthesphereisabandonedwithoutvelocityfromoutofequilibriumitwillmoveonthearcofacircleofradiusl.Thepositioncanbemeasuredwithonevariable,theangleθ

betweenthewireandthevertical.Thepotentialenergyis

(3.77)Figure3.24showsthisfunction.Thevariableθcantakeanyvaluefrom−∞

to+∞.However,thefunctionisperiodicandallthepossiblephysicalpositionsarealreadydescribedbythevaluesofθbetween–πandπ.

Fig.3.24 Theenergydiagramforapendulum

Inthefigurewehavetakentheminimumpotentialenergyasthezerooftheenergyscale.Wecanseethatthemotioncanbeunbounded(inangle),ifthetotalenergyislargerthan2mgl,whichisthemaximumpotentialenergy,asUtot2inthefigure.Theangleθgrowsindefinitelyintime,thependulumrotatesonthecircleofradiusl(inpracticethewirewouldtanglearoundthenail).Thevelocityvariesfromaminimumwhentheballisinitshighestposition(θ=π,3π,5π,..),toamaximumwhenitpassesthroughtheequilibriumposition(θ=0,2π,3π,..).

IfUtot<2mgl,asforexampleUtot1inthefigure,themotionislimited.Theballoscillatesbetweentheangles–θ0and+θ0.Ingeneralhowever,themotionisnotharmonic,becausethepotentialenergycurveisnotaparabola.Iftheoscillationsaresmall,however,thecurveisapproximatelyparabolic,asshowninFig.3.25,andthemotionisharmonic.Thesamethingcanbeseenanalytically.IfwedevelopEq.(3.77)inseriesandstopatthefirsttermweobtain

Fig.3.25 Thepotentialenergyofapendulumanditsparabolicapproximation

Noticethattheapproximationisquitegoodbecausethenextterminthedevelopment,theterminθ3isnull,hencethefirstneglectedtermistheoneinθ4.

Thelastexampleisthediatomicmolecule.TobeconcreteweconsiderHCl.Withagoodapproximationwecanconsiderthetwocomponentsaspoint-like.Theatomiccloudsofthetwoatomskeepthetwonucleiatthestableequilibriumdistancer0.Ifthedistanceisdifferent,aforceappears,whichtendstobringbacktheequilibrium.Theseforces,whichareresponsibleforchemicalbonds,areelectromagneticandofquantumnature.TheyaredifferentfromthevanderWaalsforcesweconsideredinSect.3.3.Figure3.26showsthepotentialenergyasafunctionofthedistancebetweenthenucleiofHandCl.ItisknownasMorsepotential.Thecurvehasaminimum,correspondingtotheequilibriumdistancebetweenthenuclei.Thedistancesareoftheorderofthenanometers.Theenergyisgiveninelectronvolt(eV),whichisapracticalunitforatomicenergies.Anelectronvoltisthekineticenergygainedbyanelectronfallingundertheelectricpotentialdifferenceofonevolt.Itsvalueisinroundfigures

Fig.3.26 ThepotentialenergyoftheHClmolecule

(3.78)Supposenowwecommunicateacertainenergytothesystem,forexample

bystrikingwithanothermolecule.Alsointhiscasetherearetwotypesofmotion.Iftheenergygiventothemoleculeislargeenough,asUtot2inthefigure,themotionisunbounded.Thetwoionsseparateandthemoleculedissociates.Iftheenergyissmaller,likeUtot1,themoleculeremainsboundand

performsaperiodicoscillation.AsseeninFig.3.26,thepotentialenergycurveisnotsymmetricaboutitsminimum.However,ifthetotalenergyissmallenoughandthecurvecanbeapproximatedwithaparabola,theoscillationisalmostharmonic.

Wealsoobservethatthepotentialenergycurvegrowsmorerapidlyatenergiessmallerthantheminimumthanathigherones.Inotherwords,therestoringforceislargerthanitwouldbeifelasticforcompression,smallerforexpansion.MacroscopicallythistranslatesintheasymmetryofthedeviationsfromthebehaviordescribedbyFig.3.2.

Theresonancephenomenonispresentalsointhemolecularoscillators,atquitehighfrequencies,oftheorderof1013Hz(10THz).Thesearethefrequenciesoftheelectromagneticwavesintheinfrared.Imaginedoingthefollowingexperiment.WeradiateacontainerwithtransparentwallscontainingaHClgaswithaninfraredradiation,ofwhichwecanvarythefrequencyandwemeasuretheintensityoftheradiationtransmittedbythegasincorrespondence.Takingtheratiobetweenthetransmittedandtheincidentintensitieswehavethequantityofradiationabsorbedbythegasasafunctionoffrequency.WeobtainFig.3.27.Itisaresonantcurve,becauseinresonancemuchmoreenergyistransferredfromtheradiationtothemolecularoscillatorsthanforotherfrequencies.However,twopeaks,notone,areobserved.ThereasonisthatChlorinehastwoisotopes,35Cland37Clofatomicmasses35and37respectively.Thetwoproperfrequenciessquared aredifferent,astheforcesareequal,themassesdifferent,inthetwocases.Tobecomplete,inthespectrumseveraldoubletsliketheoneinFig.3.27arepresent.Thisisbecausequantumoscillatorshaveseveral,ratherthanasingleone,properoscillationfrequencies.

Fig.3.27 AbsorptionprobabilityforHClmoleculesversusfrequency

3.1.

3.2.

3.3.

3.4.

3.5.

Fromtheexamplesinthissectionwecandrawanimportantconclusion.Thephysicalsystemsarefoundnaturallyintheir(oroneoftheir)stableequilibriumstate(s)correspondingtotheminimum(oroneofthem)ofthepotentialenergy.Asmallperturbationcantakethemoutofequilibrium.Thepotentialenergycurveisnotingeneralaparabola.However,ifthedisplacementfromequilibriumissmallenoughitcanbewellapproximatedbyaparabola.Intheseconditionsthesystemoscillatesharmonically.Consequently,thelargestfractionofnaturaloscillationsareindeedharmonic.

3.12 ProblemsConsidertheoscillatorofFig.3.5withm=0.3kgandk=30N/m.Calculatetheproperangularfrequency,theperiodandthefrequencyofitsoscillation.Writetheequationofmotionforaninitialdisplacement,withzerovelocity,of4cm.

Showthattheamplitudeofadampedoscillatorishalvedinatimeof1.39/γ.Howmuchistheenergyvariationinthistime?

Adampedoscillatorhastheproperangularfrequencyω0=300rad/sandω0/γ=50.Calculatetheangularfrequencyofthefreeoscillationsω1andtheresonancefrequencyωR.Comparethevalues.

WebuildamechanicaloscillatorasinFig.3.5.Wecanuseabodywithacertainmassandtwoidenticalsprings.Weseparatelyattachtothemass:(a)onespring,(b)twospringsinseries,(c)twospringsinparallel.Whataretheratiosoftheproperangularfrequenciesincases(b)and(c)tocase(a)?

Aperfectlyelasticspringstretches10cmwhenithangsamassof10kg.(a)whatisthevalueofthespringconstant?(b)Laythespringandthemassonahorizontalplanewithoutfriction.Movethemasssoastostretchthe

3.6.

3.7.

3.8.

3.9.

3.10.

spring5cmandletitgoatt=0.Writetheequationofmotionif(a)theinitialvelocityiszero,(b)theinitialvelocityis1m/sinthedirectionofincreasingx.

Consideraforcedoscillatorvibratingattheangularfrequencyωinitsstationaryregime.Showthatitsenergyismainlypotentialwhenω ω0,mainlywhenω ω0,exactlyhalfandhalfwhenω=ω0.

Weknowtheoscillationamplitudesofthedisplacementandthevelocityofaharmonicoscillator.Howcanweknowtheangularfrequency?

Aforcewithsinusoidaldependenceontimeactingonanoscillatormakesitoscillate,inastationaryregime,withamplitudeA1=20mm.Asecondforce,actingaloneonthesameoscillator,makesitoscillateinthestationarymotionwithamplitudeA2=40mm.Ifbothforcesacttogether,theamplitudeinthestationarymotionisA=30mm.Whatisthephasedifferencebetweentheforces?

Acarofmassm=1000kgtravelshorizontallyat100km/h.Suddenlyanobstacleappearsat100m.Thedriverbrakesimmediately(neglectingthereactiontime,whichis1–2s)andstops10mbeforetheobstacle.Assumingtheforcetohavebeenconstanthowmuchwasthemagnitudeoftheforce?Iftheroadweredownhillwithaslopeof15%atwhichspeedthecarwouldhavehittheobstacle?

Ablockofmassm=1kgliesonahorizontalplaneattachedtoarope,theotherextremeofwhichisfixedtothepointOoftheplane.TheblockundertheseconstraintsismovingonacircleofcenterOandradiusl=1mandvelocityattheconsideredinstantυ=2m/s.Thecoefficientofkineticfrictionbetweenblockandplaneisµd=0.4.Whatisthe

3.11.

3.12.

3.13.

magnitudeoftheresultantoftheforcesinthatinstant?Whatisthedirectionrelativetovelocity?

AsphereofradiusamovingwithvelocityυactsinairwithadragforceR.Thelatterdependsontheradiusas with

and .Consideraraindropfallingstartingfromnullvelocity.Thedropmovesundertheactionofitsweightandtheresistance.Whenthevelocityissmall,theweightislargerthantheresistanceandthedropaccelerates.However,atacertainvelocitythetwoforcesbecomeequalandoppositeandthevelocitybecomesconstant.Itiscalledlimitvelocity.Calculatethelimitvelocitiesforadropofradiusa=0.1mmandforoneofradiusa=1mm.Inbothcasesassumethesecondtermintheaboveexpressioncanbeneglected.Verifyaposterioriiftheassumptionisreasonable.Foradropofradiusa=1mm,nowassumethatthefirsttermisnegligibleandagainverifyaposterioriifthehypothesiswasreasonable.

AbodyofmassmisattachedtoanextremeofaropeoflengthR.Theotherextremeisfixed.Thebodyrotatesinaverticalplane.a)FindtheexpressionofthetensionToftheropewhenthebodypasses,withvelocityυ,inthehighestpointofthetrajectory.Whatistheagentofthecentripetalforceinthispoint?Studythemeaningofthefoundexpressionfordecreasingvaluesofυ.WhatdoesT>0,T=0andT<0mean?WhatdoeshappenwhenthevelocityissuchthatT=0?Repeatforthelowestpoint.

Asmallbodystartssliding,withnegligibleinitialvelocity,onafrictionlesswheelstartingfromitshighestpoint,asinFig.3.28.TheradiusofthewheelisR.(a)atwhatheighth,measuredfromthecenterofthewheeldoesthebodydetachandfallfreely?(b)howwouldtheresultchangeonthemoon?

Fig.3.28 Bodyslidingonawheel,problem3.13

(1)

©SpringerInternationalPublishingSwitzerland2016AlessandroBettini,ACourseinClassicalPhysics1—Mechanics,UndergraduateLectureNotesinPhysics,DOI10.1007/978-3-319-29257-1_4

4.Gravitation

AlessandroBettini1

DipartimentodiFisicaeAstronomia,UniversitàdiPadova,Padova,Italy

AlessandroBettiniEmail:alessandro.bettini@pd.infn.it

ThefirsttwobooksofNewton’sPrincipiaestablishthemechanicslawsforphenomenaonthesurfaceofearth.Thethirdbook,titled“Thesystemoftheword”,appliesthesamelawstointerpretthemotionsofextra-terrestrialbodies.Thegrandunificationofterrestrialandheavenlyphysics,startedbyG.GalileiandJ.Kepler,wascompleted.Intheintroductiontothevolume,I.NewtonwroteItwastheancientopinionofnotafew,intheearliestagesofphilosophy,thatthefixedstarsstoodimmovableinthehighestpartsoftheworld;thatunderthefixedstarstheplanetswerecarriedaboutthesun;thattheearth,asoneoftheplanets,describedanannualcourseaboutthesun,whilebyadiurnalmotionitwasinthemeantimerevolvedaboutitsownaxis;andthesun,asthecommonfirewhichservedtowarmthewhole,wasfixedatthecentreoftheuniverse.

ThiswasthephilosophytaughtofoldbyPhylolaus,AristarchusofSamos,Platoinhisriperyears,andthewholesectofthePythagoreans;andthiswasthejudgmentofAnaximander,moreancientstill…

Afewlinesbelow,afterhavingmentionedthecontributionsoftheRomansandoftheEgyptians,headded

ItisnottobedeniedthatAnaxagoras,Democritus,andothers,didnowandthenstartup,whowouldhaveitthattheearthpossessedthecentreoftheworld,andthatthestarsofallsortswererevolvedtowardsthewestabout

theearthquiescentinthecentre,someataswifter,othersataslowerrate.However,itwasagreedonbothsidesthatmotionsofthecelestial

bodieswereperformedinspacesaltogetherfreeandvoidofresistance.Thewhimofsolidorbswasofalaterdate,introducedbyEudoxus,CalippusandAristotle;whentheancientphilosophybegantodecline,andtogivetheplacetothenewprevailingfictionsoftheGreeks

Observationofthenightsky,withitsmoonandcountlessstarshas,sinceancienttimes,neverfailedtoastonishhumanitythroughouttheworld.Alongwithastonishment,adeepcuriosityarousedaboutthenatureoftheseheavenlybodiesandthereasonsoftheirexistence.Alongwiththemyth,trulyscientificactivitiesdevelopedintimeinseveralcultures.SincethesecondmillenniumB.C.mankindaccuratelyandsystematicallyregisteredthepositionsofthestarsinthesky.However,themysticalcharmofthestarryskycontributedtothesuggestion,inseveralperiods,thatthemotionoftheheavenlybodiesshouldhaveobeyedsymmetryrulesofahigher,oftendivine,order.ThisisthecaseofthesolidorbitsofAristotle,mentionedbyNewton,andoftheuniformcircularmotionsofPtolemyandCopernicus.Gradually,beginningintheRenaissance,theredevelopedaninquiryleadingtoestablishmentofthephysicslawsthatrulesthemotionsinthecosmos.

Inthischapterweshallstudyuniversalgravitation,thephysicallawthatdescribesmotionsoftheplanetsandtheirsatellites,ofthesolarsystemandofthegalaxiesandtheirclustersaswellasthemotionsofallbodiesuptotheboundariesoftheUniverse.WemightstartfromtheNewtonlawofgravitationandanalyzeitsconsequences.Weprefertoreachitfollowing,albeitbriefly,thehistoricalprocessthatledtodiscoveryofthelaw.Indeed,thepathleadingtothesediscoverieshasneverbeenstraight,butrathertortuous,throughlateral,sometimeswrong,paths,withsuccessesandfailures,laboriousinanycase.Universalgravitationisoneofthegrandtheoriesbuiltbyseveralscientists.Knowledge,evenifinasummary,ofthehistoricalrootsoftheprocessaddstothedepthofthephysicslaws.Asamatteroffact,physicscanbeunderstoodevenwithoutknowingitshistory.Thehistoricalpartofthechaptershouldbeconsideredasa,hopefullyinteresting,readingadventure.Thepartstorememberarethelawsandtheirexperimentalproofs.

Figure4.1showsthelifetimespansofthegreatauthorsofthedevelopmentofmechanicsandastrophysicsfromtheXVItotheXVIIIcentury,theperiodoftheconstructionofavasttheoreticaledifice.

Fig.4.1 Lifespansoftheprincipalcontributors

InSect.4.1weshallbrieflydescribethegeocentricandheliocentricmodels.InSect.4.2weshallseehowtheperiodsanddiametersoftheorbitsoftheplanetsweremeasuredfromGreekcivilizationtotheRenaissance.WeshallthenseethefundamentalcontributionofTychoBrahewithhissystematicmeasurements,withprecisionincreasedbyanorderofmagnitude,ofthepositionsoftheplanetsandhowJohannesKepler,basedonthosemeasurements,discoveredthattheorbitsoftheplanetsareellipses,ratherthancircles,andestablishedhisthreelaws.TheKeplerlawsareveryimportantbutstillphenomenological.ThedynamicaltheorywaslaterestablishedbyNewton,asdiscussedfromSects.4.4to4.6.

TheNewtonlawisasimpleandsymmetricmathematicalexpression.Inthefundamentalphysicallawstheharmonyoftheworldtakesonanabstractcharacter,appearingasthesimplicityofthemathematicalexpressionthatisabletodescribeanenormousquantityofdifferentphenomena,which,whenthatlawwasnotknown,appearedtobeuncorrelated.

TheNewtonlawcontainsauniversalconstant,whichisthesameonearthandintheCosmos.InSect.4.7weseehowitwasmeasuredinthelaboratory.

Thegravitationalforceactsbetweenbodiesthatarenotincontact,rathertheymaybeveryfarfromeachother.Theforceactsthroughavacuum.Thisisalsothenatureofalltheotherfundamentalforces,inparticularoftheelectromagneticone.Forallofthemtheconceptoffieldofforceisimportant.Thesourceoftheforce,forexamplethesun,createsafieldofforceinallthespacearoundit.Thefieldthenactsoneverymassiveobjectasaforce.WeshallseethatinSect.4.8.

InSect.4.9weshallgobacktohistoryandshowhowG.GalileidiscoveredthesatellitesofJupiter,discussingsomeofhisdata.

InSect.4.10weshallseehowtheNewtonlawdescribesthemotionsof

cosmicobjectsofthemostdifferentsizesanddistancesandhowitshowsthatthenatureofthelargestfractionofmatterisstillunknown.Itiscalleddarkmatter.

Inthefirstpartofthechapterweassumedforsimplicitytheorbitsoftheplanetstobecircular.Inthefinalthreesections,werelaxthisassumptionanddiscussfullytheproblemofellipticorbits.Thisisknownasthe“directKeplerproblem”:knowingthattheorbitisanellipsewiththecentreofforceinoneofthefoci,findtheforce.Weshalldothatfirstusingmoderncalculusformalism(Sect.4.10),then,in(Sect.4.11)weshallreadandexplain,linebyline,theoriginaldemonstrationofNewton,asabeautifulexampleofhisthought.Inthelastsection,weshallconsidertheenergyofabodyinthegravitationalfieldofacentralbody.

4.1 TheOrbitsofthePlanetsObservationalastronomyisaveryancientscience,datingbacktothemostancientcivilizationstothethirdmillenniumB.C.Thevaryingcelestialco-ordinatesofthestars,ofthemoonandoftheplanetswereaccuratelyandsystematicallymeasuredandregistered.Theproblemhasalwaysbeentounderstandwhatthedatameant.Particularlycomplicatedarethemotionsoftheplanets,whichowetheirnametotheGreekwordfortramp.

TheHeavenlybodies,includingtheplanets,aresofarawaythattheirdistancescouldnotbemeasuredinancienttimes,withtheexceptionofthemoon.Whatwemeasure,foreachbody,isthedirectionatwhichitappearsasafunctionofthetimeoftheobservation.Thedirectionsaregivenbytwoangularco-ordinates.However,itwasnaturaltothinkofthestarsaspointsonasphereofverylarge,butarbitrary,radius,whichwascalledacelestialsphere.Itscenterisontheearthanditsaxiscoincideswiththerotationaxisoftheearth.Thecirclecutonthecelestialspherebytheplanethroughtheearthequatoriscalledthecelestialequator.

TheannualmotionoftheHeavenlybodiesappearstoanobserveronearthasarotationaroundacommoncenter,theearth.Thestars,asdifferentfromtheplanets,domoveonthecelestialsphere,butkeepingallthedistancesbetweentheminvariable.Forthisreasontheyhavebeencalledfixed.Weknownowthatthestarsarenotfixedatallandthattheyareatverydifferentdistances.Theyappeartobefixedbecausethedistancesareenormous.Themoststriking(apparent)motionisthediurnaloneduetotherotationoftheearthonitsaxis.Afurtherapparentmotionofthefixedstars,duetotherevolutionoftheeartharoundthesun,isarotationwithaperiodcalledthesiderealyear.Thesidereal

yearisalsothetimetakenbythesun,initsapparentmotion,toreturntothesamepositionrelativetothefixedstars.Assuchitisalmost,butnotexactly,equaltoourcommonyear.Asweshallsee,themoonandtheplanetshavemorecomplicatedapparentmovements(whicharecombinationsoftheirownandofearth).

Asfordistances,AristarchusofSamos(310–230BC)developedabrilliantmethodtoextractthedistancesfromearthtothemoonandtothesunbyangularmeasurement.Hefoundcorrectlythatthedistanceofthemoonisabout60timestheradiusoftheearth.However,duetoaninsufficientresolutioninthemeasurementoftheangles,heconcludedthatthesunis20timesfartherawaythanthemoon,ratherthanabout400timesasitis.Thiswasenoughtoconcludethat,consideringthemoonandthesuntohavethesameapparentsize,therealsizeofthesunhadtobeenormous.Aristarchusconcludedthathisfindingconfirmedthatthesunmustbethecenterofthesystem.Hethenfoundthecorrectorderofthedistancesofthefiveplanetsaroundthesunthatarevisiblewiththenakedeyearoundthesun,whichwasstandingstillatthecenterofthesystem.However,atleasttoourknowledge,hedidnotfullydevelopaquantitativemodeloftheplanetarysystem.

ApowerfulquantitativemodelwasdevelopedthreecenturieslaterbyClaudiusPtolemy(90–168AD),wholivedatAlexandriainEgypt.Bythattimetheideathattheheavenlybodieshadtomovewithconstant(inmagnitude)velocityoncircles,orcombinationsofcircles,beingbroughtaroundonasystemofsolidspheres,hadbecomedominant,asNewtonrecallswiththewordsquotedintheintroductiontothischapter.Hisbook,originallywritteninGreekandtitled“Mathematicalsyntaxes”cametousthroughitsArabictranslationandisuniversallyknownasAlmagest.

The“planets”wereseven:sun,moon,Mercury,Venus,Mars,JupiterandSaturn.Figure4.2ashowsthebasisofthemodel.Earthisatrestatthecenterofthesystem.Thesundescribesacirclearoundtheearth.Thepathofthesunonthecelestialsphere,throughthefixedstars,istheecliptic.Themotionofeachplanet,likePinthefigure,ismorecomplicated.InafirstapproximationitisdescribedbyacircularuniformmotionaroundearthperformedbythepointCandbyasecondcircularuniformmotionoftheplanetitselfaroundC.Theformercircleisthedeferentthelattertheepicycle.Thetwomotionsare(approximately)inthesameplaneandtheircombinationisacurve,calledanepicycloid,showninFig.4.2a.Clearly,boththedeferentandtheepicyclearedifferentfordifferentplanets.Theobservedtrajectoryoftheplanetistheprojectionofitsepicycloidonthecelestialsphere,takingintoaccounttheanglebetweenitsorbitandtheplaneofthecelestialequator,whichisalsosomewhat

differentfordifferentplanets.Noticethatforthelargestfractionofitsperiodtheplanetmovesforward,fromEasttoWest.However,incorrespondencewiththesmallerloopsoftheepicycloiditmoves,forsometime,backwards.Thisisinaccordwithobservations.

Fig.4.2 Motionofanexternalplanetrelativetoatheearth,bthesun.Figuresareapproximate

Ptolemycalculated,onthebasisoftheavailablemeasurements,resultsofcenturiesofobservations,theradiioftheprimaryandsecondarycircles(withthesolarorbitradiusasunit)andthecorrespondingperiods.Hefoundhowever,thatthisrelativelysimplemodeldidnotwork,namelydidnotexplainallthedata.Tomakeitworkheaddedtwofeatures.

1.Theprimarycircle(deferent)ofeachplanetisnotcenteredexactlyonearthbutinapointnotveryfarfromheranddifferentfromplanettoplanet.Itiscalledequant,becauseitmakesthemotiononthedeferentuniform.Wenowknowthattheequantistheemptyfocusoftheellipticalorbitoftheplanet.WeshallunderstandinSect.4.3howitworks.

2.Anumberoftertiaryandquaternarycircles,allcalledepicycles.ThemodelofPtolemy,thoughevennotparticularlysimple,wasableto

reasonablyexplainalltheobservationalfactsandwouldremainsuchtilltheaccuracyinthemeasurementsoftheplanetspositionswillbeimprovedbyanorderofmagnitudebyTychoBrahe(1546–1601).

Wecannoticethattheperiodofthedeferentintwocases(MercuryandVenus)andofthe(first)epicycleintheotherthreecases(Mars,JupiterandSaturn)areallequaltoasiderealyear.Weknowthattheorbitofthefirsttwoplanetsissmaller,theorbitoftheotherthreeislargerthantheorbitoftheearth(Fig.4.2).Ptolemydidnotnoticethisfeature.Inhismodelallthecirclesareindependent.ThisimportantdiscoveryisduetoNicolausCopernicus(1473–1543).

AnotherfeaturethatisnotexplainedbythemodeliswhybothMercuryandVenusneverdepartmuchfromthesun.ThemaximumanglebetweenMercuryandthesunisθm=22.5°andVenusandthesunθm=46°.

Letusnowgobackforsimplicitytothemodelwithonlyoneprimaryandonesecondarycircle.Letuschangeourreferenceframebychoosingthesunatrestatitscenter.WeassumethattheearthmovesuniformlyonacirclearoundthesunwiththeradiusoftheepicycleoftheplanetandthattheplanetPmovesuniformlyonthecirclecenteredonthesunandradiusequaltothedeferentradius,asinFig.4.2b.Therelativepositionsofearthandplanetareexactlythesameasbefore,butthedescriptionisnowlogicallysimpler.Inaddition,thereasonforwhichtheperiodontheepicycleisthesiderealyearbecomesobvious.Figure4.2representsanexternalplanet.Thereadercaneasilyverifythatananalogueexplanationworksfortheinternalplanetsbysimplyexchangingtherolesofprimaryandsecondarycircles.

IntheheliocentricframethereasonwhyMarsandVenus,whichhaveorbitssmallerthaneartharoundthesun,cannotbeveryfarfromthesunwhenviewedbyearthisalsoclear,asshowninFig.4.3.ThisargumentwasalreadyknowntotheGreeks,inparticulartoAristarchus.

Fig.4.3 Theorbitofaninternalplanet,aviewedfromearth,bviewedbythesun

Theheliocentricdescriptionwehavesketched,initsmodernform,isduetoNicolausCopernicus.Hegaveapreliminaryversionofhismodelinthe“Commnetariolus”distributedprivatelytohisfriendsin1514,andthefinalonein“DeRevolutionibusOrbiumCaelestium”publishedin1543,theyearofhisdeath.DifferentlyfromAristarchus,Copernicusdevelopedafullmathematicalmodelabletoexplaintheobservationalfacts.

TheCopernicusmodel,aswehavepresenteditsofar,looksmuchsimplerthanthePtolemymodel.Onecanthenaskwhyittooksolongtobeaccepted.

Thereasonisthat,asforPtolemybeforehim,suchasimplemodeldoesnotwork.ThemainreasonwasthatCopernicusstillbelievedthattheorbitshadtobecirclesorcombinationsofcirclesandthemotionsonthemuniform.Thereasonofthebeliefwasdogmatic,ratherthanscientific:theheavenlybodiesbeingthecreationofGod,theirmotionmustbeperfect.Thebodiesmustbeonarotatingsphere,because,inhiswords,thesphereinitsrotationmoves

onitselfthroughthesamepoints,itexpressesitsforminthesimplestbody,inwhichitisimpossibletofindeitherabeginningoranendordistinguishthepointsfromeachother.

Theconsequencewasthat,toagreewiththedata,Copernicus,aslongbeforehimPtolemy,hadtointroduceboththeequantandaratherlargenumberofepicycles.Indeed,theCopernicusmodel,intheformhepresentedit,wasnotlessarbitrarythanthePtolemymodel.

4.2 ThePeriodsofthePlanetsandtheRadiiofTheirOrbitsAswehavealreadymentioned,twoplanets,MercuryandVenus,intheirmotionasseenfromearthnevergofarfromthesun.ThemaximumanglebetweenMercuryandthesunisθm=22.5°andVenusandthesunθm=46°.ThemodelofCopernicusallowsustocalculatetheradiioftheorbitsoftheseplanets.HerethemodelshowsitssuperioritytoPtolemy.

FromFig.4.4,whichisdrawnforVenus,wehave

Fig.4.4 TheearthandtheJupiterorbits

(4.1)

Noticethattheconditionisfortheratiooftheradiusoftheplanetwiththeradiusoftheearthorbit.Indeedthelatteristhenaturalunitinastronomicalmeasurementsandiscalledastronomicalunit(au).Tobeprecisetheastronomicalunitisthemeandistanceoftheearthfromthesun.WeshallnotdiscussthedifferentmethodstomeasurerE.Wesimplymentionthattheproblemofthescalesofthedistancesisacentraloneinastronomy.

ThevalueoftheastronomicalunitwasnotknowneventoKepler.Hewasabletodeterminealowerlimit(onthebasisoftheparallaxofMars)as1au>15Gm.ThefirstmeasurementsweremadeatthebeginningoftheXVIIcenturybyGiovanniDomenicoCassini(1625–1712)andbyEdmundHalley(1656–1742),whofoundvaluesbetween140and150Gm.

Thevalueknowntodayis

(4.2)FromtheabovevaluesofθmwehaveforMercuryrM≈0.34auandfor

VenusrV≈0.72au.Fortheexternalplanets,thethreeknowntoCopernicus,theargumentis

similar,butnowtheradiusoftheorbitoftheplanetislargerthanthatoftheearth.Figure4.4givesthegeometry.TheCopernicusinterpretationisthatthelargercircle,thedeferent,istheorbitoftheplanet,andthesmallerone,theepicycle,istheorbitofearth.Consequentlytheangulardiameterunderwhichthelatterisseenfromearthis2θm.Fromthefigureweseethat

(4.3)AlreadyPtolemyknewtheanglesforthethreeplanets,θm=41°forMars,θ

m=11°forJupiterandθm=6°forSaturn.Equation(4.3)givesfortheradiioftheirorbitsrMa≈1.5auforMars,rJ≈5.2auforJupiterandrJ≈9.5auforSaturn.

Letusseehowtoextracttheperiodsfromtheobservationaldata.Forthatwemusttakeintoaccountthattheobservationsaredonefromaframemovinginthesolarsystem.Thisproblemissolvedalittledifferentlyfortheinternalandfortheexternalplanet,asinthecaseoftheradiioftheorbits.Forthesakeofbrevityweshallconsideronlyoneexternalplanet,forexampleJupiter.

ConsiderthetwosituationsrepresentedinFig.4.5.Inbothofthemtherelativepositionsofearth,sunandJupiteristhesame.Itisalsosuch,beingthethreebodiesonthesameline,tobeeasilyandpreciselyrecognized.Thisisdone,foragivenobserver,bytakingthedateatwhichJupitercrossesthecelestialmeridianatmidnight.Thecelestialmeridianistheprojectionofthelocalmeridianonthecelestialsphere.

Fig.4.5 TwoconsecutivetransitionsofJupiteronthecelestialmeridian

Theintervalsbetweentwoconsecutiverecurrencesofthephenomenonareallequalandcalledthesynodicperiod.Consequently,wecanaverageonseveralmeasurementsandincreasetheprecision.ThesynodicperiodofJupiterisτ=399d.InthisperiodJupitertravelsthroughtheangleθ(Fig.4.5),theearthtravelsthatplusarevolution,namely360°+θ.ThenumberofrevolutionsofJupiterperunittimenJ=1/TJ,whereTJisitsperiod.Similarlyforearth,nE=1/TE.Wecanthenwrite thatis andalso

QuestionQ4.1.Findtheequivalentexpressionforaninternalplanet.Table4.1givesthevaluesoftheorbitradiiinastronomicunitsandofthe

periodsofthefirstsixplanetsasknowntoCopernicusandasitistoday.

Table4.1 Orbitradiiandperiodsofthefirstsixplanets

Planet Orbitradius(au) Orbitradius(au) Period PeriodCopernicus Modern Copernicus Modern

Mercury 0.376 0.387 87.97day 87.97dayVenus 0.719 0.723 224.70day 224.70dayEarth 1.000 1.000 365.26day 365.26dayMars 1.520 1.524 1.882year 1.881yearJupiter 5.219 5.203 11.87year 11.862yearSaturn 9.174 9.539 29.44year 29.457year

WeseethatthevaluesknowntoCopernicus,inparticularfortheperiods,werealreadyclosetothemodernones.WeaddthatthevaluesthatcanbeextractedfromthedataofPtolemyarequitesimilartoo.Amillenniumofobservationsbefore150ADdidallowgreatprecision.

4.3 TheKeplerLawsAswehaveseenthe(almost)heliocentricCopernicussystemwasnotmuchsimplerthanthePtolemy(almost)geocentricone.Bothsystemsmakeuseoftheequant.Tobeprecise,thecenteroftheCopernicussystemisnotthesun,buttheequantoftheearth(whatwenowknowtobetheemptyfocusofherellipticalorbit).Inbothcases,beyondaprimarycircle,severalsecondaryandtertiaryoneswerenecessarytofitthedata.Sincehisyouth,TychoBrahe(1546–1601)startedhisstudyoftheastronomicaltextsandhisobservationsofthenightsky.HesoonfoundoutthatneitherthetablesofPtolemynorthoseofCopernicuswereveryaccurate.Bothofthemwereincontradictionwiththefacts.Whenhewas17yearoldhehadtheopportunitytoobserveanotveryfrequentphenomenon,theconjunctionofJupiterandSaturn(thetwoplanetsappearveryclosetoeachother).BrahecalculatedtheconjunctiontimepredictedbythePtolemytablesfindingittobeoffbyaboutonemonth(whichisnotreallysomuchconsideringitisbasedonobservations1400old)andthatpredictedbytheCopernicustablesfindingitoffbyseveraldays(beinganextrapolationoverafewdecennia,therelativeerrorofCopernicusismuchlargerthanthatofPtolemy).BrahewasnowsurethatacorrectmodeloftheCosmos(thenthesolarsystem)couldbefoundbyplanningandperformingasystematicseriesofmeasurementsasaccurateaspossible,ratherthaninterpretingtheclassictexts.

Theobservationsstillhadtobedonewiththenakedeyebecausethetelescope,asascientificinstrument,willnotbeinventedbyGalileiuntil1609.OneofhisfirstinstrumentsisshowninFig.4.6.Thestarunderconsiderationmustbeseenthroughtwosmallholes(DandEinthefigure)fixedattheextremesofabarthatcanrotateoverthearcofacircle.Theangleofthebarrelativetothevertical,definedbytheplumblineAH,ismeasuredwithagoniometeronascalegivingthearcminute.Toincreasethesensitivitytheinstrumenthadtobelarge.Thegraduatedcirclewasalmostsevenmetersindiameter.Theinstrumenthadtoberobustandaccuratelybuilttoreducesystematicerrors.Theinstrumentwasbuiltoftimberandwassoheavythattwentymenwereneededtoinstallitinagarden.

Fig.4.6 InstrumentofBrahetomeasurethepositionofthestars

SomewhatlaterBrahesucceededtobefundedbykingFredericIIofDenmarkandNorwayfortheconstructionofabigastronomicobservatoryontheislandofHveennearCopenhagen,theUraniburgobservatory.Thecastleinwhichtheobservatorywasbuilthadarichlibrary,bedrooms,kitchensanddiningrooms.Brahedesigned,builtandinstalledadozendifferentinstruments,aptatvarioustypesofobservation.Forthenext20years,atUraniburgandlaterinPrague,Brahecontinuedhissystematicobservations.BeforeBrahetheangularresolutionhadnotimprovedfromGreciantimes,beingabout10′.Hewasdeterminedtoimprovedownto1′orbetter.Hegatheredthedatainaseriesoftables,whichbecamethedatabasethatallowedKeplerandNewtontosolvetheproblemoftheheavenlybodies’motions.

JohannesKepler(1571–1630)startedhisstudiesintheschoolofTychoBrahein1600.Hebeganbysearchingthroughalargeamountofavailabledatatodetermineifhecouldfindanysimplerelation.TablessuchasTable4.1pointedtoexistenceofarelationbetweenorbitradiiandperiods.Thelargertheradiusthelargeristheperiod.But,istherereallyamathematicallysimplerelation?Keplerfinallyfounditandpublisheditinthebook“Harmonicemundi

”in1618.Hewriteswithconfidence:

initiallyIthoughtIwasdreaming…butitisabsolutelycertainandexactthattheratioexistingbetweentheperiodictimesofanypairofplanetsisexactlytheratioofthemeandistances[fromsun]tothepower2/3.

Wecandothecalculationsourselves.StartingfromTable4.1weobtainthedatainTable4.2.WecaneasilyunderstandKepler’sprideandsatisfactionwhenhefoundsuchasimplerelation.Weknowitasthe3rdKeplerlaw,becauseitcame10yearslaterthanthediscoveryofthefirsttwo.Thefirsttwolawsregardtheorbitsofasingleplanet,thethirdgivesarelationbetweendifferentplanets.

Table4.2 Ratiosofthecubesoftheorbitradiiandthesquaresoftheperiodforthefirstsixplanets

Planet r3/T2(au3d–2)Mercury 7.64×10–6

Venus 7.52×10–6

Earth 7.50×10–6

Mars 7.50×10–6

Jupiter 7.49×10–6

Saturn 7.43×10–6

LetusnowbrieflyseehowKeplerestablishedthattheorbitsoftheplanetsarenotcomplicatedcombinationsofcircles,but,simply,ellipses.Itsgreatdiscoverywasbasedonthestudyofasingleplanet,Mars.ThechoicefellonMarsbecauseitsdeviationsfromthepredictionsofbothmodelsbasedoncircleswherelargerthanfortheotherplanets.Itsstrangebehaviorwastheobjectofstudyofseveralastronomers,butitsanomaliesremainedunexplained.BrahehadtakenKeplerashisassistantin1600andchargedhimwithasolutiontothisproblem.Keplerworkedontheproblemfor6years,inwhichpartialsuccessesalternatedtopartialfailures,wrongpathswerefollowedandretracedback,beforereachingthesolutionthatweknow.

Keplerfullyacceptedfromthestartaheliocentricviewwiththeguidingideathattheorbitsshouldbeasimplecurvearoundthesun,butnotnecessarilyacircle.Theproblemtofindthecurvewasmadedifficultbythefactthatthepositionsoftheplanet,Marsinhisanalysis,weremeasuredinaframefixedtotheearth,whichmovesinanon-uniformandunknownmotionaroundthesun.Ittookseveralyearstosolvethisfirstproblem,tofindaccuratelyenough,the

motionofearth.Weshallnotdescribeherethevariousmathematicalmethodsheemployed,someofwhicharereallyelegant.Wesimplystatethathefoundthattheearthorbitisindistinguishablefromacircle.However,itscenterisnotthesunanditsangularvelocityaboutthesunisnotuniform.ThedogmathathadresistedfromAristotletoCopernicusincludedwasbroken.

WithreferencetoFig.4.7,disthedistancefromthecenterofthesuntothecenterofthecircleandRitsradius.FromthedataofBrahe,Keplerfoundthatd/R=0.018.Theangulardiameterofthesun,asseenfromearth,variesperiodicallyduringtheyearbetweenaminimumandamaximum.KeplerhadBrahe’smeasurementsforthat.Withtheabovevalueofd/R,Keplercalculatedthevariationsofearthsundistanceduringtheyearandtheconsequentvariationsoftheapparentsundiameter.Hefoundhisresultsinagreementwiththedata.Hegainedconfidencethathewasonthecorrectpath.

Fig.4.7 Schemeoftheearth’sorbit.FirstapproximationbyKepler.Continuouslineisacircle,dottedlineanellipse;thedifferencebetweenthemisexaggerated

Inretrospectweknownow,andKeplerhimselfwastolearnthatinawhile,thatthismodeloftheearthorbitisnotcorrect,becausetheorbitisanellipse.However,theeccentricityoftheearthorbitissosmallthatthemaximumdifferencebetweenthepreliminaryKeplermodelandthetrueorbitwassmallerthantheexperimentaluncertainty.Tofixtheordersofmagnitude,thedistanceNN′isaboutonehalfofapercentofR.Inconclusiontheerrorintroducedintheanalysisbythepreliminarymodelisirrelevant.

Havingdefinedthegeometryoftheorbit,Keplerhadtofindthemotion.HedidthatusingatrickinventedbyPtolemy,andthatwehavealreadyquoted,theequant.ThisisthepointQinthefigure,lyingonthelinejoiningthecenterofthesunandthecenterCofthecircle,atthesamedistancedasthesunbutonthe

1.

2.

3.

otherside.ThentheangularvelocityofthepositionvectorfromQtotheearthisconstant.Itiscalledequantforthisreason.Weshallseesoonwhyitworks.

Keplernowknewthemotionofearthinareferenceframeinwhichthesunstoodstill.HecouldthencalculatethepositionsofMarsatalltimes.Itwasanenormousamountofcalculations(byhandobviously).Oncemore,heassumedtheorbitoftheplanettobeaneccentriccircleandauniformangularvelocityaroundanequant(differentfromthatofearth).Hecalculated40pointsontheMarsorbitandcompareditwiththeBrahedata.Themaximumdisagreementwasonly8′,averysmallone,butlargerthantheuncertaintiesintheBrahemeasurements.KeplerknewhecouldtrustBrahe.Themodelhadtobewrong.

Keplerhadtofindanothercurve.Finally,hisenormouscomputingeffortshowedthelight.Suddenly,everythingbecameclear:thecurveistheellipse.ThefirsttwoKeplerlawswerefound.Keplercontinuedhisworkfindingtheparametersoftheellipseoftheorbitsoftheotherplanets,includingearth,calculatingtheirpositionsandfindingtheminagreementwiththerichandpreciseBrahedata.

WenoticenowthatthereasonwhyaneccentriccirclehadworkedfortheearthandnotforMarsistherelativelylargeeccentricityofitsorbit,whichis0.09,whichisfivetimelargerthanthatoftheearth.

Hepublishedhisresultsin1609inhisbookAstronomianova.ThethreeKeplerlawsare:

Theorbitsoftheplanetsareellipses,thesunoccupyingoneoftheirfoci.

Thepositionvectorfromthesuntotheplanetsweepsoutequalareasinequaltimes

Theratioofthesquaresoftheperiodsofanytwoplanetsisequaltotheratioofthecubesoftheiraveragedistancesfromthesun.

Wecannowshowthereasonsthatmaketheequantworkinafirst

approximation.Indeed,thereasonisinthesecondKeplerlaw.ConsiderFig.4.8whereanellipse,infactmuchmoredifferentfromacirclethantherealcases,isshown.Theequant,whichisthecenterofacirclethattriestorepresentthe

ellipse,isjusttheemptyfocusoftheellipse.InFig.4.8theareasSCDandSABaretravelledinthesametimebytheplanetandareequalforthesecondKeplerlaw.ConsequentlythearcCDislongerthenABproportionallyatitsdistancefromthesun.However,thereisasecondeffect.Agivenpathlengthontheorbitappearsfromthesuntobesmaller,initsangularspan,whenitiscloserthanwhenitisfarther,oncemoreproportionallytothedistance.Thetwoeffects,oneduetothelawoftheareasandthegeometricaloneareidentical.Consequently,ifwelooktotheplanetfromtheotherfocus,theformereffectremainswhilethesecondinvertsandthetwocanceleachother.

Fig.4.8 Geometryplusarealawexplaintheequant

ThecontributionofBrahehadbeenasystematicandaccurateexperimentalwork,theworkofKepleraningeniousandsuperbanalysisofthedata.Bothwereneededtodiscoverthreesimplelaws,whichwereabletointerpretalltheavailabledata.Theworkwasnotyetcompletehowever.ThemarvelousKeplerlawswerestillpurelyphenomenological.Afundamentalstepwasmissing:theirdynamicalinterpretation,whichwasgoingtoleadtouniversalgravitation,oneofthehighestcreationsofhumangenius,thegeniusofIsaacNewton(1642–1727).

4.4 TheNewtonLawWebeginbyshowingthataconsequenceoftheKeplerlawsisthattheangularmomentum,L,ofanyplanetPaboutthepositionofthesunisconstant.WithreferencetoFig.4.9,letrbethepositionvector,vthevelocityandmthemassoftheplanet.Itsangularmomentumisthen

Fig.4.9 Theelementaryareasweptbytheradiusvectorindt

(4.4)Lisalwaysperpendiculartobothrandv,hencetotheplaneoftheorbitthat

isconstantforthefirstKeplerlaw.HencethedirectionofLisconstant.InadditionLisconstantalsoinmagnitudeforthesecondlaw.Indeed,

considertheareadAsweptbythepositionvectorinthetimedt,whichistheareaofthetriangleinFig.4.9.Twoofitssidesarevdtandr.Rememberingthegeometricmeaningofthevectorproductwehave

(4.5)or

(4.6)ThequantitydA/dtistheareasweptbythepositionvectorintheunitoftime

andiscalledarealvelocity.ItisconstantforthesecondKeplerlaw.Weimmediatelyrecognizethatthesecondmemberisproportionaltothemagnitudeoftheangularmomentum,namely

(4.7)Thearealvelocitybeingconstant,themagnitudeoftheangularmomentumis

constanttoo.Inconclusiontheangularmomentumvectoraboutthesunisconstant.Ontheotherhand,theplanetiscertainlysubjecttoaforce,becauseitaccelerates,butthisforcedoesnotvarytheangularmomentumaboutapointfixedinaninertialframe.Consequently,itsmomentaboutthatpolemustbezero,namelyitsdirectionmustbeparalleltothepositionvectorfromthesuntotheplanet.Itmustbetowardsthesunbecauseinacurvedmotiontheforceisalwaysdirectedonthesideofthecurvaturecenter.

Inconclusion,theforceoneveryplanetmustbedirectedtowardsthesun.Theconclusionsuggests,betterforces,ustothinkthesuntobethesourceoftheforcesactingonalltheplanets.

Wenowconsiderthemagnitudeoftheforce.Thesymmetryoftheproblemsuggestschoosingareferenceframewithorigininthesunandpolarco-ordinates

withanarbitrarypolaraxis.Letrbethemagnitudeandθtheazimuthofthepositionvectoroftheplanetr.Datashowthatthemotionoftheplanetsdoesnotslowdownthroughthecenturies,hencetheforceshouldbeconservative.Havingjustshownthatitisalsocentral,forthetheoremwedemonstratedinSect.2.15,itsmagnitudecannotdependonθ,butdependsonlyonthedistancefromthecenterofthesunr(Fig.4.10).

Fig.4.10 Thereferenceframetostudythemotionoftheplanet

Tomakethedemonstrationassimpleaspossibleweshallassumetheorbitstobecircumferencesratherthanellipses.InSects.4.11and4.13theproblemoftheellipsewillbetreatedexactly.

Ifthemotioniscircular,thearealawimpliesthattheangularvelocityωisconstant.Theforceshouldbethecentripetalforceofsuchamotion

(4.8)wheremisthemassoftheplanetandTisitsperiod.ThethirdKeplerlawstatesthat

(4.9)whereKSistheproportionalityconstant,thesameforalltheplanetsofthesolarsystem(butnotnecessarilyforothersystems)andthat,substitutedinEq.(4.8),gives

(4.10)Wehavefoundtwofundamentalpropertiesoftheforce:1.Itisinversely

proportionaltothedistancefromthesun,whichisitssource,2.isproportionaltothemassoftheplanet.Wenowshowthethirdproperty:theforceisproportionaltothemassofitssource.Tofindit,observationaldataonsystemssimilartothesolarone,butwithadifferentcentralbody,areneeded.Newton,hadalreadycomparedtheforceexertedbyearthonbodiesonitssurface,namelytheweight,andonthemoon,asweshallseeinSect.4.5.Hehadestablished

that,takingintoaccountthedifferenceinthedistancesfromthecenter,theforceisthesame.Thecharacteristicsofthegravitationalforceareuniversal.

Twosmall“solarsystems”wereknown,Jupiterwithitsfourprincipalsatellites(Io,Europa,Ganymede,Callisto),whichhadbeendiscoveredbyGalileoGalilei(1564–1642)(weshalltellofthediscoveryinSect.4.9),andSaturnwithitstwolargersatellites,whichhadbeenobservedbyChristiaanHuygensandbyGiovanniDomenicoCassini.Theseobservationshadestablishedthevalidityofthe3rdKeplerlawforthesystems(inbothcasesmore,smaller,satelliteswerediscoveredinrecenttimeswiththespacemissions).

Gravity,Newtonconcluded,isofalltheplanetsandsatellites,andcontinued:

Andsinceallattraction(byLawIII)ismutual,Jupiterwillthereforegravitatetowardsallhisownsatellites,Saturntowardshis,theearthtowardsthemoon,andthesuntowardstheprimaryplanets.

Tobeconcrete,consideroneoftheJupitersatellites,Callisto.JupiterisattractedbythesunwithaforceproportionaltoitsmassandattractsCallistowithaforceproportionaltothemassofCallisto.Forthe3rdNewtonlawCallistoattractsJupiterwithaforceequalandopposite.But,thislatterforcehasthesamecharacteristicsastheforcethatJupiterreceivesfromthesun,includingbeingproportionaltothemassofJupiter.WecanconcludethattheforcethatJupiterexertsonCallistoisproportionaltothemassofJupiter(beyondthatofCallisto).Thepropertyisgeneral,namelythegravitationalforcebetweenanytwo(point-like)objectsofmassesmandMisproportionaltotheproductofthemasses.Wewrite

(4.11)whereGNisauniversalconstant,theNewtonconstant,thatweshallsoondetermine.ThisequationgivesthemagnitudeofboththeforcesofmassMonmandofmonM.Theirdirectionsareequalandopposite.IfristhepositionvectorfromMtoofmandurisitsunitaryvector,theforceexertedbyMonmis

(4.12)ThisistheNewtonlawofuniversalgravitation.Wefirstobservethat,aswritten,thelawisvalidforpoint-likeobjects.Inthe

casesofthesolarsystemandinthesystemsofJupiterandSaturn,allthebodies,sunincluded,canbeconsideredaspointsbecausetheirdistancesarealwaysvery

muchlargerthantheirdiameters.However,alsotwoextendedobjects,forexampletwobricksoneclosetotheother,attractgravitationallyoneanother.Tofindtheforcewemustideallydivideeachbodyininfinitesimalparts.EverypairofinfinitesimalelementsattractseachotherwiththeforceofEq.(4.12)whereristhepositionvectorofoneelementrelativetotheotherandthemassesarethoseofthetwoelements.Thetotalforceisobtainedbytakingthevectorsum(integrating)ofallthepairs.Thereiscertainlyacaseinwhichsuchanintegrationisneeded,namelytheweight.Indeed,westatethattheweightofanobjectonthesurfaceofearthisthegravitationalforceoftheearthconsideredasapointinitscenter.Whyisthispossible?TheanswerisinSect.4.6.

AsecondobservationisonthemassesintheNewtonlawEq.(4.11).Theyareclearlygravitationalmasses.However,inourdemonstrationwehavestartedfromEq.(4.8)wherethemassistheinertialone.AswehaveseeninSect.2.9,theequalityofinertialandgravitationalmasseshadbeenestablishedbytheexperimentsofGalilei,whichNewtonhadrepeated.However,theexperimentshadbeendoneonterrestrialbodiesandthequestionarises:doesthesamerelationholdforcelestialbodies?NewtonshowedthistobetrueconsideringthesystemofJupiteranditsfourGalileiansatellites.Thesystemisasmallreplicaofthesolarsystem,butispartofthesolarsystemtoo.Observationshadshownthatthesatellitesperform“exceedinglyregularmotions”.TheradiusesoftheorbitsaboutJupiterandtheperiodshadbeenmeasured.Theperiodsturnedouttobeproportionaltothe3/2rdpoweroftheorbitsradiuses.Consequently,theforceexertedbyJupiterisinverselyproportionaltothedistance.SupposenowtheratiobetweengravitationalandinertialmassofJupiterandanyofitssatellites,Callistoforexample,tobedifferent,sayas

whereεisapositivesmallnumber.Then,Newtonargues,theforcesofthesunonJupiterandonCallisto,atequaldistancesfromthesun,willdifferby±εalso,andthiswouldhaveaneffectontheorbitofCallistoaboutJupiter.Thecalculationoftheeffectneedstosolveathree-bodyproblem,Jupiter,Callistoandthesun,whichcannotbedoneanalytically.ButNewtonwasabletofindthat,iftheforcesofthesunonJupiterandCallistowoulddifferinacertainproportion,thenthedistancesofthecenteroftheorbitofCallisto(callitrCS)aboutthesunandthecenterofJupiter(rJ)fromthesunwoulddiffer“nearly”asthesquarerootofthesameproportion“asbysomecomputationsIhavefound”,namely,

Hewrites

Thereforeif,atequaldistancesfromthesun,theaccelerativegravity(hemeansthegravitationalforce)ofanysatellitetowardsthesunweregreaterorlessthantheaccelerativegravityJupitertowardsthesunbutbyone1/1000partofthewholegravity,thedistanceofthecentreofthesatellite’sorbitfromthesunwouldbegreaterorlessthanthedistanceofJupiterfromthesunbyone1/2000partofthewholedistance;thatisthefifthpartoftheutmostsatellite(Callisto)fromthecentreofJupiter;aneccentricityoftheorbitwhichwouldbeverysensible.ButtheorbitsofthesatellitesareconcentrictoJupiter,andthereforetheaccelerativegravitiesofJupiter,andofallitssatellitestowardsthesun,areequalamongthemselves.

Newtonaddsthatiftheratiosofgravitationaltoinertialmassoftheearth,,andofthemoon, ,wouldbedifferent,theabove-described

effectshouldbepresentandadeformationofthemoonorbitshouldbeobservable.Today,themoon-earthdistanceismeasuredwithextremeprecisionwithLASERrangingtechniques.In1969theApollo11astronautsandlaterotherlunarmissionsdeployedonthesurfaceofthemoonsystemsofmirrorsabletoreflectbackaLASERpulsesentfromearth.Themeasurementoftheround-triptimeofthepulsegivesthemoondistancewithafewmillimeterprecisionasafunctionoftime.Theextremelysensitivetechniquedidnotdetectanyeffect,providingtheverylowupperlimit

WenowcomebacktotheuniversalityoftheNewtonlaw.Ifitisso,theconstantGNmustbethesameinanycircumstanceandisoneofthefundamentalconstantsofphysics,calledthegravitationalNewtonconstant.Atlaboratoryscale,betweeneverydaylifesizeobjects,theNewtonlawisverysmallanddifficulttomeasure.ThiswasfirstdonebyHenryCavendish(1731–1810)(seeSect.4.7)leadinghimtoalaboratorymeasurementofGN(whichisalsocalledaCavendishconstant).

TheuniversalityoftheNewtonlawneedstobeverifiedexperimentally.Thishasbeendoneatallthelengthscalesinmanydifferentconditions,findingitvalid.Weshalldiscussafewexamplesfurtherinthechapter.However,alimitofvalidityexists,asweshallsee.

Equation(4.12)ismathematicallyverysimpleandsymmetricinitselements.

Itinterpretsahugeamountofphenomena,fromthemotionofplanetstothefreefallofobjectsonearth,fromthemotionofthesatellites,tothatofthestarsandthegalaxies.TheexpressionshowsushowNaturecanbedescribedinitsmostfundamentalaspectsinsimpleandelegantmathematicalform.TheharmonyoftheworldthatuptotheMiddleAge,andtoCopernicus,wasbelievedtobesubstantiatedintheexistenceofamechanismofsolidspheres,symmetricobjects,thatrotateuniformly(simplemotion),comesback,inanabstractform,intheharmony,sotospeak,ofthephysicallaw.

WefinallycomebackontheconstantKSinEq.(4.9).FromEq.(4.12)wecanwrite,forthesolarsystem

(4.13)Weseethattheconstantdependsonthemassofthesun,namelythemassof

thecentralbody.Itisnotuniversal.ForexamplefortheJupitersystemitisthemassofJupiter,fortheearth-moonsystemsitisthemassoftheearth,etc.

4.5 TheMoonandtheAppleIfEq.(4.12)isuniversal,theforcethatearthexertsonthemoon,thecentripetalforcecorrespondingtohermotion,mustbethesameastheforcesheexertsonabodyonhersurface,forexampleanapple,whichisitsweight.InparticulartheconstantGNshouldbethesame.AsNewtonhimselfrecalls,in1665hestartedtoaskhimselfthisquestion.Hedevelopedthefollowingargument.Indeed,inhercircularmotionthemooncontinuouslyfallsacceleratingtowardsearth.Thisissimplyanotherwaytolookatcentripetalacceleration.

SupposethatthemoonisatthepointAofherorbit,asinFig.4.11,atacertaininstant.InthefigurewehavetakenareferenceframewiththeoriginOinthecenterofearthandy-axisdirectedtowardsthemoonintheconsideredinstant.Afteracertainshorttime,sayafteronesecond,ifnoforcewerepresent,themoonwouldhavemovedtopointB.Ontheotherhand,ifthemoonwouldbeabandonedstillinB,shewouldfallinasecond,undertheactionofgravity,fromBtoP.PointPisatthesamedistancerfromthecenterofearthasA.Letuscalculatethedroph,takingintoaccountthattheangleθisverysmall.ThePythagoreantheoremforthetriangleONPgives

Fig.4.11 Howthemoonfalls

Ifθisinfinitesimal,h2isaninfinitesimalofsecondorderandcanbeneglected.Wecanalsoconsiderxequaltosandwrite

(4.14)Toevaluatethedisplacementofthemooninonesecondwecanusethe

proportions:2πr=1:T,whereTistheperiodofthemoonrevolution,T=27.3d=2.4×106sandr=3.8×108m.Wehaves=2πr/T≈1000mand

Inasecondthemoonfallsalittlemorethanamillimeter.Wenowcomparethiswiththedroplengthofanobjectonearth,thefamousappleforexample,whichis

(4.15)Theratioofthetwodropsinonesecondisequaltotheratiooftheir

accelerations.Thelatter,iftheNewtonlawisvalid,shouldbeintheinverseratioofthesquaresoftheirdistances.Theratioofthedropsis

.Newtonknewthattheratioofthedistanceofthe

moonisabout60timestheradiusoftheearthandwhatwehavejustfoundisabout602.

However,Newtonhadstilltheproblemthatwealreadymentioned.Whilemoonandearthcanbeconsideredaspoints,consideringtheirlargedistance,forwhatreasonweshouldconsidertheapple,onavisuallyflatground,shouldbeattractedtowardsapoint6380kmunderthegroundasifallthemassofearthwouldbeconcentratedthere?

Thisisa“miracle”trueonlyforforcesinverselyproportionaltothedistancesquare.Inthenextsectionweshallprovethefollowingtheorem:theforceexertedbyahomogeneoussphericalmassinanypointoutsideitssurfaceisequaltotheforcethatwouldbeexertedifallthemasswereinapointatitscenter.

Newtondidnotpublishanyresultuntilhehadmadeeverythingclear,completeandperfect,inthePrincipiapublishedin1687.

4.6 TheGravitationalForceoftheHomogeneousSphereWeshallcalculatetheforceofasphereofmassMonapoint-likeparticleofmassmoutsidethesphereatadistancerfromthecenter.Weassumethatthedensityofthesphere,ifvariable,dependsonlyonthedistancefromthecenter(sphericalsymmetry).WeshallprovethattheforceisequaltothatwhichwouldexertallthemassMconcentratedinthecenter.

Westartbyobservingthatisenoughtoprovethethesisforasphericalshellofinfinitesimalthickness.Indeedifitistrueforoneshellitisalsotrueforthesphere,whichcanbeconsideredascomposedofshellswiththesamecenter.

Considerthesphericalshell,ofradiusRandcenterO,showninFig.4.12,havingradiusRandonittheringAA′limitedbyconeswiththeirvertexinOandsemi-vertexanglesθandθ+dθ.Letϕbethesemi-vertexangleoftheconewithvertexinPandtheringAA′asbase.

Fig.4.12 Elementsforcalculationoftheforceofasphericalshellonanexternalpoint

AlltheelementsoftheringAA′areatthesamedistancefromPandconsequentlytheyexertonPforces,callthemd2F,equalinmagnitude,butnotindirection.Thesymmetryoftheproblemtellsusthattheresultantoftheseforces,dF,isdirectedasOP.Thecontributionsnormaltoitcanceleachother.ThecomponentinthedirectionOPoftheforceisproportionaltothemassoftheelement,tocosϕandinverselytothesquareofthedistances2.Theresultantof

theforcesonminPduetotheringbeing

wheredMisthemassofthering.Nowthemassoftheringistothemassoftheshellastheareaoftheringistotheareaoftheshell:

whichgivesusdM=(M/2)sinθdθ.Theforceoftheringonthemassmisthen

(4.16)Theforceoftheshellistheintegralofthisexpressionforθvaryingfrom0to

π,namely

(4.17)

Bothsandϕarefunctionsoftheintegrationvariableθ.Itisconvenienthowevertoexpresseverythingasfunctionsofs.TheCarnottheoremappliedtothetriangleOAPgives

(4.18)Wedifferentiatethefirstequation,rememberingthatrandRareconstant,

obtaining

WesubstitutethisexpressionandthesecondEq.(4.18)intheintegralofEq.(4.17)andtakeintoaccountthatnowthevariableissandthelimitsmustbechangedinaccord,obtaining

Theintegraldoesnotpresentdifficulties.Theindefiniteintegralgives

which,evaluatedinitslimits,gives4R.InconclusiontheforceoftheshellonapointPofmassmis

(4.19)whichis,inparticular,independentoftheradiusRoftheshell.Thisprovesthetheorem.

ConsidernowapointPofmassminsidetheshell.WhatistheforceonPexertedbytheshell?Thereasoningremainsexactlythesame,butforthelimitsontheintegrationons.Nowtheangleθvariesbetween0and2πandcorrespondentlysbetweenR+randR−r.Thedefiniteintegraliszero.Thegravitationalforceexertedbyasphericalshellonapointinsideitiszero.Thisisanotherpropertyoftheinversesquarelawforces.

Newtongaveanotherproofofthelastpropertyusingasimplegeometricargument.ConsiderpointPinsidetheshellasshowninFig.4.13andtheconewithvertexinPofverysmallvertexangle.Thetwonapesinterceptontheshell’stwosurfaces∆S1and∆S2.Asthedensityisconstant,themassesofthetwosurfacesareproportionaltotheirareas.ThelatterareproportionaltothesquaresoftheirdistancesfromP,say and .ButtheforcestheyexertinPareproportionaldirectlytothemassesandinverselytothesquaredistances.Thetwoforcesareequalinmagnitude.Astheirdirectionsareopposite,theirresultantisnull.Astheshellcanbedividedinpairsgivingnullcontribution,theresultantiszero.

Fig.4.13 Thegeometrytocalculatethegravitationalforceofasphericalshellonaninternalpoint

4.7 MeasuringtheNewtonConstantTheNewtongravitationalforceEq.(4.12)exertsbetweeneverypairofpoint-like,orspherical,masses.Itisimportanttocontrolexperimentallyitsvaliditynotonlyattheastronomicalscales,butalsoatthelaboratoryscale.Thelaboratoryexperimentsaredifficultbecausetheforceis,atthesescales,verysmall.Anydisturbancesuchassmallaircurrents,spuriouselectricforces,the

movementoftheexperimenteritself,isapossiblecauseoferrorsandmustbeeliminated.

However,ifwewanttoknowtheNewtonconstant,wemustmeasuretheforcebetweentwoknownmassesataknowndistance.Inthecaseoftheheavenlybodiesinfactwedonotknowapriorythemasses,butweinferthemfromtheNewtonlaw.

ThegravitationalforcewasfirstmeasuredbyHenriCavendish(1731–1810)in1798.HisexperimentisshownschematicallyinFig.4.14.Arigidmetalbarsuspendedonaverythinmetalwire,carriestoequalleadspheresatequaldistancefromthewire.Thesystemisinequilibriumandfreetorotateaboutthewire.ThistypeofarrangementiscalledtorsionbalanceandwillbefurtherdiscussedinSect.8.9.

Fig.4.14 TheCavendishexperiment

Twomorelargerandheavierequalspheres,ofmassM,arearrangedsymmetrically,eachatthesamedistancefromoneofthesmallones.Consequentlyeachofthelargespheresattractsthesmallonenearbywithan(equal)gravitationalforce.Thearmofthecoupleisthedistancebetweenthecentersofthesmallspheresandcanbeaccuratelymeasured.Themomentofthecoupleinducesarotationtothebar.Thewirereactswithanelastictorsionmoment,whichisproportionaltoitsrotationangle.Theequilibriumisatanangleatwhichthetorsionmomentandthemomentofthegravitationalcoupleareequal.Hence,themeasurementofthisanglegivesthemomentofthecoupleand,thearmbeingknown,theforces.

Therotationangleismeasuredwiththetechniqueoftheopticallever.Anarrowlightbeamissenttoaverylightmirror,fixedtothewire.Themirrorreflectsthebeamonascalelocatedatacertaindistance.Thedeviceisverysensitive.Evenaverysmallchangeintheorientationofthemirrorcausesasizeablemovementofthelightspotonthescale.Indeed,themomentsareverysmall.Thewiremusthaveaverysmallelasticconstantandconsequentlybe

verythin,butstillcapableofholdingtheweightofthesmallspheresandbar.Alltheapparatusmustbeclosedinacontainertoavoidaircurrents.Thepresenceofelectrostaticchargesmustbeavoided,etc.

ThevalueofthegravitationalconstantobtainedbyCavendishwas

(4.20)Thepresentvalueis

(4.21)Tohaveaquantitativeidea,considerthatthelargespheresofCavendishhad

amassM=158kg,thesmallonesm=0.73kgandthatthedistancebetweenonesmallandonelargewasr=0.225m.Thetwoforcestobemeasuredareabout10–7N.Thisisabouttheweightofahair.

4.8 TheGravitationalFieldWeinterruptinthissectionourdiscussionofexperimentalproofsoftheNewtongravitationallaw,todiscussanimportantpropertyofthegravitational,andoftheotherfundamentalforces.Namelytheyareactionatadistance.Otherexamplesaretheelectricforce,whichoperatesbetweenelectricallychargedbodies,andthemagneticforce,forexamplebetweenamagnetandapieceofiron.

Inallthesecasesanextremelyusefulconceptisthefieldofforceorsimplyfield(gravitational,electric,magneticfield).

Considerthegravitationalforceexertedbytheearth,ondifferentobjects.Itdependsonthemassoftheobject(isproportionaltoit)andonthepositionoftheobject.Ifweconsidertwoparticlesofdifferentmassesinthesamepositionanddividetheforcesactingoneachofthembyitsmass,wefindthesameresult.Thisvectorfunctionofthepositionisthegravitationalfield.

Thegravitationalfieldgeneratedbyadistributionofmassesisavectorfunctionoftheposition.Itisequaltotheforceactingontheunitmassinthatposition.Themassesgivingorigintothefieldarecalledthesourcesofthefield.

Inparticulartodescribethefieldoftheearthwecantakeareferenceframewithorigininthecenteroftheearth.ConsiderapointPatthepositionvectorrwiththeunitvectorur.IfweputamassminP,itfeelstheforce

(4.22)whereMisthemassoftheearth.Thegravitationalfieldisthevectorfunctionoftheposition

(4.23)Thisexpressionisvalidforpointsoutsidetheearthintheapproximationof

earthbeingsphericalandwithasphericallysymmetricaldistributionofmasses.Thephysicaldimensionsofthegravitationalfieldareaforcedividedbyamass,hencethedimensionsoftheacceleration.Asamatteroffact,itisjustthegravityaccelerationg.

Theconceptoffieldeliminatesfromourreasoningtheideaofactionatadistance.Wecanthinkasfollows.Theearth,oranydistributionofmasses,createsinallthespacearounditaphysicalentity,thegravitationalfield,whichextends,evenifwithdecreasingintensity,toinfinity.Thefieldexistsindependentlyofbeingperceivedasaforce.Butifweplaceinapointofthefieldatestbodyofmassm,itwillfeellikeaforceequaltotheproductofmtimesthegravitationalfieldinthatpoint.Bymeansofthefieldthegravitationalactionbecomeslocal.

Wecannowconsiderthepotentialenergyofourtestmassinthefieldoftheearth.Definingthepotentialenergytobezeroatinfinitedistance,wehave

(4.24)Thephysicalmeaningis:thepotentialenergyofthemassminthepointPis

theworktobedoneagainsttheforcesofthefieldtomovethemassmfrominfinitytoP.

Obviouslythepotentialenergy,astheforce,isproportionaltom.Ifwedivideitbymwefindafunctionoftheposition,independentofthebody

(4.25)Thisfunctionisthegravitationalpotential.Therelationshipbetween

potentialandfieldisthesameasbetweenpotentialenergyandforce.Thegravitationalpotentialinapointistheworktobedoneagainsttheforcesofthefieldtocarryfrominfinitytothatpointaunitarymass.Thephysicaldimensionsofthegravitationalpotentialareavelocitysquared.Itismeasuredinm2/s2.

Considernowourmassmmovingonacircularorbitofradiusrwithvelocityυ.Itmightbeforexampleourmoon.Thereisasimplerelationbetweenkineticandpotentialenergy.Recallingthatυ=2πr/T,whereTistheperiod,thekineticenergyis

andforthe3rdKeplerlawEq.(4.13)

(4.26)Thisresult,validforcircularorbits,isthatthekineticenergyisonehalfof

thepotentialenergyinabsolutevalue.Consequently,themagnitudeofthegravitationalpotentialinthepointsoftheorbitisequaltothesquareofthevelocityofthebodyonthatorbit

(4.27)ThisexpressionwillbeusefulinSect.7.13.Toappreciatetheordersofmagnitude,considerthemotionoftheearth

aroundthesun.Thevelocityis ForEq.(4.27),thepotentialofthefieldofthesuninthepointsoftheearthorbitisϕ≈109m2s–2.

WehavealreadymentionedthatlimitsofthevalidityoftheNewtonlawexist,whenitmustgiveplacetogeneralrelativity.Moreprecisely,theeffectsthatareincontradictionwiththeNewtonlaw,andthatareexplainedbygeneralrelativity,areoftheorderofthegravitationalpotentialcomparedtothesquareofthespeedoflight,namelyϕ/c2.Consideringthatc2≈9×1016m2s–2,theseeffectsareusuallyverysmall(oftheorderof10–8ontheearthorbit),butcanbedetectedwithhighprecisionobservations,asinthecaseoftheanomalousprecessionoftheMercuryperihelion(seeSect.4.11).Theeffectsbecomelargeatveryhighgravitationalpotentials,nearmassiveandcompactobjects,likeblackholes.

Thegraphicrepresentationofthegravitationalfieldisveryusefultohaveavisualideaofitsmainfeatures.Itisdonewiththelinesofforceandwiththeequipotentialsurfaces.

AlineofforceisdrawnasshowninFig.4.15.Westartfromapoint,1inthefigure,whereweevaluatethevectorofthefield.Thenwemakeasmallstepδsinthedirectionofthefield,reachingpoint2.Wecalculatethefieldinthispointandproceedanotherstepasabove,etc.Inthiswayweobtainabrokenline.Itbecomesafieldlineforδstendingtozero.Itisacontinuouscurve,inallthepointsofwhichthefieldistangent.Obviouslythefieldlinesareinfiniteinnumber.However,thereisonlyonelinethroughanygivenpoint.Iftheywere,say,two,thefieldshouldhavehadtwodirectionscontemporarily.Graphically,wedrawanumberoflines,whichisenoughtoseethefeaturesofthefield.

Fig.4.15 Constructionofalineofforce

Theequipotentialsurfacesarethelociofthepointsthatsatisfytheequationsϕ(x,y,z)=constant,oneforeachvalueoftheconstant.Theseareinfiniteinnumbertoo.Itisconvenienttodrawasetofsurfacesatconstantstepsofthepotential.Ananalogyarethegeographicmapsinwhichthelevelcurvesaredrawnevery,say,onehundredmetersofelevation.Intheregionswherethelevelcurvesaredenser,theelevationvariesmorerapidlyandtheslopeofthesurfaceissteeper.Thesituationisanalogousforequipotentialsurfaces.

Figure4.16ashowssomelinesofforceandequipotentialsurfacesforasphericalmassM.Thelinesofforceareradialandpointtothemass,becausetheforceisattractive.Theequipotentialsaresphericalandbecomedensergettingclosertothemass,whichisthesourceofthefield.

Fig.4.16 EquipotentialsandfieldlinesforaasphericalmassM,btwomassesonetwicetheother

Figure4.16brepresentsthefieldoriginatedbytwosphericalmasses,onedoublethemassoftheother.Ineverypointthefieldisthevectorsumofthefieldsofthetwomassestakenseparately,thepotentialissimplythesumofthepotentials.Noticethe“saddle”pointonthelinejoiningthetwocenters.Herethereisaminimummovinginthatdirection,amaximummovingperpendicularlytoit.

Oneseesthatthelinesofforcearealwaysperpendiculartotheequipotentials.Thisisageneralproperty.Indeed,supposewearemovingwiththeinfinitesimaldisplacementds.Thepotentialdifferencebetweenthetwopointsis Ifthedisplacementisontheequipotential,dϕ=0bydefinition,henceGmustbeperpendiculartods.Thelinesofforcethathavethe

directionofGareperpendiculartotheequipotential.IfwecallGstheprojectionofGonthedirectionofthedisplacementwecan

write

(4.28)whichcanbealsowrittenas

(4.29)Wereadthisexpressionas:thecomponentofthefieldinagivendirectionis

thedirectionalderivativeofthepotentialinthatdirection.DirectionalderivativeisjustthenameofthederivativeinEq.(4.29),itistherateofchangeofthefunctioninthatdirection.Aswehavejustseenthedirectionalderivativeisnullfordirectionsontheequipotentials.

ConsiderinfinitesimaldisplacementsasthoseinFig.4.17,whichareindifferentdirectionsbutallleadingfromtheequipotentialϕtoϕ+dϕ.Thedirectionalderivativeisdifferentforeachofthembecausedϕisthesameanddsisdifferent.Thederivativeisamaximumwhenthedirectionisnormaltothesurfacesbecausedsisthereaminimum.Thevectorhavingthemagnitudeofthemaximumdirectionalderivativeandthedirectionofthenormaltotheequipotentialtowardsincreasingpotentialiscalledthegradientofthepotential.Itssymbolisgradϕ.Inconclusionwehave

Fig.4.17 Differentstepsbetweenthesameequipotentials

(4.30)Ifwethinkofthelevelcurvesofageographicmap,thegradientisdirected

asthelineofmaximalslopeoftheground;itsmagnitudeisgreaterthegreateristheslope.

Onearth,theequipotentialsurfacesarematerializedbythesurfacesofthelakesandoftheseas(neglectingthewaves).

Wenowseehowtocalculatethegradientstartingfromthepotential.WestartfromEq.(4.28)andusethetotaldifferentialtheorem

(4.31)wheredx,dyanddzaretheCartesiancomponentsofδs.ItimmediatelyfollowsthattheCartesiancomponentsofthegradientarethepartialderivativesofthepotential

(4.32)Obviously,similarrelationsexistbetweengravitationalpotentialand

gravitationalforceofamassm.Itisjustamatterofmultiplyingbym,

(4.33)and

(4.34)

4.9 GalileiandtheJovianSystemG.Galilei(1564–1642)wasthefirsthumantoexplorescientificallytheskyusingthetelescope,whichhehaddeveloped.Asamatteroffact,combinationsoftwolensesputoneaftertheotheratacertaindistancehadexistedforatleast30years.Thefirstwrittenmentionisin1589,byGiovanniBattistaDellaPorta(1535–1615).AtthebeginningoftheXVIIcenturytelescopeswerebuiltintheNetherlandsbyeyeglassesmanufacturers.Theyweretoyssoldinexhibitionsatlowprices.GalileinewoftheDutchtelescopein1609.Hequicklyenvisionedawaytotransformthedeviceintoascientificinstrumentandimmediatelystartedhisexperimentalwork,withoutasolidtheoreticalbasis.Whatisknowntodayasgeometricalopticswasdevelopedonlyin1611byJohannesKepler(motivatedbythedesiretoexplainhowthetelescopeworks).LenseshadalreadybeenproducedsincetheXIIIcentury,buttheirqualitywasnotadequateforascientificinstrument.

Animportantpropertyofthetelescopeisangularmagnification,whichistheratiobetweentheangleunderwhichanobjectisseenthroughthetelescopeandtheangleunderwhichitisseenwiththenakedeye.Thesecondpropertyistheresolvingpower,namelytheabilityofthetelescopetoresolve,toseeseparated,twopointimagesverycloseonetotheother.Toincreasebothpropertiesthediameteroftheobjectivelens(theonefartherfromtheeye)mustbeincreased.However,thelargerthelens,themoredifficultisitsproductionwithoutanydefect.Withaseriesofimprovements,andthehelpoftheVenetianglassmakers,Galileidevelopedthetechniquetothepointthathecouldbuilda

telescopewithmagnification10and,sometimelater,onewithmagnification30,withlensesofperfectopticalquality.Withthismagnificationthelightreachingtheeyein302=900timesaswiththenakedeye.

Galileipublishedhisfirstobservationinthebooklet“Sidereusnuncius”(astronomicalnotice)in1610.Inaddition,thelogbooksofhisobservationshavecomedowntous.OneofhisgreatdiscoverieswasthataroundJupiter’sfoursatellitesorbit,makingasmallreplicaofthesolarsystem.AviewofthesystemwithamoderntelescopeisshowninFig.4.18.LetusseehowhedescribeshisdiscoveryintheSidereusNuncius.

Fig.4.18 Jupiterandhissatellites.Image©NASA

Onthenightofthe7thofJanuary1610,lookingtoJupiter,Galileiobservedthreesmall“starlets”.TheyattractedhisattentionbecausetheywereperfectlyalignedbetweenthemandwithJupiterandontheecliptic.HedidnotcorrelatethestarletswithJupiter,thinkingtheywerefixedstarsinthebackground.Hetooknoteoftheirpositionsinthelogbook,aswetrytoreproduceinFig.4.19a.

Fig.4.19 SketchesoftheGalileiobservationsinJanuary1610inthenightsofa7th,b8th,c10th,d13th

Thefollowingnightherepeatedtheobservationsandnoticedthattherelativepositionshadchanged,asinFig.4.19b.HethoughtthechangetobeduetothemovementofJupiterrelativetothestars,thathebelievedtobefixed,withsomedoubts,becausethemotiondidnotmatchthecalculations.Heanxiouslywaited,ashewrites,thefollowingnight,buthishopewasfrustrated,becausealltheskywascloudy.Thenightofthe10ththestarswereonlytwoandhadagainchanged

position,butstillonaline,asinFig.4.19c.Thethirdone,hethought,shouldbehiddenbyJupiter.Galileihadnomoredoubts.Hewrites(translatedbytheauthor):

myperplexitychangedtoastonishmentandIbecamesurethattheapparentmovementwasnotofJupiter,butinthestarsIobserved;henceIdecidedtocontinuemyinvestigationwithincreasedattentionandscrupulosity.

The13thhesawforthefirsttimethefourthsatellite,whichhadenteredthefieldofviewofthetelescope,asinFig.4.19d.

Afterseveralmorenightsofobservationshepublishedthediscovery,togetherwithotherimportantonesonthemoonandtheMilkywayintheabovequotedbookinMarch1610.

Thenexttaskwasthemeasurementoftheperiods.Themeasurementwasextremelydifficult,asmuchthatKeplerhaddeclareditimpossible,becausetheimagesofthefourstarletswereindistinguishable.GalileiunderstoodthattheprecisiononhismeasurementsoftheangulardistancesfromthecenterofJupiterhadtobeimproved.Hehadmeasuredthem“byeye”withaprecisionofbetterthanonearcminute(1/60°).Itwasnotenough.Hedevelopedthemicrometer,withwhichhewastomeasurethepositionswithaprecision“betterthanveryfewarcseconds”(onearcsec=1/3600°).

Galileicontinuedhissystematicmeasurementsforseveralyears,butalreadyin1611hehadbeenabletoidentifyeachofthesatellitesandtocalculatetheirperiodandtheapparentdiametersoftheorbits.InFig.4.20wereportasubsetofhismeasurementsmadeinspring1611,astakenfromhishandnotes.Forsimplicitytheyareforthetwomoreexternalones,CallistoandGanymede.Theplanesoftheorbitsarealmostonthelineofviewfromearth.Consequently,iftheorbitisanellipse(or,inparticularacircle)themotionappearsassinusoidalfunctionsoftime.Withacomputeritistodayeasytofindthesinusoidthatbestinterpolatesthedata,theonesshowninthefigure.Clearly,thedataareinagreementwiththehypothesis.Theprocedurealsogivesusavaluefortheamplitudeandtheperiod.Galileihadnocomputerandmadehiscalculationsbyhand.

Fig.4.20 ThedistancesfromJupiterofhistwofarthersatellitesasmeasuredbyGalileiinspring1611.Thesinusoidsarefrommycalculations

Table4.3reportstheperiodsasmeasuredbyGalileiandhowtheyareknowntoday.Oneseesthathismeasurementswerequitegood.

Table4.3 PeriodsoftheJupitersatellites(indays)

Io Europa Ganymede CallistoGalilei 1.76 3.53 7.16 16.3modern 1.77 3.55 7.17 16.75

Accuratemeasurementsoftheapparentamplitudesaremoredifficult.NoticethatthesequantitiesaremeasuredrelativetotheapparentdiameterofJupiter,namelytheyare,say,n=r/rJ.Table4.4reportsthevaluesofnasmeasuredbyGalileiinsubsequentyears,showinghowtheprecisionisincreasing,approachingthepresentlyknownvalues.

Table4.4 AngularradiioftheorbitsrelativetotheradiusofJupiter

Io Europa Ganymede Callisto1610? 3.5 5.7 8.8 15.31611 3.8 6.2 8.4 151611? 4 7 10 151612 5.7 8.6 14 “almost25”modern 5.58 8.88 14.16 24.90

TheJoviansystemisasmallsolarsystem.IsthethirdKeplerlawverified?Galileididnotcheckthat,butNewtondid.Fromthedatainthetwotables,wecandoitourselvesobtainingthefollowingtable(Table4.5).

Table4.5 The3rdKeplerlawintheJoviansystem

Galilei ModernT(d) n=r/rJ n3/T2 T(d) n=r/rJ n3/T2

Io 1.76 5.7 59.8 1.77 5.91 65.8Europa 3.55 8.6 50.5 3.55 9.40 65.9Ganymede 7.16 14.0 53.5 7.16 14.97 65.8Callisto 16.3 24.9 58.1 16.69 26.33 65.5

The3rdKeplerlawissatisfied,betterobviouslybythemoderndata,forwhichtheexperimentaluncertaintiesaresmaller.

Wecanfinallychecktheuniversality,namelyifthegravitationalconstanthasthesamevalueintheJovianandinthesolarsystems.WecheckifEq.(4.13),namely, isvalidwiththesameGN,wherenowMisthemassofJupiter,randTareorbitradiumandperiodofanyofthesatellites.Forthatweneedabsolutevalues.WenowknowthedistanceofJupiterandthentheradiir.TheJupitermasshasbeenevaluatedfromhisperturbingeffectsontheotherplanets.Withthesevalueswefindthat,indeed,thegravitationalconstantisthesame.

4.10 Galaxies,ClustersandSomethingElseInthissectionweshallgivetwoexamplesofstructuresoflargerscalesthanthesolarsystems.TheNewtonlawisvalidalsoatthelargestscales.However,weshallalsoseethatthesamelawgivesusevidencethatthemassoftheUniverseismadeforitslargestfractionofcomponentsthatarenotvisible,becausetheydonotemitorabsorblight.Thisistheso-calleddarkmatter,whosenaturewedonotknow.

AfirstexampleisshowninFig.4.21.Itisaglobularcluster,asystemcontainingmillionsofstars,whichareveryold,havinganagecomparablewiththeUniverseitself.Theeffectofthegravitationalforcekeepingthosestarstogetherisspectacular.

Fig.4.21 TheglobalclusterNGC2808.Image©ESA

Figure4.22.showstheimageofaspiralgalaxy,asystemofhundredsandmillionsofstarskepttogetherbythegravitationalattraction.Allthisenormoussystemisrotating,asevidentbytheimage.Theangularmomentumofthehugegascloudfromwhichthegalaxyoriginatedbillionsofyearsagoremainedconstant.

Fig.4.22 ThegalaxyM74fromtheHubbleSpaceTelescope.Image©NASA

Letusmorecloselytotherotation.Letusstartbyconsideringhowtheorbitalvelocityυ(r)ofabodyofmassmorbitingaroundacentralbodyofmassM(likeaplanetaroundthesun)varieswiththedistancefromthecenterr.Assumeforsimplicityacircularorbit.Westatethatthecentripetalforcemustbeequaltothegravitationalattraction

(4.35)or

(4.36)Thevelocityisinverselyproportionaltothesquarerootofthedistancefrom

thecenter.Thevalidityofthelawcanbetestedontheplanetsofthesolarsystem.

Fiveplanetsarevisiblewiththenakedeyeandhavebeenknownsinceancienttimes.Inorderofdistancefromthesun,includingearth,theyare:Mercury,Venus,Earth,Mars,JupiterandSaturn.In1781,WilliamHerschel(1738–1822)discovereda“star”,theimageofwhichinthetelescopehadanon-zerodiameter.Itwastheseventhplanet,Uranus.TheobjecthadbeenalreadyobservedbyGalileiandbymoreastronomersinthefollowingyears.Theyhadnotrecognizeditasaplanet,duetothelimitationsoftheirtelescopes,buthadmeasureditscoordinates.Onthebasisofthesemeasurements,HerschelcouldreconstructtheparametersoftheorbitofUranus.ThemotionofUranusshowedsomeanomalies,whencomparedtotheNewtonlawpredictions.Thesewereinterpretedin1846,independentlybyUrbainLeVerier(1811–1877)andbyJohanCoucheAdams(1819–1891),aspossiblyduetoaneighthplanet.Whenhiscalculationswerecomplete,LeVerriersentaletter,withthecalculatedcoordinates,totheastronomerJohanneGrottfriedGalle(1812–1910)inBerlin,askinghimtoverify.Thefollowingnight,GallefoundNeptunewithin1°ofthepredictedposition.Similarly,in1930Plutowasdiscovered,havingitsexistencepredictedfromtheanomaliesoftheNeptunemotion.

Figure4.23showstheorbitalvelocityoftheplanetsasafunctionoftheirdistancefromthesun.Equation(4.36)isfullysatisfied.

Fig.4.23 Inversesquarerootdependenceoforbitalvelocitiesoftheplanets

Considernowthegalaxy,atypicalone,shownschematicallyinFig.4.24.Theimageshowsthatitsluminositydecreasesforincreasingdistancerfromthecenter,tillitdisappears.Thismeansthatthestardensitydecreasesdepartingfromthecenter.WeindicatewithM(r)thetotalmasscontainedinasphereofradiusr.Wewouldguessithavingthesamebehaviorastheluminosity.Butitisnotso.Letυ(r)bethe(average)velocityofthepointsofthegalaxyatthedistancerfromtherotationaxis.Wecanconsiderwithareasonableapproximationthemassdistributionassphericallysymmetrical.Then,thegravitationalforceactingonabody,astaroragasparticle,atthedistanceristhesameastheforceofallthemassinsider,concentratedinthecenter,exactlyasfortheweightofanapple.Differentlyfromtheapple,thereisnowalotofmassoutsider,but,aswehaveproveninSect.4.6,itsgravitationalforceinsideasphericalshelliszero.

Fig.4.24 Asphericalmassdistribution.M(r)isthemassinasphereofradiusr

Theforceatthedistanceris

(4.37)andtherotationvelocityatthedistanceris

(4.38)Theimageofthegalaxyshowsthattheluminosityendsatacertaindistance.

Thevisiblepartofthegalaxyhasaradiusthatwecallrvis.Typicalvaluesvaryfrom10kpcto100kpc(1pc,parsec,1is3×1016m=3.3lightyears)fromthecenter.WethenexpectthefunctionM(r)toincreasewithrandtobecomeconstantataboutrvis,becausethereisnomoremassafterthat,asrepresentedinFig.4.24.Consequently,thefunctionυ(r)forvaluesofrlargerthantheradiusofthegalaxyrvisshoulddecreaseas1/√r.

Howcanwemeasuretherotationvelocitiesofthegalaxiesatdifferentdistancesfromtheaxis?Themotionofthesinglestarsisnotobservablefromearth.However,eachoftheelementsinnatureemitslighthavingawell-definedspectrum,whichischaracteristicoftheelement.Ifthesourceismoving,thespectrumisshiftedinaknownwaydependentontherelativevelocitybetweensourceandobserver(itiscalledtheDopplereffect).

Consequently,wemeasurethevelocitiesofthedifferentelementsofagalaxybymeasuringthespectraofthelighttheyemit.Inpracticethelightemittedbythehugecloudsofgases,suchashydrogenandheliumthatextendfartherthan

thestarsfromtheaxis,butdonotcontributesubstantiallytothemass.Figure4.25showsthevelocitiesrelativetousofthegalaxyNGC2998as

functionsoftheapparentdistancefromitscenter.Wecandeducethatthegalaxyhasanaveragevelocity(thevelocityofitscenter)ofabout4700km/s.However,ontheleftthevelocitiesaresystematicallysmaller,higherontheright.Thisisbecauseweareobservingtherotationofthegalacticdiskatanangledifferentfrom90°.Consequentlythediskisapproachingononeside,withdrawingontheother.Tohavetherotationcurveofthegalaxy,namelytheorbitalvelocitiesatdifferentdistancesfromitscenter,wesubtracttheaveragevelocity.Thedistanceofthegalaxybeingknown,wecanconverttheapparentdistancesfromaxesinabsolutedistances.WeobtainthediagraminFig.4.26.

Fig.4.25 RotationcurveforthegalaxyNGC2998

Fig.4.26 TherotationcurveofthegalaxyNGC2998,theorbitalvelocityversusdistancefromcenter

Wewouldexpecttheorbitalvelocitytodecreaseas1/√ratdistanceslargerthanthevisibleradius,whichisinthiscaseabout8kpc.Itisnotso;thevelocityremainspracticallyconstantuptothemaximumdistanceexplored,muchbeyondthedistanceatwhichnomorestarsarepresent.

ThebehaviorofNGC2998isnotanexception,ratheristhenorm.Thesamephenomenonwasfoundinallthespiralgalaxies.WeneedtoconcludethateithertheNewtonlawisnolongervalidinthesecircumstances,orthatthereismuchmorematterinthegalaxiesthanthevisibleone,whichextendsmuchbeyondthevisibleone.Ithasbeencalleddarkmatter(butinvisiblematterwouldbeabettername).Wenowknowthattherightalternativeisthelatter.Theconclusion

comesfromalargenumberofobservations,atdifferentlengthscales,forphenomenaruledbydifferentphysics,atdifferenterasoftheUniverse.Allpointconsistentlytotheconclusionthatdarkmatterisaboutfivetimesmoreabundantthanthematterweknow.Thesearchfordarkmatterisoneofthefrontiersoftoday’sphysics.

4.11 EllipticOrbitsInSect.4.4wehaveseenthesolutionoftheso-calleddirectKeplerproblem,namelyhowtofindtheforcefromknowledgeoftheorbit.Wehavedonethathowever,intheparticularcaseofcircularorbits.Itisinstructivetosolvetheproblemingeneral,forellipticorbits.Weshalldothatinthissectionusingthemoderncalculus.Inthenextsectionweshallshowthesame,followingtheNewtondemonstration.

WestartfindingtheexpressionsofvelocityandaccelerationofagenericmaterialpointP,movinganarbitraryplanecurve,inpolarco-ordinates.Weintroduceapolarco-ordinateframewithoriginOandpolaraxisx(seeFig.4.27).Wecallθtheazimuthofthepositionvectorr,anduθandurtheunitaryvectorsrespectively.ThetimederivativesofthelatterisgivenbythePoissonformula(1.59)

Fig.4.27 Theunitvectorsofthepolarco-ordinates

(4.39)Wenowfindthevelocity,whichisthetimederivativeofthepositionvector

r=rur

which,forthefirstofEq.(4.39)is

Wenowderiveoncemoretohavetheacceleration

andfinally

(4.40)

Wehavenowthekinematicexpressionsweneed.Payattentiontothefactthatυrandararethecomponentsofthevectorsonthepositionvectorrfromthefocus,notfromthecenteroftheellipse.

Wenowconsiderthemotionoftheplanet.The1stKeplerlawstatesthattheorbitisanellipsewiththesuninoneofthefoci.

Westartbyrecallingthemainpropertiesoftheellipse(oneoftheconicsections,togetherwiththehyperbolesandtheparabola).Wechoosethepolarco-ordinateframeshowninFig.4.28withtheorigininthefocuswherethesunisandthemajoraxisaspolaraxis.(Noticethattherearealsopolarco-ordinateswiththeorigininthecenterO).Theangleθiscalledanomaly(tobeprecise,itissometimescalledtrueanomaly,todistinguishitfromthecaseinwhichtheoriginisinthecenter),aandbthesemi-majorandsemi-minoraxes.

Fig.4.28 Thegeometryoftheellipseanditsmainparameters

Theequationoftheellipse,inits“canonical”form,is

(4.41)whereeistheeccentricityandpisthesemi-latusrectumwhichisthe

positionvectorforθ=90°.Therelationbetweeneccentricityandsemi-axesis

(4.42)Thecirclecanbeconsideredadegenerateellipsewithe=0.Thesmallerthe

eccentricitythesmalleristhedifferenceoftheellipsefromthecircle.Asamatteroffact,theeccentricitiesoftheplanetsareinanycasequitesmall,muchsmallerthaninFig.4.28.

Considernowtheforce.Firstweobservethat,beingtheforcedirectedtothesun,theFθcomponentiszero.Thisstatementisequivalenttothe2ndKeplerlawandtotheconservationofangularmomentum.Indeed,fromFig.4.29weseethattheinfinitesimalareasweptbythepositionvectorisHencethearealvelocityis

Fig.4.29 Elementaryareasweptbythepositionvectoroftheplanet

(4.43)and

(4.44)Inaddition,callingLtheangularmomentumandrecallingEq.(4.7)wecan

write

(4.45)Thisexpressionwillbeusefulinthefollowing.WearenowreadytogototheaccelerationarandtheforceFrtowardsthe

sun.Wealreadyfound,Eq.(4.40),that

(4.46)

Thepolarco-ordinatesrandθarenotindependent,butlinkedbytheellipseEq.(4.41).Takingthetimederivativeofthisequation,rearrangingthetermsandusingEq.(4.45),wehave

(4.47)Wederivethisagain,becauseEq.(4.46)containsthesecondderivative,and

useagainEq.(4.45),obtaining

WenowsubstitutethisinEq.(4.46),useoncemoreEq.(4.45)andget

Lookingbacktotheequationoftheellipsewerecognizethattheexpressioninparenthesisinthelastmemberisjust–1/p.Finallywehave

(4.48)wheretheminussigntellsusthattheforceisoppositetor.Weseethattheaccelerationisinverselyproportionaltothesquareofthedistancefromthesun.Thesameistrueobviouslyfortheforce

(4.49)Thiscompletestheproof.Wehaveproventhatiftheorbitisanellipsewith

thesuninoneofthefoci,theforceisinverselyproportionaltothesquareofthedistance.TheremainingpartoftheargumenttoreachtheNewtonlawisthesamewealreadydidforcircularorbits,withtheconclusion

(4.50)Wedidnotneedthe3rdKeplerlawtoreachthisconclusion,asithadbeen

thecaseintheparticularcaseofcircularorbits.Indeed,inthatcaseEq.(4.41)reducestor=p=constantandnotalloftheargumentsofthissectionanylongerhold.

BeforeconcludingwestressoncemorethatthereisauniquedependenceonrofacentralforceFr(r)thatproducesellipticorbitswiththesuninafocus,

.AsNewtonshowed,eventhesmallestdifferenceintheexponent,

wouldproduceanorbitofthetypeshowninFig.4.30,whichis,sotosay,aslowlyrotatingellipse,calledarosette.Weshallnotreproducetheargumenthere,butonlygiveahint.Inamotiononanellipseoronarosette,bothpolarco-ordinates,randθ,varyintimeperiodically.Theperiodofthe

latterisinanycasethetimetoincreaseθby2π.Theperiodofrdependsontheforce.Onlyiftheforceisinverselyproportionaltor2isitequaltotheperiodofθandthetrajectoryisclosed.Iftheexponentof1/risnotexactly2,thetwoperiodsaredifferent,theorbitdoesnotcloseandwehaveasituationlikeFig.4.30.Thiseffectcannotbeseeniftheorbitiscircular,becauseacirclerotatingonitselfisnotdifferentfromacircle.

Fig.4.30 A“rosette”orbit,showinga“snapshot”every15°ofprecession

Astronomershaveobservedforcenturiestheapparenttrajectoriesoftheplanetsintheskywithhighaccuracy.Theabsolutetrajectoriesareobtainedsubtractingthenowwell-knownmotionofearth.Theapheliaandtheperihelia,inparticular,canbeaccuratelyidentified.Iftheforceisproportionaltotheinversesquaredistancefromthesun,thesepointsshouldremainfixed.Indeed,thisisalmostthecase,butnotquite.Veryslowmovements,calledprecessions,oftheperiheliaareobserved.Theyaretheeffectsoftheforcesoftheotherplanets,thelargeronesinparticular,thatactontheminadditiontothesun.Observationsandcalculationsagree,withanexception,whichwasfoundbyLeVerrierin1849.HecalculatedtheprecessionoftheperihelionofMercury,thenearesttosun,in10arcminutespercentury.Thelargestfractionofthatisexplainedbythejustmentionedeffectsofotherplanets.Butnotcompletely;43arcsecpercenturyremainedunexplained.Anumberofhypothesiswereadvanced,butallofthemfailed.ThiswasthefirsthistoricalexampleofthelimitsoftheNewtonlaw.TheexplanationoftheanomalousprecessionoftheMercuryperihelionbyAlbertEinstein(1879–1955)in1915markedthesuccessofgeneralrelativity.

The3rdKeplerlawisaconsequenceofthe1stone.Letusprovethat.WestartwiththeconsiderationthattheperiodTistheareaoftheorbitAdividedby

thearealvelocityandexpressingthelatterintermsoftheangularmomentumLusingEq.(4.45).

WenowwritetheaccelerationarEq.(4.48)usingthisequationandwritingtheparameterpintermsoftheaxes,Eq.(4.42)

TheforceontheplanetistheNewtonforce,andwecanwrite

andfinally

(4.51)Thatisthe3rdKeplerlaw:thesquaresoftheperiodictimesareproportional

tothecubesoftheellipsesemi-majoraxis,forallthebodiesorbitingthesamecentralbody(ofmassM).

4.12 TheNewtonSolutionInthissectionweshallnotintroduceanynewconcept,ratherweshallshowhowIsaacNewtondemonstratessomeofthosewediscussedintheprevioussections.Readingpagesofthegiantsis,infact,veryinstructive,evenif,asisthecasewithNewton,itisnotalwayseasy.Afterhavinggiventhenecessarypreliminaryinformation,weshallreadonepageofthePrincipia,explainingtheirmeaninglinebyline.Asweshallsee,theNewtonargumentsaremainlygeometrical.Thenovelty,withrespecttowhatwasalreadyknowntotheGreeks,isthefinalpassagetothelimitforthelengthoftheconsideredorbitarcgoingtozero.

AfterhavingstatedthelawsofmotioninSect.1ofthePrincipia,NewtondedicatesSect.2to“Thedeterminationofthecentripetalforces”.Hereheconsidersorbitsofvariousgeometricalshapesundertheactionofaforcedirectedtoaimmovablecenter(i.e.centripetal).Thecaseweshalltakeistheellipsewiththecenterofforceinafocus.Inthefirsttwo“Propositions”ofSect.2,heshowsthat,inanycase,ifthearealvelocityisconstanttheforceisdirectedtothecenterandviceversa.

Subsequently,inPropositionVI,Newtonlaysdownthebasicschemehe

shallusetosolvetheabove-mentionedproblems.TheschemeisshowninFig.4.31.

Fig.4.31 TheschemeofPropositionVI

AbodymovesonthearcPQofitsorbitintheshorttimeinterval∆t.Iftherewerenogravitationalforcefromthesun,theplanetwouldmoveofrectilinearuniformmotiononthedisplacementPR.Ontheotherhand,ifabandonedstillinPtheplanetwoulddropinthetime∆t,underthegravitationalattraction,bythedisplacementPX.Iftheforceisconstant,themotionisuniformlyacceleratedandPXisproportionalto∆t2.Ifbothconditionsarepresent,thedisplacementisthediagonalPQ.WenowdrawthesegmentQRparalleltoPX.QRtouchesthetrajectoryinQ.

Whatwejuststatedwouldbetrueiftheforcewereconstantduring∆t,whichisnottrue.However,thesmalleris∆tthesmalleristhevariationoftheforceinthatinterval.Thismeansgoingtothelimitof Thelimitgeometricallycorrespondstoapproximatethesegmentofthetrajectorywithasegmentofparabola.ThemotionisthenequaltowhatwasfoundbyGalileifortheprojectilesonearth.

Ontheotherhand,QRisalsoproportionaltotheaccelerationandtotheforceFwearelookingfor,namely ,or .

Fortheconstancyofthearealvelocity,thetimeintervalisproportionaltotheareasweptbythepositionvectorinthatinterval,whichistheareaofthetriangleSQP.Thelatter,inturn,isproportionaltotheproductofitsbaseSPanditsheightQT,andwehave

(4.52)Thisexpressionisvalidforanycurve.Weshallseehowitsimplifiesinthe

caseoftheellipticorbit,withthecenterofforceinafocus.Todothat,weshallneedtoknowsomedefinitionsandfourpropertiesoftheellipse.Wegivethem

herewithoutproof.Adiameterisachordgoingthroughthecenteroftheellipse.Considerthe

tangenttotheellipseinanygivenpointPonit(seeFig.4.32).LetbePP′thediameterpassinginPandDKthediameterparalleltothetangentinP.ThediametersPP′andDKarecalledconjugatediameters.

Fig.4.32 PP′andDKareconjugatediameters

Noticethattheconjugatediametersbisecteachotherbut,ingeneral,donothaveequallengths,neithert0theycrossatrightangles.

Property1.Thesumsofthedistancesofanypointoftheellipsefromthetwofociareequalandareequaltothemajoraxis,2a.Property2.(Fig.4.33).Alltheparallelogramshavingconjugatediametersassideshavethesamearea.Itisequaltotheareaoftheparallelogramhaving,inparticular,theaxesassides,namely4ab.

Fig.4.33 Property2.Parallelogramswithconjugatediameterssides

Property3(Fig.4.34).ThetwofocallinesthatjoinanypointPoftheellipseformequalangleswiththetangentinthatpoint.

Fig.4.34 Property3.Twofocallinesandtheirangleswiththetangent

Property4(Fig.4.35).Everydiameterbisectsalltheconjugatechords.Foranygivendiametertheratiobetweentheareasoftherectanglesmadebythetwosegmentsofthediameterandthesquareofthecorrespondingsemi-chordareequal.Namely

Fig.4.35 Property4

(4.53)

WehavenowthepropertiesoftheellipseweshallneedandwecanreadPropositionXI.

PropositionXIstates:

ifabodyrevolvesinanellipse;itisrequestedtofindthelawofthecentripetalforcedirectedtothefocusoftheellipse.

TheproofshowsthattheratioQR/QT2inEq.(4.52),intheparticularcaseoftheellipsewiththecenterofforceinafocus,isequaltothelatusrectum,whichwecalled2pandhecallsL.Weshallusehissymbolinthissection(noriskofconfusionwiththeangularmomentum).

Figure4.36reproducesthediagramonwhichthetheoremisdeveloped.ThefirstlinesofthePropositionare:

Fig.4.36 TheNewtondiagramforPropositionXI

LetSbethefocusoftheellipse.DrawSPcuttingthediameterDKoftheellipseinE,andtheordinateQVinX;andcompletetheparallelogramQXPR

Thesun(thecenterofforce)isinthefocusS;Histheotherfocus,Cisthecenter,CA=aandCB=barethesemi-majorandthesemi-minoraxesrespectively.AtacertaininstanttheplanetisinP,SP=risthepositionvectorfromthesun.WedrawthetangentRPZtotheellipseinPandthelineQVparalleltoit.BeXandVthepointswereitcutsSPandPCrespectively.WealsodrawthelinesofQRPTasinFig.4.31.TocompletethediagramwedrawtheperpendicularfromPtothediameterDKandcallFthepointinwhichtheymeet.

TheNewtonlanguageisextremelysynthetic.Whatisevidentforhimisnotalwaysevidentforus.Weshallexplainhislinesimmediately.

ItisevidentthatEPisequaltothegreatersemiaxisAC:fordrawingHIfromtheotherfocusHoftheellipseparalleltoEC,becauseCSandCHareequal,ESandEIwillbealsoequal;andhenceEPishalfthesumofPSandPI,thatis(becauseoftheparallelsHIandPR,andtheequalanglesIPR,HPZ)ofPSandPH,whichtakentogetherareequaltothewholeaxis2AC.

ThegeometricelementsofFig.4.36thatarerelevantforthisstepareredrawninFig.4.37.Westartfromtheequation(Property1)

Fig.4.37 EPisequaltothemajoraxis

(4.54)ThetriangleIPHisisosceleswithvertexinP.Thisisbecause:

theanglesRPIandPIHareequal,asalternateinterioranglesofthetwoparallellinesJLandRZtheanglesHPZandIHPareequalasalternateinterioranglesofthesamelinestheanglesPIRandHPZareequalfortheProperty3oftheellipse

ConsequentlytheanglesPIHandIHPareequal,whichprovesthestatement.HencePH=PIandwecansimplifyEq.(4.54)as

(4.55)ThetrianglesISHandESCaresimilarbecausetheyhavethesameanglein

thevertexSandthesidesoppositetoit(ECandIHrespectively)areparallel.Inaddition,SHistwiceSCandconsequentlySI=2ES,thatisalsoES=IE.SubstitutinginEq.(4.55)weobtain

(4.56)andfinally

(4.57)NowNewtonworksonQR:

DrawQTperpendiculartoSP[wedidthatalready],andputtingLfortheprincipallatusrectumoftheellipse(orfor2BC2/AC[seeourEq.(4.41)])weshallhave

IsnotsosimpletofollowtheNewtonlanguage.Heusesproportions,whichweshallwriteasfractionstomakethemmorereadable.Inaddition,whenhetakesaroutehedoesnottellusthereasons,whichareunderstoodonlyattheend.Letustrusthimandfollow.HestartsfromtheratioQR/PVwithnumerator

anddenominatormultipliedbyL,becauseattheendthiswillbeuseful.TherelevantgeometricalelementsaredrawninFig.4.38.

Fig.4.38 WorkingonQR

Thefirststepistrivial

ButQR=PXbyconstruction.LetusfindPX.ThetrianglesPXVandPECaresimilarbecausetheyhaveacommonvertex

inPandthetwooppositesides,XVandEC,areparallel.ConsequentlyPE/PC=PX/PVandalso

UsingEq.(4.57)namelyPE=ACwehave

andinconclusion

(4.58)ThenextstepisworkingonPV.ThesinglelineofNewtonis:

also and

OncemoreNewtonworksonaratio,L/GV,andmultipliesnumeratoranddenominatorbythesamequantity,whichisPV,thequantitywearenowlooking

for.TherelevantgeometricalelementsareshowninFig.4.39.

Fig.4.39 WorkonPV

Thefirststepisagaintrivial,

(4.59)WeusetheProperty4oftheellipseappliedtothediameterPGandtothe

semi-chordsQVandDCconjugatedtoit,getting

(4.60)Newtoncontinues,findingafourthproportion.Finallyhewillputthefour

together.Wetakeabreath,abandonhimforamomentandputimmediatelytogetherthethreeEqs.(4.58),(4.59)and(4.60)wefound.Wemultiplythemmemberbymemberandobtain

Simplifying,butkeepingLthatwillbeuseful,wehave

(4.61)Weneedanotherproportion,thelastone.

ByCor.II,Lem.VII[istheruleforgoingtothelimit],whenpointsPandQcoincide,QV2=QX2andQX2orQV2:QT2=EP2:PF2=CA2:PF2,and(byLem.XII)=CD2:CB2.

Wenowneedtoexpress1/QT2.Asusual,Newtonworkswithproportions,andweshalldothesamewithratios.ThistimeitisQX2/QT2.AttheendweshalltakethelimitforthelengthofthearcPQgoingtozero,namelytohavethepointsPandQcoincident.Inthislimit,pointsXandVcoincidetooanditisthen

convenienttoconsiderQV2/QT2inplaceofQX2/QT2.TherelevantgeometricalelementsaredrawninFig.4.40.

Fig.4.40 Workon1/QT2

ThetrianglesEPFandXQTaresimilar,because

theanglesinFandTareequalbecausearebothright,thesidesoftheanglesrespectivelyinPandQ(inevidenceinthefigure)aremutuallyperpendicular,hencetheyareequal.

Hence

and,asEP=CA,

(4.62)TofindPFweuseProperty2.Figure4.41ashowsthatPFisonehalfofthe

heighttoDKofthedrawnparallelogramonconjugatediameters.

Fig.4.41 UsingProperty2

Property2gives: or

which,substitutinginEq.(4.62)andsquaring,gives

(4.63)Thenextstepistomultiplythefourproportions.Newtonwrites:

Multiplyingtogethercorrespondingtermsofthefourproportions,andsimplifying,weshallhave

since

WehavealreadymultipliedthefirstthreeratiosobtainingEq.(4.61).HencewemultiplynowitsmemberswiththoseofEq.(4.63)

andsimplify

Recallingtherelationbetweenlatusrectumandaxes andsimplifyingwefinallyhave

(4.64)RememberthatthefactorinEq.(4.52)wewanttoexpressisQR/QT2.We

haveitnowinEq.(4.64).Thefinalstepistakingthelimitfor RememberthattheconceptoflimitwasnotknownbeforeNewton.Hewrites:

ButthepointsQandPcoinciding,2PCandGVareequal.Andthereforethequantities andQT2,proportionaltothese,willbealsoequal.Let

thoseequalsbemultipliedby and willbecomeequalto

InthelimitinwhichthearcPQbecomesinfinitelysmall,pointVcoincides

withP.Consequently,GVbecomesequalto2PC,andthesecondmemberofEq.(4.64)goestoone,becoming

NowmultiplybothmembersbySP2/QRandget

Finally,Newtonconcludes:

Andtherefore(byCor.Iandv,Prop.VI)thecentripetalforceisinverselyas,thatis,inverselyasthesquareofthedistanceSP.

Q.E.D.

Namely:

(4.65)and,giventhatL,our2p,isaconstantforagivenellipse,

(4.66)Theforceisinverselyproportionaltothesquareofthedistancefromthe

center.Thatiswhatwehadtoshow.

4.13 TheConstantsofMotionWenowgobacktothemainstreamandconsiderthepotentialandthekineticenergyofabodyofmassminthegravitationalfieldofabodyofmassM,movingonanellipse.

Westartwithitsangularmomentum,whichwecallLandgobacktoourformalismcallingpthesemi-latusrectum.FromEq.(4.50)wecanwrite

(4.67)

which,usingEq.(4.42)is

(4.68)Inwords,thesquareoftheangularmomentumisproportionaltothemajor

axis.Foragivenmajoraxis,theangularmomentumisthelargestfore=0,whichisthecircle.Itdecreasesforincreasinge,i.e.fortheellipsebecoming

moreandmoresqueezed.Considernowthepotentialenergy,andmakeuseforroftheellipse

equation

(4.69)Forthekineticenergy,rememberEq.(4.39)

(4.70)

Usingtheexpressionofdr/dtgivenbyEq.(4.47)andusingEq.(4.52),wehave

WeusenowtheEq.(4.41)oftheellipsetoexpress1/r2and,takingintoaccountthat obtain

(4.71)Bothpotentialenergy,Eq.(4.69),andkineticenergyEq.(4.56)dependon

thepositionoftheplanetandconsequentlyontime.Notsothetotalenergy,whichistheirsum

(4.72)whichwecanalsowrite,inequivalentmanner

(4.73)Inconclusion,thetotalenergyoftheplanetdependsonlyonthesemi-major

axis.Differentorbits,suchasthoseinFig.4.42,whichhavethesamesemi-majorbutdifferentsemi-minoraxeshavethesametotalenergy.However,aswehaveseenabove,theangularmomentumgrowsfordecreasingeccentricity.

4.1.

4.2.

4.3.

4.4.

Fig.4.42 Orbitsofthesameenergyanddifferentangularmomenta

Payalsoattentiontothefactthatthetotalenergyisnegative.However,thisisnottheonlypossibilityforabodymovingaboutthesun,oranyothersourceofgravitationalforce.Asamatteroffact,inourdemonstrationwehaveusedonlyEq.(4.41).Thisisnotonlytheequationoftheellipse,but,moregenerallyofalltheconics,ellipseife<1,parabolaife=1,hyperbolesife>1.Thethreecasescorrespond,fromthephysicalpointofview,tototalenergy(4.73)negative,nullorpositiverespectively.Thepotentialenergyisalwaysnegative,tendingtozeroatinfinitedistancefromthecenter.Thekineticenergycanbepositiveorzero.Consequently,atinfinitedistancethetotalenergyispositiveor,asaminimum,zero.Ifthetotalenergyofabodyisnegative,itmustremainatfinitedistances.Theorbitissaidtobound.Theellipse(includingthecircleasaparticularcase)istheonlyconicthatdoesnotreachinfinity.Ifonthecontrary,thetotalenergyofabodyispositive,itwillbeabletogofartherandfarther;atinfinitedistances,ormorerealisticallyatdistanceslargeenoughtohavenegligiblepotentialenergy,allitsenergyiskinetic,positiveinfact.Theintermediatecaseiswhenthebodyreachesinfinitywithzerokinetic(andtotal)energy.Thetrajectoryisaparabola.

4.14 ProblemsApendulumhaving1speriodonthesurfaceofearthisbroughtonthesurfaceofaplanethavingthesameradiusofearthandmassfourtimeslarger.Whatistheperiodofthependulum?

Thegravitationalpotentialdifferencebetweentwopointsontheearthsurface(atthesamelatitude)is1000m2s–2.Whatisthedifferencebetweentheheightsweretheyarelocated?

Weabandonabodyatthedistancefromearthofthemoonorbitwithnovelocity.Willitfallwithconstantvelocity?Withconstantacceleration?

Wemoveabodyfromthesealeveltothetopofamountain5000mhigh

4.5.

4.6.

4.7.

4.8.

4.9.

4.10.

(samelatitude).Howdoesitsmassvary?Howdoesitsweightvary?

Doesthevelocityatwhichasatellitemovesinacircularorbitaroundtheearthdependonthemassoftheearth?onthemassofthesatellite?ontheradiusoftheorbit?

Theapparentdiameterofthesunasseenfromearthisapproximatelyα=0.55°.Whatwouldbetheperiodofahypotheticalplanetorbitingjustoutofthesun?

Wewanttoputanartificialsatelliteinorbitaroundtheearthhavingaperiodof2h.KnowingthegravityaccelerationgonthesurfaceofearthanitsradiusRE,findtheheightoftherequestedorbitabovethesurface.

Consideraspring(ofaballpointpen)withrestlength3cmandelasticconstantk=50N/m.WefixtoitstwoextremestwoequalPbspheres(densityρ=11×103kg/m3),ofmassm=104kgeach.Assume,unrealistically,thatallfrictionscanbeneglected.Howmuchwillthespringshrinkundertheactionofthegravitationalattractionofthetwospheres?

Knowingthevaluesofg,ofGNandoftheradiusofearth(RE=6.4×106

m),makeanestimateofthemassandofthemeandensityofearth.

KnowingthevaluesofGNandoftheradiusofearthorbit(rE=1.5×1011m)andofitsperiod,makeanestimateofthemassofthesun.Knowingthatitsapparentdiameterfromearthis0.55°,estimateitsmeandensity.

4.11.

4.12.

4.13.

4.14.

ThesunmovesonanorbitthatwecanconsidercircularaboutthecenteroftheGalaxy.TheradiusofthesunorbitisRS=25000lyear=2.5×1020m,hisvelocityisυS=250km/s.Comparethesedatawiththoserelativetothemotionofearthaboutthesun(rE=1.5×1011m,υE=30km/s).MakeanestimateofthetotalmassMtotaroundwhichthesunorbits;giveitasamultipleofthesolarmassMS.

Io,oneoftheJupitersatellites,hastheorbitalperiodTI=1.77dandtheorbitradiusrI=4.22×108m.Comparethesedatawiththoseofthemotionoftheearthaboutthesun(rE=1.5×1011m,υE=30km/s).DeterminethemassofJupiterinsolarmasses.

Findaproceduretodeterminethemassofearth.

KnowingthattheearthmovesaroundthesunwiththevelocityofυE=30km/s,findthegravitationalpotentialofthesunϕS(E)inthepointsofearthorbit.ThegravitationalpotentialinapointoftheearthisthesumofthejustconsideredϕS(E)duetothesunandofthegravitationalfieldsoftheearthitself,sayϕE(E),andofalltheGalaxy,sayϕG(E).CalculatethevaluesofthelattertworelativetoϕE(E),knowingthatthemassesinthethreecasesareapproximatelyMS=2×1030kg,ME=6×1024kg,MG=2×1041kgandtakingasdistances,fromearthtosunrES=1.5×1011

m,radiusofearthrE=6.4×106m,distancefromsuntothecenterofGalaxyrSG=2.5×1020m

Footnotes

1 Aparsecisthedistanceatwhichthediameteroftheearthorbitisseenundertheangleofasecond.

(1)

©SpringerInternationalPublishingSwitzerland2016AlessandroBettini,ACourseinClassicalPhysics1—Mechanics,UndergraduateLectureNotesinPhysics,DOI10.1007/978-3-319-29257-1_5

5.RelativeMotions

AlessandroBettini1

DipartimentodiFisicaeAstronomia,UniversitàdiPadova,Padova,Italy

AlessandroBettiniEmail:alessandro.bettini@pd.infn.it

Inourstudyofthekinematicsofthematerialpoint,wehavealreadyseenthattheequationsofmotiondependonthereferenceframe.Thelawofmotions,andmoregenerallyallthelawsofPhysics,transform,aswesay,fromoneframetoanother.Thischapterisdedicatedtothestudyofthesetransformations.

Tworeferenceframesmaydifferindifferentways.Thetwoframeshavenorelativemotion,theirco-ordinatehomologousaxes

areparallel,buthavedifferentorigins;theframesdifferforarigidtranslation.Thetwoframeshavenorelativemotionandcoincidentorigins,butthe

directionsoftheaxesaredifferent;theframesdifferforarigidrotation.Oneframecantranslaterelativetotheotherintimewithuniformorvarying

velocity,oritcanrotate,againwithconstantorvaryingangularvelocity,oritcantranslateandrotatecontemporarily.

InSect.5.1,weshallconsidertwostationaryframesrelativetooneanother,witharelativetranslationorrotation.WeshallseethatthelawsofPhysicshavethesameform,namelythesamemathematicalexpressions,inbothframes.Aswesay,thelawsarecovariantundertranslationsandrotations.Themeaningofthetermwillbeexplained.

Weshallthenconsiderframesinrelativemotionandlearnthat,whentherelativemotionisatranslationwithconstantspeed,thelawsofmechanicsarealsocovariant.Thisistherelativityprinciple,afundamentalprincipleofphysics,establishedbyGalilei.Forexample,experimentsdoneinsideaclosedroomina

shipcannotestablishwhethertheshipismovinginuniformmotionorisstandingstill.Oneoftheconsequencesisthatoncewehavefoundaninertialframe,anyotherframemovinginauniformtranslationmotionrelativetoitisalsoinertial.

InSect.5.3,weshalldealwiththerelativetranslatoryacceleratedmotion.Asalreadyanticipated,inanyreferencethatacceleratesrelativetoaninertialframe,theNewtonlawsarenotvalid.Forexample,abodyatrestcanstartmovingwithoutanyforceactingonit.Themotioncanbedescribedintroducingfictitiousforces,whichareknownbyseveralequivalentnames,apparentforcesoftherelativemotions,pseudo-forcesandinertialforces.Wefeelsuch“force,”forexample,whenwebrakesuddenlyinacar.InSect.5.4,weshalldealwiththegeneralcase(translationandrotation)andweshallseetherelationsbetweenvelocitiesandbetweenaccelerationsintwoframesofanyrelativemotion.InSect.5.5,weshalldiscussseveralexamplesofmotioninframesrotatingrelativetoaninertialframe.

Anyframeatrestinalaboratoryonearthdoes,infact,movewithearth.Ininitial,andquitegood,approximation,theseframescanbeconsideredtobeinertial.Notcompletely,however,becauseearthrotatesonitsaxisandmovesalongitsorbitaroundthesun,andeventhesunmovesalongitsorbitinthegalaxy.InSect.5.7,weshallstudyafeweffectsoftheinertialforcesinframesatrestrelativetoearth:thevariationwithlatitudeofthemagnitudeoftheweight,therotationoftheoscillationplaneofpendulums,thedeviationfromtheverticaloffreefallandthecirculationofwinds.

Theinertialforcesactingonabodyareproportionaltoitsinertialmass,whilethegravitationalattractionofearthisproportionaltoitsgravitationalmass.Thisobservationallowsfortherealizationofverydelicateexperimentstocheckwhetherthetwomassesaredifferentorequal.WeshalldescribesuchanexperimentinSect.5.8.

5.1 CovarianceofthePhysicalLawsUnderRotationsandTranslationsConsidertwoCartesianreferenceframes,whicharestationaryrelativetoeachother,S(coordinatesx,y,z,originO)andS′(coordinatesx′,y′,z′,originO′).

Aphysicallawisamathematicalequationbetweenphysicalquantities.Therelationbetweenthetwoframescanbearigidrotationorarigidtranslation.Letusstartwithrotations.

Wechoosetheoriginsofthetwoframesascoincident.Forsimplicity,we

considertheirz-axesalsotobecoincident.Theframesdifferforarotation,byanangleθ,aroundthisaxis.Therotationisinthecommonplanexy,asshowninFig.5.1.

Fig.5.1 Tworeferenceframesdifferentforarigidrotation

SupposenowthatanobserverinSmakesaverysimpleexperiment.Hemeasures,usingabalance,themassesoftwoobjects,findingthevaluesm1andm2.Hefindsthatthesecondmassisthreetimesthefirst.Hewritestherelation(letuscallitthe“law”)

(5.1)AnotherobserverinS′performsthesameexperiment.Weindicatewitha

primethehomologousquantitieshefinds.Inthisverysimplecase,consideringthattheprocedureofmeasuringthemasswithabalancedoesnotdependonthedirectionoftheaxes,wecanconcludethathewillfindthesameresult,namely

(5.2)Thesecondobserveralsostatesthat

(5.3)Equation(5.3)hasthesameformasEq.(5.1).Namely,thetwoobservers

describethesamephenomenonwithlawsofthesameform.Indeed,massisascalarquantity,whichisinvariantunderrotationsoftheaxes.Ingeneral,arelationbetweenscalarquantities,ifvalidinaframe,isalsovalidinanyframerotatedrelativetothefirstone,becausebothsidesoftheequationdonotvarygoingfromoneframetotheother.

Butitisnotalwaysso.SupposethattheobserverinSmeasurestwocomponentsofthevelocityofapoint,findingthevaluesυxandυy.Hefinds,again,thesecondquantitythreetimeslargerthanthefirstandwritestheequation

(5.4)WhatwouldtheobserverinS′find?Weknowtheanswerbecauseweknow

therelationsbetweenvelocitiesinthetwoframes:

(5.5)Wecalculatetheratiobetween and ,alsoemployingEq.(5.5):

Inconclusion,inS′,wehave

(5.6)Theformofthe“law”isdifferentthistimeinthetwoframes,being(5.4)in

Sand(5.6)inS′.Thisisanobviousconsequenceofthefactthatthecomponentsofavectortransformdifferentlyonefromanother.

Butwaitamoment,alawmaybevalidinbothframes,evenifitssidesarenotinvariant,asinthecaseofthemasses;rather,itissufficientthat,iftheyvary,inthesameway.Letusseewhathappensforalawlinkingvectorquantities.

TheobserverinS′,whichweassume,forthesakeofthisexample,tobeinertial,studiesthemotionofamaterialpoint.Hemeasurestheaccelerationa(namelyitsthreecomponents),theforceactingonthepointF(again,thethreecomponents)andthemassm.Hefindstherelation

(5.7)Moreexplicitly,thisvectorrelationcorrespondstothreeequations:

(5.8)Weknowhowthecomponentsofthevectors,suchasFandaare,transform

fromoneframetotheother,namely

(5.9)

andwecanwrite

and,forEq.(5.9),

(5.10)whichhasthesameformasEq.(5.8).Bothsidesoftheequationsare

different,varyingfromoneframetotheother.However,theyvaryinthesameway,becausebothsidesarevectors.Thus,wesaythattheequationiscovariant.

Inconclusion,thelawsofPhysicskeepthesameformunderrotationsoftheaxes,or,inotherwords,arecovariantunderrotations.Andyet,fromanotherperspective,itisimpossibleexperimentallytoestablishanyprivilegeddirectionsofthereferenceaxes.Spaceshouldbeconsideredisotropic,withoutpreferentialdirections.

Thecaseofthetranslationsisverysimple.Scalarquantitiesobviouslyhavethesamevaluesintwoframesdifferingforatranslation.Thisisalsovalidforvectors,whicharesimplytranslated;hence,theyarethesamevector.

5.2 UniformRelativeTranslation.RelativityPrincipleConsidernowtworeferenceframes,SandS’,whichareinrelativemotion.WearbitrarilycalloneofthemS(originOandcoordinatesx,y,z)fixedandtheotheroneS’(originO’andcoordinatesx’,y’,z’)mobile.WeconsiderthecaseofauniformtranslationofS’.AllthepointsofS’movewiththesamevelocityrelativetoS,whichisconstantinmagnitudeanddirection.Theframes,forexample,mightconsistofonefixedontheground,theotheronacarriagemovingonstraightrails,oraframefixedattheshoreandoneonashipmovingstraight,inbothcaseswithuniformmotion.

Theaxesofthetwoframesdonotchangetherelativedirectionsandwecantakethemasbeingparallel.Fig.5.2showsthetwoframesatacertaininstant.Atalatertime,themobileframewillbeinadifferentposition,moreontheright,butitsaxeswillstillbeparalleltotheaxesofS.Wechooset=0asthetimeatwhichtheaxesofthetwoframesoverlap.

Fig.5.2 Tworeferenceframesinrelativeuniformtranslationmotion

Figure5.2showsthematerialpointPanditstrajectory.Thepositionvectors

randr’ofPinthetwoframeshavethewell-knownrelation

(5.11)whererO’isthepositionvectoroftheoriginO’ofthemobileframeS’inthefixedframeO,namelyOO’.

AfixedandamobileobserverseethepointPmovingwithdifferentvelocities,vandv’.Tofindtheirrelation,wetakethetimederivativesofEq.(5.11),obtaining

(5.12)wherevO’isthevelocityoftheoriginO’ofthemobileframe,andalsoofallitspoints(becausethemotionisatranslation)asseenbyS.Thevelocityofaninsectflyingintheshipintheaboveexamplerelativetotheshoreisthevectorsumofthevelocityoftheinsectrelativetotheshipandthevelocityoftheshiprelativetotheshore.

Afurthertimederivationgivestherelationbetweenaccelerations

(5.13)whereaO’isthevelocityoftheoriginO’ofthemobileframe,andalsoofallitspoints.

WenowconsidertheimportantparticularcaseinwhichthetranslationofS’relativetoSisuniform,namelythevelocityofitsorigin,andofallitspoints,seenbySisconstantintime

(5.14)Then,obviously,

(5.15)andEq.(5.13)becomes

(5.16)Theaccelerationsinthetwoframesareequal.Theimplicationsofthis

simpleconclusionareextremelyimportantconsideringinertialframes.IfSisaninertialframe,anymaterialpointPnotsubjecttoforcesmovesat

constantvelocityv(orremainsatrest).Inotherwords,itsaccelerationiszero,a=0.Inthemobileframe,itsaccelerationa’,whichisequaltoa,isalsozero.Consequently,S’isinertialtoo.

Weconcludethat,givenaninertialreferenceframe,anyotherframemovingrelativetoitbyauniformtranslationisalsoinertial.

WhataboutthesecondNewtonlaw?ItisvalidintheframeS,whichisinertialbyassumption.IsitalsovalidinS’?InS,wehave

(5.17)

TheobserverinS’measuresthesamemass(m’=m)andthesameforce(if,e.g.,heusesadynamometer,thespringstretchesbythesameamount),F=F’.Theaccelerationa’thathemeasuresisalsoequaltoa,butonlyinthecaseweareconsideringofrelativetranslationatconstantvelocity.Then,inS’,therelationbetweenforce,massandaccelerationis

(5.18)Inotherwords:thelawsofmechanicsarecovariantunderthe

transformationsthatlinktworeferenceframesinrelativeuniformtranslationmotion.

Asanexample,considerareferenceS’fixedonasailingshipmovingontheseaatconstantvelocityandSaframefixedtotheshore.Asabove,wechoosetheaxesofthetwoframesmutuallyparallelandwithcoincidentoriginsatt=0.Anexperimenterclimbsontopofthemastanddropsastone.Fig.5.3showsthetrajectoriesofthestoneasseenbyanobserverontheshore,a),andontheship,b).

Fig.5.3 Trajectoryofastonedroppedfromthetopofthemastofaship,asseenfromtheshipandtheshore

FortheobserverinS,thestonefallsundertheactionofitsweight,aconstantforce(F=mg),directeddownwards,oppositetothez-axis(thatwehavetakentobeverticalupwards).Theinitialvelocityofthestoneisthevelocityoftheship,andwehavetakenthex-axisinthatdirection.Hence,themotionofthestoneinthezdirectionisuniformlyaccelerated,whileinthexdirection,itisuniform(neglectingtheairresistance).Thetrajectoryisaparabola.Inthefigure,wemarkedthepositionsofthestoneintimeinstantsseparatedbythesametimeinterval.

InS’,theforcesarethesame,buttheinitialconditionsaredifferent;theinitialvelocityofthestoneiszero.Hence,itfallsverticallyalongthez’-axiswithauniformlyacceleratedmotion.

Summarizing,inthetwoframes,thetrajectoriesaredifferent.Thereasonforthedifferenceisinthedifferentinitialconditionsofthemotion.Onthecontrary,

bothobserversdescribethemotionwiththesamelaw,F=ma.Thetwoframesareperfectlyequivalentforeverydynamicexperiment.Eachofthemcanbeconsideredasfixedormovable.

Thisconclusionisimportantandisknownastherelativityprinciple.Theprincipledoesnotdealdirectlywiththephenomenabutratherwiththelawsthatdescribethephenomena.Itstatesthat:thelawsofPhysicsarecovariant,namelyhavethesameform,inanyreferenceframemovingoftranslationaluniformrelativemotion.

Inourdiscussion,wehaveseenthattherelativityprincipleisvalidforthelawsofmechanics,whichisthephysicschapterwearestudying.However,itsvalidityiscompletelygeneral,including,inparticular,allfundamentalinteractions,gravitational,electromagnetic,nuclearstrongandweakinteractions.Inotherwords,itisimpossibleexperimentallytoestablishtherelativemotion,provideditisasuniformtranslation.Historically,theprinciplewasestablishedbyG.Galilei.Hedidnotusethatname,whichwasgiventoitbyHenriPoincaré(1854–1912)in1904,butGalileiestablisheditincompletegenerality,describing,inabeautifulpage,aseriesofexperiments,someofwhichwereofanelectromagneticnature,belowthedeckofalargesailingship.ThepageoftheDialogue(transaltedfromItalianintoEnglishbytheauthor)is:

Shutyourselfwithafewfriendsinthelargestroombelowdecksofsomelargevessel,andhavewithyouflies,butterfliesandsimilarsmallflyinganimals.Letalargebowlofwaterwithseveralsmallfishinitbethecabintoo.Hangalso,atacertainheight,abucketpouringoutwaterdropbydropintoanothervasewithanarrowmouthbeneathit.Whentheshipstandsstill,carefullyobservehowthoseflyingsmallanimalsflywithequalspeedtowardsallsidesofthecabin;youwillseethefishswimindifferentlyinalldirections;allthedropswillfallintothevesselbeneath;andyou,whenthrowingsomethingtoafriend,willnotneedthrowitmorestronglyinonedirectionthananother,whenthedistancesareequal;andjumpingupfeettogether,youwillpassequalspacesinalldirections.

Onceyouhaveobservedallthesethingscarefully,thoughthereisnodoubtthatwhenthevesselisstandingtheymusthappenlikethat,letthevesselmovewithspeedashighasyoulike.Then(providedthemotionisuniformandnotunevenlyfluctuating)youwillnotdiscovertheslightestchangeinanyofthenamedeffects,noryouwillbeabletounderstandfromanyofthemwhethertheshipismovingorstandingstill.Injumpingyouwillpassontheplankingthesamespacesasbefore,noryouwillmakelongerjumpstowardthesternthantowardtheprow,asaconsequenceof

thefastmotionofthevessel,despitethefactthatduringthetimeyouareintheairtheplankingunderyouisrunninginadirectionoppositetoyourjump.Inthrowingsomethingtoyourcompanion,nomoreforcewillbeneededtoreachhimwhetherheisonthesideoftheprowandyouofthesternoryourpositionsareinverted.Thedropswillfallasbeforeinthelowerbowl,withoutasingleonedroppingtowardsthestern,although,whilethedropisintheair,thevesselrunsmanypalms.Thefishintheirwaterwillswimtowardtheforwardpartoftheirvasewithnomoreeffortthantowardthebackwardpart,andwillcomewithequaleasetofoodplacedanywhereontherimofthevase.Andfinallythebutterfliesandtheflieswillcontinuetheirflightsindifferentlytowardseveryside,norwilleverhappentofindthemconcentratedclosetothewallonthesideofthestern,asiftiredfromkeepingupwiththecourseoftheship,fromwhichthey,remainingintheair,willhavebeenseparatedforalongtime.Andifsomesmokewillbemadeburningabitofincense,itwillbeseenascendingupwardand,similartoalittlecloud,remainingstillandindifferentlymovingnomoretowardonesidethantheother.Thecauseofallthesecorrespondencesofeffectsisthatthemotionoftheshipiscommontoallthingscontainedinit,andtotheairalso.

WenoticeherethatthedevelopmentofelectromagnetisminthelastpartoftheXIXcenturyledtodoubtsconcerningthegeneralvalidityoftheprinciple.Theprocessofindepthanalysisofthephysicallawsthatfollowed,leadingtotherelativitytheory,showedthattheGalileirelativityprinciplewas,aswehavestated,validingeneral.However,itwasfoundthatthetransformationsoftheco-ordinates,andofthetime,betweenreferenceframes,validatasmallvelocityrelativetothespeedoflight,donotholdathighspeeds.WeshalldiscussthatinChap.5.Here,wesimplyanticipatetherootoftheissue.Considerthetransformationequationsthatlinktheco-ordinatesinS′andinS

(5.19)wherewehaveincludedtherelationbetweentimestandt′measuredbythetwoobservers.Indeed,themeasurementofatimeintervalshouldbe,wethink,thesameontheshoreasontheship(tocontinuetheexample).However,thisconclusion,comingfromoureverydayexperienceandfromexperimentsattheusualvelocities,iswrongatvelocitiesnottoosmallcomparedtothespeedoflight.Twoobserversintwoframesmovingatthosespeedsmeasuredifferenttimeintervalsbetweenthesametwoevents;inotherwords,tandt′arenotequal.Aconsequenceisthattherelationsbetweenco-ordinatesaredifferent

fromthoseofEq.(5.19).ThetransformationEq.(5.19),calledGalileitransformations,failathighvelocitiesandmustbegeneralizedintotheLorentztransformations,asweshallseeinChap.5.Buttherelativityprincipleremainscompletelyvalid.

5.3 Non-uniformTranslation.PseudoForcesWenowconsiderthecaseinwhichthemotionofthereferenceS′relativetoSisstillatranslation,butwithvariablevelocity.Consider,forexample,S′tobefixedonatrolleymovingonstraightrailswithanaccelerated(ordecelerated)motionrelativetoSfixedontheground.WestillconsiderthemotionofthepointPinFig.5.2asseenbytwoobserversinthetwoframes.Therelationbetweentheaccelerationsis

(5.20)Asintheprevioussection,aO′istheaccelerationinSoftheoriginofS′and

alsoofallthepointsfixedinit(itsmotionbeingatranslation).SupposenowthatSisaninertialframe.IfnoforceactsonP,itsaccelerationiszero,a=0.InS′,however,a=–aO′≠0.Namely,inS′,abodynotsubjecttoforcesmayaccelerate.Thelawofinertiadoesnothold.S′isnotinertial.Considerthetrolleyintheaboveexampleinitiallymovingatconstantspeed.Ifweputaballonahorizontalplane,itwillnotmove.Ifthetrolleynowsuddenlyslowsdown,weshallseetheballacceleratingforward,withoutanyforceacting.ThisistheinterpretationoftheobserverinS′.TheinertialobserverinSthinksthatthereisnoforceactingontheball(supposefrictiontobenegligible),andthatitisjustcontinuingitsuniformmotion(Fig.5.3).

IfaforceFactsonthepointPofmassm,theinertialobserverinSfindstherelation

(5.21)TheobserverinS′measuresthesamevalueofthemass,m′=m,thesame

force,F′=F,butadifferentaccelerationa′,andfinds ,or

(5.22)WealsoseethatthesecondNewtonlaw,notonlythefirstone,doesnothold

forthenon-inertialobserverS′.However,theobserverinS′canplayatrick.Indeed,heisaccustomedto

thinkingthatanyaccelerationwillbeduetoaforceandwillimaginethataforcehassuddenlystartedtoactontheballonthetable.Formally,thetrickisbyJean

Baptisted’Alembert(1717–1783);wecanre-writeEq.(5.22)movingm′aO′totheleft-handside,as

(5.23)andcall−m′aO′aforce,or,moreaccurately,afictitiousforce,orinertialforce

(5.24)andEq.(5.23)becomes

(5.25)Namely,ifweaddtothe“real”forcesthefictitious,orinertial,ones,were-

establishthevalidityofthe1stand2ndNewtonlaws.However,theseforcesare,aswesaid,fictitious,notreal,becausetheyarenotproducedbyanyphysicalagent.Consequently,thereisnocorrespondingreaction.The3rdNewtonlaw,theaction-reactionlaw,doesnotholdfortheinertialforces.

Letusgobacktotheexampleofasphereonatableonthetrolley.Theresultantofthetrueforces,weightandnormalreactionoftheplane,iszero.Whenthevelocityofthetrolleyisconstant,thefictitiousforceFinisalsozerobecausesoisaO′.Butwhenthetrolleyslowsdown,theobserverintheS′seestheeffectofafictitiousforceasinEq.(5.24).Itisdirectedforward,oppositetoaO′.Hecanmeasurethefictitiousforceattachingthespheretoaspringandmeasuringitsstretch.Inthisway,heverifiesthatEq.(5.24)iscorrect.

5.4 RotationandTranslation.PseudoForcesConsidernowastationaryframeS(originOandcoordinatesx,y,z)andamobileframeS′(originO′andcoordinatesx′,y′,z′),themotionofwhichiscompletelygeneral.Itmaybeatranslation,withconstantorvariablevelocity,arotation,againwithconstantorvaryingangularvelocity,orbothofthemtogether.Figure5.4representsthetwoframesatacertaintime.Atanothertime,forexample,abitlater,boththepositionofO′andthedirectionoftheaxesofS′willbe,ingeneral,different.

Fig.5.4 ReferenceframeS′movesinanarbitrarymotionrelativetoS

Webeginbyfindingaformulathatwillbeusefulinthefollowing.ConsideravectorA,whichdoesnotvarywithtimerelativetoS′.ExamplesarethepositionvectorinS′andthevelocityofapointmovinginrectilinearuniformmotionrelativetoS′.ThevectorAisnotconstantinS.Wenowfinditstimederivative.WenoticethatAvariesrelativetoSonlyindirection,notinmagnitude.Moreprecisely,Arotateswiththesameangularvelocityatwhich,inthatinstant,themobileframeS′rotatesrelativetoS.Weindicateitwithω.Noticethatωcanvaryintime,whichiswhywespecify“inthatinstant”.Undertheseconditions,thetimederivativeofAisgivenbythePoissonformulaandwehave

wherethesubscriptSspecifiesthatitistherateofchangeinthereferenceS.IfthevectorAalsovariesinS′,wehavetosumtherateofchangeinS′,and

finallywehave

(5.26)whichistheformulawewerelookingfor.Noticethatintheprecedingsections,wedidnottakecaretospecifyinwhichframeweweretakingthederivatives.Thiswasallowedbecause,beingtheconsideredtransformationstranslations,theCartesiancomponentsofthevectorswerenotmodified.ThiscanbeimmediatelyverifiedinEq.(5.26)inwhich,ifω=0,thederivativesinthetwoframesareequal.

WeshallnowfindtherelationsbetweenthekinematicquantitiesinSandinS′.Weshallcalltheformerabsoluteandthelatterrelative,butwenoticethatthedefinitionisarbitrary;wecouldhavestartedcallingS′stationaryandSmobile.

Seethattherelationbetweenthepositionvectorsisalways

(5.27)Toobtaintherelationbetweenrelative(inS′)andabsolute(inS)velocities,

weneedthetimederivatives.Todothat,weneedtohaveoneachsideoftheequationonlyvectorsinoneframe.Hence,were-writeEq.(5.27)as

(5.28)andderivethevectorr–rO′usingtherule(5.26),obtaining

(5.29)Themeaningoftheleft-handsideofthisequationisclear:itisthedifference

betweentheabsolutevelocitiesofthepointP,sayv,andofthepointO′,sayvO′.WesubstituteEq.(5.28)ontheright-handside,obtaining

(5.30)Now,weseethatthefirsttermontheright-handsideistherateofchangein

S′ofthepositionvectorinS′,namelythevelocityofPinS′,whichwecallrelativeandindicatewithv′.Wethenwrite

(5.31)Inotherwords,thevelocityvofthepointPinSisthesumofitsvelocityv′

inS′andoftwomoretermsthatwehavegroupedinvt.ThemeaningofthelatterisunderstoodconsideringthecaseinwhichthepointdoesnotmoveinS′,namelyifv′=0.Then,vtistheabsolutevelocityofthepoint.WecanthenstatethatvtisthevelocityofthepointfixedintheframeS′,andcallitQ,throughwhichthemovingpointPpassesattheconsideredtime.Wecanthinkofvtasthevelocityofthemovingspace.Itiscalledthevelocityoftransportation.Itcontainstwoterms,

(5.32)whichwediscusslookingatFig.5.5.ThefirstoneisthevelocityoftheoriginofS′andcorrespondstothetranslationalcomponentofitsmotionrelativetoS.ThesecondtermisduetotherotationofS′.WecanthinkofthisastakingplaceaboutaninstantaneousrotationaxispassingthroughO′withangularvelocity,intheconsideredinstant,ω.Indeed,thevelocityofthepointQstationaryinS′wherePispassingisjust .

Fig.5.5 Therelativevelocityintherotatingframe

Weshallnowfindtheaccelerations,byafurtherderivative.Weshallmeetmoreterms.WestartfromEq.(5.31)intheform

(5.33)andderivetheleft-handsideusingEq.(5.26),obtaining

(5.34)Similarlytoabove,theleft-handsideisthedifferencebetweentheabsolute

accelerationsofP,saya,andO′,sayaO′.Stillanalogously,weuseEq.(5.33)tosubstitutev–vO′ontheright-handside,obtaining

(5.35)

Thelasttermlooksabitcomplicated,butitstermshavewell-definedphysicalmeanings.Letusexaminethem.ThefirsttermistheaccelerationofPinS,namelytherelativeacceleration,saya′.Inthesecondterm,theangularaccelerationofthemotionofS′relativetoSappears.Weshallnameit

(5.36)Thenexttwotermsareequal.Weputthemtogetherandalsogroupsome

otherterms,writing

(5.37)whichexpressestheCoriolistheorem,afterGustavedeCoriolis(1792–1843).Wenowdefine

(5.38)whichiscalledtheaccelerationoftransportationand

(5.39)whichiscalledtheCoriolis.Finally,wewriteEq.(5.37)as

(5.40)Themeaningoftheaccelerationoftransportationatisanalogoustothatof

thevelocityoftransportationvt.Indeed,ifbothvelocityandaccelerationofPinS′arezero,thenitsabsoluteaccelerationisat,astheothertwotermsontheright-handsideofEq.(5.40)arethenzero.ThetermatistheabsoluteaccelerationofthepointstationaryinS′throughwhichthepointP(callitQagain)ispassingattheconsideredinstant.Itisthesumofthreeterms.ThefirstistheaccelerationrelativetoSoftheoriginofthemobileframeS′.ThesecondtermistheabsoluteaccelerationofQduetotherotationofS′relativetoS.ThesituationisshowninFig.5.6.Indeed,thevelocityofQ(ofpositionvectorr′)duetotherotationis .Inturn,thisvelocityvariesintime,anditsrateofchangeis,bythesameformula .ThisissimplythecentripetalaccelerationofthepointQ.Indeed,asweunderstandlookingatFig.5.6,wehave

Fig.5.6 Geometryinarotatingframe

wheredisthecurvatureradius(theradiusoftheosculatingcircle)ofthecurveQisdescribing.Andfurther

whichisthecentripetalaccelerationofQ.ConsidernowthethirdterminEq.(5.36).Iftheangularvelocityωis

constant,theabsolutevelocityofQvariesonlyindirection,andthistermiszero.Ifωisnotconstant,themagnitudeoftheabsolutevelocityofQalsovaries.Thisaccelerationisgivenbythethirdterm, .

AsfortheCoriolisaccelerationaCo,weseeinEq.(5.39)thatitiszerointhreecases:when,intheconsideredinstant,thepointPdoesnotmoveinS′(v′=0),whenthemobileframedoesnotrotate(ω=0)andwhenthevelocityofthepointPisparalleltotheangularvelocity.TheCoriolisaccelerationdoesnotdependonthepositionofPbutdoesdependonitsrelativevelocityandbecomeslargerforlargerrelativevelocities.Itisalwaysdirectedperpendicularlytothemotionandconsequentlyisacauseofchangeinitsdirection,ratherthanofitsmagnitude.Weshallseeexamplesinthenextsection.

WeshallnowassumethatSisaninertialframe.Aswehaveseenintheprevioussection,ifS′acceleratesrelativetoS,itisconsequentlynotinertial.Inotherwords,theNewtonlawsinSdonothold.Letuslookatthedetails.

IntheinertialframeS,thelawofmotionofthemassmundertheactionoftheforceFisF=ma.Thiscanbewritten,usingEq.(5.40),as

TheobserverinS′measurestheaccelerationa′andwantstohavethatontheright-handside.Wemovetheothertermstotheleft-handside,obtaining

(5.41)WegettheNewtonlawbackformallybydefiningtwofictitiousforces

(5.42)and

(5.43)whichiscalledtheCoriolisforce,andwesubsequentlyget

(5.44)Wecanthenstatethat,inaframemobilewithanarbitrarymotionrelativeto

aninertialframe,theproductofthemasstimestheaccelerationisequaltotheresultantofbothtrueandfictitiousforces.However,asalreadystated,thefictitiousforcesarenotrealandarenotduetoanyphysicalagent.Consequently,

theaction-reactionlawisnotsatisfied.

5.5 MotioninaRotatingFrameConsidernowthesimplecaseinwhichthereferenceframeS′rotatesrelativetotheinertialframeSwithangularvelocityωconstantinmagnitudeanddirection.Forexample,S′maybefixedonarotatingplatform,forexample,amerry-go-round,andSstationaryonearth.Asweshallseeinthenextsection,suchaframeisnotexactlyinertialduetotherotationofearthonitsaxisanditsrevolutionaroundthesun,buttheeffectsofthedifferencearequitesmallandweshalldisregardthemhere.

Wechoosetheoriginofbothframesinthecenteroftheplatform,theirzandz′axesverticalupwardsand,consequently,x,yandx′,y′inthehorizontalplaneoftheplatform,asshowninFig.5.7.Theaxesxandyarestationaryrelativetotheground,whilex′andy′rotate.Withourchoiceofco-ordinates,thepositionvectorsinthetwoframescoincide,r=r′.

Fig.5.7 TheSreferenceframeisstationarytotheground,S′rotateswithconstantangularvelocity

Intheparticularcaseweareconsidering,therelevantexpressionsforvelocitiesandaccelerationssimplifyin

(5.45)

(5.46)

(5.47)

(5.48)

(5.49)Letusconsiderthevelocities.Ingeneral,thepointPisnotnecessarilyonthe

platform.InFig.5.8,wehavedrawnitsomewhathigherup.Ingeneral,thevectorsωandrarenotparallel.RecallingthatvtisthevelocityofthepointQstationaryinS′intheinstantpositionofP,weseethatitistangenttothecirclethoroughPnormaltotherotationaxisandwithitscenterontheaxis.ThiscircleisthetrajectoryofQ.Themagnitudeofthevelocityoftransportationisthen

Fig.5.8 ThevelocityvtofthepointQ

(5.50)wheredistheradiusofthecircle,namelythedistancefromtherotationaxis.vtisthensimplythevelocityofQinitscircularmotion.

Wenowconsidertheaccelerations.WeimmediatelyseethattheattermissimplythecentripetalaccelerationofthepointQasseenintheinertialframeS.

LetusnowconsiderapointPofmassmtobestandingstill,relativetoS′,ontheplatformatthedistancerfromtheaxis.SupposethatthefrictionisnegligibleandthatPiskeptinpositionbyarubberbandattachedtoasmallringaroundtheaxis.

TheinertialobserverinSseesPmovinginuniformcircularmotionwithvelocityωr.Heknowsthatthemotionhasanaccelerationtowardsthecenter,thecentripetalacceleration,ofmagnitudeω2r(thisistheabsoluteaccelerationinthiscase).The(centripetal)forcecausingtheaccelerationisduetotherubberband.Theobservercancheckthatmeasuringthestretchoftherubberband.

Thenon-inertialobserverinS′,ontheplatform,alsoseesthattherubberbandisstretched,determiningthatacentripetalforceisactingonP.Hemeasuresitandfindsthesameresultastheinertialobserver.ThemobileobservernowinsistsonhavingthefirstNewtonlawbevalidandconcludesthatasecondforce,equalandoppositetothatoftherubberband,mustexist.Thisistheinertialforce,duetotheaccelerationoftransportation,–mat,thedirectionofwhichisoppositetothecentripetalforce.Inthiscase,theforceiscentrifugal.

Inthiscase,andalways,thecentrifugalforcesarenotrealforces,butpseudoforcesoftherelativemotion.Theyappearonlywhenwepretendtodescribethemotioninanon-inertial,rotatingframeasifitwereinertial.However,thecentrifugalforceisfeltasarealforce,suchas,forexample,inafastrotatingmerry-go-round.

WenowdiscusstheCoriolisacceleration(Eq.5.49)andtheeffectsofthecorrespondingfictitiousCoriolisforce

(5.51)ConsideragainthepointPlyingontherotatingplatform.IfPdoesnotmove

relativetotheplatform,theCoriolisaccelerationisnull,asinthecasejustdiscussed.Letv′bethisvelocity,whichweassume,forsimplicity,tobeparalleltotheplatform.Aswehavealreadynoticed,theCoriolisacceleration,andconsequentlytheCoriolisforce,doesnotdependonthepositionofPontheplatformandisinanycaseperpendiculartotherelativevelocity.ConsiderFig.5.9.Iftheangularvelocityωisdirectedoutoftheplaneofthefigure,asinFig.5.9a,weseetheplatformturningcounter-clockwise.Inthiscase,theCoriolisaccelerationisdirectedtowardstheleftofthemotion,andtheCoriolisforcetotheright.SupposeyouarethepointPwakingorrunningontheplatform.Youwillfeelapushtotherightofyourspeed,inwhateverdirectionyoumove.Contrastingly,ifωisdirectedinsidethedrawing,asinFig.5.9b,andtherotationisclockwise,theCoriolisforcepushestotheleftofthespeed.

Fig.5.9 Coriolisaccelerationand(pseudo)forceonaplatformrotating.aCounter-clockwise,bClockwise

Ifweweretolookattheearthfromsomedistancefromitssurfaceontheaxis,wewouldseethenorthernhemisphererotatingcounter-clockwiseifwewereabovetheNorthpole,andthesouthernoneclockwiseifwewereabovetheSouthpole.TheCoriolisforcesarethedominantcausesofthecirculationof

windsintheatmosphereandcyclonicandanticyclonicphenomena.Weshalldiscussthatinthenextsection.

Considernowanotherexample,namelyamaterialpointP,standinginequilibriumabovetheplatforminafixedpositionrelativetoS,i.e.,totheground.Wemightthinkaboutaflylocatedjustabovetheplatform.TheobserverinSseesPatrest.Knowingthatitissubjecttoitsweight,heunderstandsthatanotherforce,equalandoppositetotheweight,shouldexist.Theforceisexertedbythebeatingofthefly’swings.

FortheobserverinS′,thedescriptionismorecomplicated.HeseesPmovinginacircularuniformmotiononacircleofradiusrwithvelocityωr.Themotionisacceleratedwithacentripetalaccelerationω2r.Hededucesthataforcemω2rshouldactonthefly.However,healsoknows,astheresultofexperimentshehasdoneinthepast,suchastheonewejustdiscussed,thatacentrifugalforceexistsontheplatform,namelyaforceofmagnitudemω2rdirectedoutwards.Consideringthatthepointmovesonacircle,heconcludesthatthecentripetalforceontheflymustbetwiceaslarge,namely2mω2r.Fromwhereisthisforceiscoming?ItistheCoriolisforce.Inthiscase,ωandv′aremutuallyperpendicular;Eq.(5.49)saysthatthemagnitudeofthisforceisjust2mω2randthatitsdirectionisradial,towardsthecenter.Physicsisdifficultinnon-inertialframes,butthefactortwoisneeded!

Asafinalexample,letusgobacktothefirstone,inwhichthepointPiskeptstillontheplatformbyarubberbandattachedtotheaxis.ThemotionseenbySiscircularuniform.Atacertaininstantwhenwecuttheband,SwillseePslidingontheplatformofastraightuniformmotionatthevelocityithadatthemomentofthecut,directedasthetangenttothecircleinthatmoment.Indeed,thereisnonetforceactingonP.

HowdoestheobserverinS′describethemotion?Tobeconcrete,assumetherotationtobecounter-clockwise.Whentherubberbandiscut,theforcethatisneededintherotatingsystemtokeeptheobjectsstandingdisappears,andwemightexpecttoseethepointPmovingoutsidealongtheradiusoftheplatform.Butthisisnotwhatweobserve;rather,thepointmovesoutsidedescribingacurve.ThereasonistheCoriolisforce.Beforetherubberbandwascut,Pdidnotmoveontheplatform,andtheCoriolisforcewasnull,butitisnotsoanylongersincePhasstartedmoving.TheCoriolisforceacts,pushingPtotherightallalongitstrajectory.Observingfromoutside,wecanbetterunderstandwhatisgoingon.Whentherubberbandiscut,Pmoveswiththesamevelocityasthepointoftheplatformonwhichitisseated.Whilemovingoutwards,Preachespointsoftheplatformhavinghigherspeeds,becausetheyarefartherfromthe

axis,andconsequentlyisleftbehindbythem.

5.6 TheInertialFrameAswehavealreadystated,areferenceframeisdefinedasinertial,ifinthatframethefirstNewtonlawisvalid.Wehavealsoseenthatifareferenceframeisinertial,anyotheronemovinginuniformtranslationmotionrelativetoitisalsoinertial.Indeed,therelativityprinciplewesawinSect.5.2statesthatnoexperimentcandistinguishbetweenthem.Inotherwords,thereisnoabsolutereferenceframe.Finally,wehaveseenthattheNewtonlawsarecovariantundertheGalileitransformations.

However,naturedoesnotnecessarilybehaveaccordingtoourdefinition,andinertialreferenceframesmightjustnotexist.Theanswermustcome,asalways,fromtheexperiment.Basically,weneedtocheckifwecanfindonereferenceframeinwhichmaterialpointsnotsubjecttoforces,or,betteryet,subjecttoforcesofnullresultant,alwaysmoveinarectilinearuniformmotion.Asamatteroffact,asisoftenthecaseinphysics,weproceedthroughsuccessiveapproximations.Wecanfindreferenceframesthatcanbeconsideredinertial,withinacertainapproximation,namelyforexperimentsofacertainsensitivityorprecision.Formorepreciseexperiments,wemustsearchforframesthatareclosertotheinertialone,andwecanfindthem.

Indeed,thelargestfractionoftheexperimentstakesplaceonearth,andisdescribedinastationaryframerelativetothewallsofourlaboratory.Theseframescanbeconsideredinertialwithinaquitegoodapproximation,althoughnotperfect.Indeed,earthrotatesonitsaxis,makingaturn(2πangle)inaday(84600s).Thecorrespondingangularvelocity,directedfromtheSouthtotheNorthpole,isωrot=7.3×10–5s–1.Figure5.10a,forexample,showsastationaryreferenceframeonearthatacertainlatitudeλ.Inthisframe,thetransportationandCoriolisaccelerationarepresent.

Fig.5.10 Threereferenceframeswith,accelerationtowardsatherotationaxisofeartha1=2.4×10–2

ms–2,bthesuna2=5.9×10–3ms–2,cthecenteroftheGalaxya3=10

–10ms–2

Letusanalyzethefirstone,towhichthecentrifugal(pseudo)forcecorresponds.ThemagnitudeofthisforceonapointPofmassmistheproductofthemass,thesquareoftheangularvelocity(equaleverywhereonearth)andtheradiusofthecircleonwhichPmoves.Thelatteristhedistancefromtheaxis,Rcosλ,whereRistheearthradiusandλisthelatitudeofP.Callinga1theacceleration,themagnitudeoftheforceis

(5.52)Letuslookatthenumbers.RecallingthatR=6.4×106mandtaking,for

example,λ=45˚,theacceleration,whichisalsotheforceperunitmass,is

(5.53)whichisquitesmall,lessthanapermilleofthegravitationalacceleration.However,forprecisemeasurements,itcanberelevant.TheCoriolisforceisusuallysmaller.However,itisimportantforlarge-scalephenomena,asweshallseeinthenextsection.

However,astationaryreferenceframeonearthdiffersfromaninertialframeforasecondreason,toevensmallereffect.Indeed,earthmovesalongitsorbit,turningaroundinayear,withanangularvelocityofωriv=2×10–7s–1onanorbitofradiusRorb=1.49×1011m(Fig.5.10b).Thecentripetalaccelerationis

(5.54)whichisanorderofmagnitudesmallerthana1.Theeffectsofthecorrespondingpseudoforcearenegligible,ifnotforthemostprecisemeasurements.Usually,theCoriolisforceisevensmaller.

Eventhesesmalleffects,however,canbeeliminatedbychoosingareferenceframewithitsorigininthesunanddirectionsoftheaxesstationarytothefixedstars.Thisframeisinertialtoanextremelygoodapproximation,althoughnotperfect.Indeed,thesunislocatedattheperipheryofourspiralgalaxy(1011starsinorderofmagnitude).ThesunturnsaroundthecenterofthegalaxyinanorbitofradiusRS≈2.4×1020moveraperiodofabout150millionyears,correspondingtotheangularvelocityofωS=7.9×10–16s–1.Thecorrespondingcentripetalaccelerationis

(5.55)Thisisverysmallindeed.However,experimentsexistthataresosensitive,

theyareabletodetectdeviationsfromthestateofinertiaevenattheseextremelysmalllevels.Asamatteroffact,ourgalaxymovestoo,inanon-uniformmotion.However,whenneeded,weknowhowtoeliminatetheeffects.

Inconclusion,inertialreferenceframesexistinnatureateverylevelofapproximationweneed.

5.7 Earth,asaNon-inertialFrameAswejustsaw,therotationofearthonitsaxis,withtheangularvelocity,ωrot=7.3×10–5s–1.ThisimpliesthatinreferenceframestationaryonearthdynamicaleffectsofthetransportationandCoriolisfictitiousforcesexist.Weshalldiscusstheprincipalonesinthissection.

WetakeareferencesystemSwiththeorigininthecenterofearthandstationarywithit.Weshallusethesymbolsvandaforvelocitiesandaccelerationsinthisframe,omittingtheprimeweusedintheprevioussections.

Theaccelerationoftransportationis

whereaOistheaccelerationoftheearth’scenter.Thisisthecentripetalaccelerationofitsmotionaroundthesuninaverygoodapproximation,andrEistheradiusoftheorbit,asshowninFig.5.11a.TheCoriolisaccelerationonapointmovingwithvelocityvinSis

Fig.5.11 aForcesandpseudoforcesonmatterpointP;bDisplacementtoeastinthefreefall(exaggerated)

InS,theequationofmotionofapointwithmassmsubjecttotherealforceFtrueisthen

(5.56)WecandistinguishthefollowingcontributionstothetrueforceFtrue:the

gravitationalattractionofearthFE,thegravitationalattractionofalltheotherheavenlybodiesFO,andofanyotherforcethatmightbepresent(airresistance,tensionofawire,etc.),withresultantF.Were-writeEq.(5.56),groupingthetermsaccordingtotheircauses,

(5.57)ThegravitationalforceFOisduetoalltheheavenlybodiesdifferentfrom

earth,butislargelydominatedbythesun.Asthediameterofearthismuchsmallerthanthedistancefromthesun,inafirstapproximation,wecanconsiderFOequalinallthepointsoftheearth.However,thesmalldifferencesthatarepresentareoneofthecausesofthetides,asweshallseeinSect.6.4.TheaccelerationproducedbyFOoneverybodyisproportionaltothemassofthebody.Consequently,itisthesameonthesurfaceoftheearthandinitscenter.Inotherwords,itistheaccelerationaO,oftheearthherself.Hence,FO–maO=0.

Wehavereachedanimportantconclusion,whichistrueaslongasFOcanbeconsiderednottovaryonthepointsoftheearth,thatthegravitationalforcesofthesun,themoonendoftheotherheavenlybodiesdonotappearintheequationsofmotioninreferenceframesstationaryonearth.Theseforcesareexactlybalancedbytheinertialforcesresultingfromtheaccelerationthatthoseagentsimparttotheearth.

WecansimplifyEq.(5.57)as

(5.58)Now,wearereadytoconsiderseveralimportantexamples.Thefirstcaseisofabodyatrest,andFissimplyitsweight.Thisistheforce

wemeasurewithabalanceandthatwehavewrittenas

(5.59)wheregisavectorquantity,whichisequalforallthebodiesinagivenposition.Uptonow,wehavetalkedofitasgravitationalacceleration,butwearenowreadytoseethatitisonlyapproximatelyso.Equation(5.58)indicatesthattheforcepushingabodydownwardsthatdoesnotmove(v=0,a=0)is

.Wecansaythatthegravitationalforceoftheearthonthebodyis

(5.60)andwrite

(5.61)whereGisthegravitationalfieldofearth,and

(5.62)Theaccelerationisthesameforallthebodiesinthesamelocation.

Equation(5.58)showsthatabodydroppedinabsenceofanyforceotherthanitsweight,fromapositionofrest,v=0,moveswithanaccelerationa=g.Wecansaythatgistheaccelerationofthefree-fallofanybody,provideditsvelocityisnullintheconsideredinstant.Ifv≠0,theCoriolisaccelerationis,ingeneral,presenttoo.

Inanycase,Eq.(5.61)tellsusthattheweightisthesumoftwocontributions:thegravitationalattractionmGoftheearth,whichlargelydominates,andthecentrifugalforceduetotherotationofearth,whichismuchsmallerandvarieswiththeposition.Wewillnowdiscusstheobservableconsequencesofthat.

Thelocalvalueofg.Supposewetakeaplumbandfixitatasupport.Intheequilibriumposition,itsweightFw,givenbyEq.(5.59),andthetensionofthewireareequalandopposite.Thedirectionisgivenbythewire.ThedistancefromtherotationaxisofapointPonthesurfaceatthelatitudeλisrE=Rcosλ,whereRistheearthradius(Fig.5.11a).TheweightFwcanbedecomposedinacomponent,letuscallitFw,r,directedtothecenterofearth,andacomponent,Fw,θ,inthedirectionofthemeridian,totheNorthinthenorthernhemisphereandtotheSouthinthesouthernone.Thetwocomponentsare

(5.63)

Thecentrifugalterm,thefirstone,iszeroatthepolesandmaximumattheEquator.ThetangentialcomponentiszerobothatthepolesandattheEquator.Intheselocations,butnotelsewhere,theweightispreciselydirectedtothecenterofearth.Asforthemagnitude,themeasuredvaluesareg=9.832ms–2atthepolesandg=9.780ms–2attheEquator.Ifweapproximatetheshapeoftheearthsurfacewithasphere,allitspointsareatthesamedistancefromthecenter,andifthemassdistributioninsidetheearthissphericallysymmetric,thegravitationaltermGisequaleverywhere.Itshouldbeequaltogatthepoles,G=9.780ms–2.Letuscheckbygivinganestimate,startingfromgattheEquator.

Thisvalueisclose,butstillabitsmallerthanwhatwefoundfromgatthepoles.Themainreasonforthatisthatearthisnotreallysphericalbutsomewhatsqueezedatthepole,aneffectofthecentrifugalforces.Consequently,thepolesareabitclosertothecenterthantheEquator.

Noticehowever,thatsmalldifferencesonthevalueofginthedifferentpointsofthesurfacearepresent,duetothelocalgeology.

Absenceofweight.Ifwemeasuretheweightofanobjectwithabalanceonthespacestation,wefindittobezero.Suchisalsotheweightofalltheobjectsinthestation,andineveryartificialsatellite.Theargumentswejustmadearestillvalid,ifweputthestationintheplaceofearth,andconsidertheearthasanexternalbody,asthesun,themoonandtheotherplanetsare.Thespaceshipissmallenoughforthegravitationalforceofthosebodiestobeconsideredequalatallthepointsoftheship.Thisforceisexactlybalancedbytheinertialforcetotheaccelerationofthespaceship.Ifitsenginesareshot,theshipfreelyfallsundertheactionofgravitation.Inthiscase,theequivalentoftheweightonearth,namelythegravitationalattractionoftheshiponthebodyinsideit,iscompletelynegligible,Fw=0.Thecentrifugaltermtotheweightinthespaceshipisalsonegligiblebecausetheshipdoesnotrotateappreciably.Theweightintheshipiszero.

Eastwardsshiftinthefree-fall.IfamaterialpointPofmassmisdroppedwithnullinitialvelocityataheighthfromtheground,itinitiallyfallsundertheactionoftheweight,Fw.However,assoonasitsvelocity,v,isappreciablydifferentfromzero,asecondinertialforce,theCoriolisforce,entersintoaction.Itis

(5.64)Thevelocityvrelativetoearthisintheplanecontainingtheearth’saxisand

pointP,namelytheplanePONinFig.5.11a.Consequently,theCoriolisforceisperpendiculartothisplane.ConsideringthatthedirectionoftheangularvelocityisfromSouthtoNorth,andthatvisdownwards,weseethattheCoriolisforceistowardEastinbothhemispheres.ThesituationisshowninFig.5.11b,whereABisthedirectionoftheplumb,i.e.,thedirectionofFw(noCoriolisforceontheplumbthatdoesnotmove)andCisthepointinwhichthebodyreachestheground,fallingfromtheheighth.TheshiftfromtheverticalBCisverysmall,andexaggeratedinthefigure.Letuscalculateit.

Wetakeareferencewiththez-axisvertical,i.e.,inthelocaldirectionoftheplumb,andthex-axishorizontaltowardstheEast.Withinagoodapproximation,

wecantakethemagnitudeofthevelocitytobeυ=gt,asintheverticalfall.Itsdirectionisoppositetothez-axis.Theequationofthecomponentofthemotiononthex-axisis

Wesolvetheequationbyintegratingtwiceontimeandimposingtheinitialconditionsx(t)=0,(dx/dt(0))=0,obtaining

(5.65)

Thetimeofthefallis,withgoodapproximation, ,andwehave

(5.66)

Forexample,atthelatitudeof45˚andafallfromh=50m,theeastwardshiftisx~5mm,whichisquitesmall,buthasbeenmeasured,carefullyeliminatingperturbingeffects.

Horizontalwindcirculation.Asiswellknown,theearth’satmosphereinacertaininstantcontainszonesofhighpressureandzonesoflowpressure.Naively,onewouldexpectwindstoblowfromtheformertothelatterinthedirectionofthepressuregradient.However,thedirectionofthewindsissubstantiallyperpendiculartothat,movingalongtheisobars,asyoucanseewatchingweatherforecastsonTV.TheeffectisduetotheCoriolisforce.

Figure5.12summarizesthesituation.Histhepressuremaximum,Lapressureminimum,intheNorthernhemisphere.Hence,theearth’sangularvelocitydirectionisoutofthepaperandtheCoriolisforceisdirected,perpendiculartothevelocity,totheright.Consider,forsimplicity,ahorizontalwindatconstantvelocity(inmagnitude).Supposeweinsulateasmallmassofairwithinanidealfilmandfollowitsmotion.Twoverticalandtwohorizontalforcesactonourmass.TheverticalonesaretheweightandtheArchimedesforce.Asthemotionishorizontal,theyareequalandopposite.Thehorizontalforcesarethepressure(true)forceandtheCoriolis(pseudo)force.Thepressureforceactsonthesurfacesofourgasmass.Thepressureonitsleft-handfacepushestotheright,whilethepressureontherightfacepushestotheleft.Ifthepressurewereequalonthetwosides,theneatforcewouldbenull.However,ifthereisapressuremaximumontherightofthegasmasswearefollowing,asinFig.5.12a,thereisaneatpressureforceF(P)pushingtotheleft.TheCoriolisforcehasanequalandoppositedirection.Consequently,thetwoforcesmay

balanceeachother,orresultintherightvaluebeingthecentripetalforceforthecurvatureofthewindtrajectory.Thiscanhappenonlyifthewindcirculatesinacounter-clockwisedirectionaroundapressuremaximum(anticyclone).Contrastingly,itmustcirculateclockwisearoundaminimum(cyclone),asinFig.5.12b.Thetwosituationsareinvertedinthesouthernhemisphere.

Fig.5.12 Isobarsaroundpressuremaximum(left)andminimum(right-hand),intheNorthernhemisphereandtheforcesonamassofair

Letuslookattheordersofmagnitudes.ThemagnitudeoftheCoriolisforceonanairmassmmovingwithhorizontalspeedυatthelatitudeλis

(5.67)Forexample,theforceonakilogramofair,whichisabout1m3,movingat

10m/sat45˚isabout10–3N.Thisshouldbecomparedtothepressureforcesonthesamevolume.Tobeofthesameorderofmagnitude,thepressureforcesontwooppositesidesofourcubicmetervolumeshouldbedifferentby10–3N.Thiscorrespondstoapressuredifferenceof10–3Pa,beingthesurfaceunitary.Hence,thepressuregradientshouldbeof10–3Pa/m,corresponding,say,toadistanceof100kmbetweentwoisobarsof100Padifference.Thisisreasonable(havealookattheweathermaps).

TheFoucaultpendulum.Asimplependulumabandonedinanon-equilibriumpositionwithnullvelocityoscillatesinaverticalplane.However,ifwewatchcarefullyforalongenoughtime,alongtheorderofonehour,wecanseethattheoscillationplanerotatesrelativetothelaboratory,i.e.,relativetoareferencefixedonearth.Thereasonfortherotationis,oncemore,thattheframeisnotexactlyinertial.Asamatteroffact,theoscillationplaneisfixedinaninertialframe,relativetowhichtheearthrotates,asinFig.5.13.

Fig.5.13 TheFoucaultpendulum

WhiletheeffecthasbeenknownsinceitsfirstobservationbyVincenzoViviani(1622–1703)in1661,themainexperimentanditscorrectinterpretationweredonebyLéonFoucault(1819–1868)in1851inthePantheonofParis.Hispendulumwas67mlongandhada28kgmass.

Asimilarsituation,showninFig.5.14,helpsinourunderstanding.There,wehaveapendulum,supportedonaturningplatform.Ifweputthependuluminoscillationandtheplatforminrotation,weobservetheoscillationplaneremainingfixed,asexpected,andtheplatformrotatingunderthependulum.Wecaneasilyimaginewhatanobserverontheplatformwouldsee,namelytheplaneofoscillationrotatingintheoppositedirection.

Fig.5.14 Pendulumonarotatingplatform

Inthisway,weeasilyunderstandwhathappensonearthifweareonapole.Here,theangularvelocityvectorωrotisnormaltothe“platform”,theearth

surface,exactlyasinthatexperiment.Fromthepointofviewofaninertialobserver,theoscillationplaneisconstant,andheseestheearthturningrelativetoit.Heunderstandswhyanobserveratthepoleseestheoscillationplanerotatingandmakingacompleteturnin24h.Inthereferencefixedtoearth,theequationofmotionis,oncemore,(5.58),withFthetensionofthewire(neglectingairresistance).TheCoriolisforce isnormaltotheoscillationplane,andcausesitsrotation.

Iftheexperimentisdonenotatthepolebutatalatitudeλ,wemustpayattentiontothevectorcharacteristicofωrot.Wedecomposeitinahorizontalcomponent,namelyparalleltothegroundinourposition,ωh,andaverticaloneωv:ωrot=ωh+ωv.Wefurtherdecomposethehorizontalcomponent,whichisstillavector,initscomponentsparallel,ωp,andnormal,ωn,totheoscillationplaneandwriteEq.(5.56)as

(5.68)Theterm hasthedirectionofthewire.Itseffectistochangethe

tensionabit,butithasnoeffectontheoscillationplane.Theterm isperpendiculartotheoscillationplaneandcausesitsrotation.Thethirdterm

isalsoperpendiculartotherotationplane,butisverysmall.Indeed,aswecanseeinFig.5.13b,itisproportionaltosinθ,whereθistheanglebetweenthewireandtheverticalandissmall,forsmalloscillations.WecanthensimplifyEq.(5.68)bywriting

(5.69)Inconcussion,themotionissimilartothatatthepolewiththesole

differencebeingthatinplaceoftheangularvelocityω,wemustconsideritscomponentalongthelocalvertical,ofmagnitude

(5.70)Theoscillationplanemakesacompleteturnintheperiod

(5.71)At45˚latitude,inonehour,theplanerotatesby10.6˚.Figure5.13cshowstheprojectiononthehorizontalplaneofthetrajectoryof

theFoucaultpendulum.Thevectorωvisnormaltothedrawingtowardstheobserver.TheFoucaultforceisalwaysdirectednormallytothevelocitytotherightofthedirectionofmotion.Theforcebendsthetrajectory,asshownwithexaggerationinFig.5.13c.SupposethatthependulumisinitiallyinAandabandonedwithnullvelocity.Initially,whentheCoriolisforceisverysmall,the

pendulumheadstoA′.Butassoonasthevelocitybecomesappreciable,theCoriolisforcepushestotheright,bendingthetrajectory.ThependulumreachespointB,whereitstops.Whenthevelocityhasagainsufficientlyincreased,butintheoppositedirection,theCoriolisforcepushesintheoppositedirectiontoo,althoughstilltotherightofthemotion.ThependulumreachesC,etc.

IntheFoucaultexperiment,thelengthofthependulumwaslarge,l=67m,correspondingtoaperiodT=16.4s.Withsuchalongperiod,thelateralshiftcanalreadybeobservedinasingleoscillation.TheoscillationamplitudewasA=3m.AttheParislatitude,sinλ=0.753andtherotationperiodisTrot=3.8h=14480s=31,8h=14480s.InanoscillationperiodT,theplanerotatesattheangle2πT/Trot.Hence,theshiftoftheoscillationextremeinoneperiodiss=2πAT/Trot=2.7mm.

Moreover,thelengthisimportantforanotherreasontowhichwecanonlyhint.Inpractice,ithappensthatthestressforcesalwayspresentinthewireandinthehooksupportingthependulumresultinaspuriousrotationoftheoscillationplane.Theeffectisslow,butimportantforobservationsofseveralhours.Itcanbeshown,however,thatitissmallerforlongerlengths.

5.8 TheEötvösExperimentInSect.2.5,wehaveseenhowGalileiandthenNewtonexperimentallyestablishedtheidentitybetweeninertialandgravitationalmass.Thisisaveryfundamentalissue,andexperimentshavebeendone,andarestillbeingdone,toincreasetheprecisionwithinwhichtheequalityisverified.WediscussherethebeautifulexperimentsconductedbytheHungarianphysicistLorándEötvös(1848–1919)inthelastyearsoftheXIXcentury.

Inthischapterwegaveanumberofexamplesoftheeffectsoftheinertialforces,thepseudoforcesthatappearinnon-inertialframes.Theinertialforcesactingonamassareproportionaltothemass,justlikethegravitationalforce.Thereisanimportantdifference,however,asinertialforcesareproportionaltotheinertialmassmi,andthegravitationalforceisproportionaltothegravitationalmassmg.Supposetheratiobetweenthetwotypesofmasstobedifferentfordifferentsubstances.Wecouldthenhangspheresmadeofthetwosubstancestotwowiresandlookforanysmalldifferenceinthedirectionsofthewires.Inthissection,weshallusedifferentsymbols,miandmg,forthetwotypesofmass.

ConsiderabodyhangingfromawirefixedinΩ,asinFig.5.15,atapointat

thelatitudeλ.ThedistancefromtheaxisisrERcosλ,whereRistheearth’sradius.Thecentrifugalforcehasadirectionperpendiculartotheaxisoutwardsandmagnitude

Fig.5.15 ThebasisoftheEötvösexperiment

(5.72)andthegravitationalforce

(5.73)Iffortwosubstances,miandmgaredifferent,theanglebetweenthetwo

forcesisalsodifferent,andsoisthedirectionofthewire.AswesawinSect.5.6,thecentrifugalaccelerationontheearth’ssurfaceisoftheorderofthepermilleofthegravityacceleration.Correspondingly,thesought-aftereffectscanbeverysmall.

TheEötvösexperimentdirectlycomparestheanglesofwirestowhichspheresofdifferentsubstancesareattached.Thetwowiresareattachedtotheextremesofarigidbar.Thebarissuspendedbyametalwirethatactsasatorsionbalance,asshowninFig.5.16,similartowhatwedescribedinSect.4.7.

5.1.

5.2.

Fig.5.16 TheschemeoftheEötvösexperiment

Figure5.16ashowsthesysteminperspective,withFig.5.16blookingatitparalleltothebar.Iftheratiomi/mgisdifferentforthetwospheres,thedirectionsαandβofthetwotensionsareabitdifferent.Thisproducesamomentonthebar,duetothehorizontalcomponentsofthetwotensions,thatrotatesitaboutthewirefromwhichithangs.Underrotation,thewiredevelopsanelasticmoment,whichincreaseswiththeangle.Attheequilibriumangle,thetwomomentsareequalandopposite.Measuringtheangle,thetorsionbalancegivesthemoment.

TheresultoftheverysensitiveEötvösexperimentwasnull,allowinghimtogivetheupperlimit ,namelythatthedifference,ifany,islessthan5partsperbillion.AnexperimentofthesametypebyRobertHenryDicke(1916–1997)inthe1960sestablishedtheevensmallerlimitof

.

5.9 ProblemsAkidsitsinacarriagemovingonstraightrails.(a)Ifthespeedofthecarriageisconstant,inwhichdirectionshouldhelaunchaballtotakeitbackinhishandwithoutmoving?Inwhichdirectionifthecarriageacceleratesforwards?

Atraintravelsonstraighthorizontalrailsatthevelocityυ0=30m/s.

5.3.

5.4.

5.5.

5.6.

5.7.

Reachingastation,itsstops,withconstantacceleration,ins=150m.Asuitcaseofmassm=10kgliesonthefloor,withdynamicalfrictioncoefficientµd=0.20.Duringthebraking,itslidesalongthecorridor.(a)Howmuchisitsaccelerationrelativetothegroundduringthistime?(b)Whichisthevelocityofthesuitcasewhenthetrainstops?(c)Afterthetrainstops,thesuitcasecontinuestoslideforawhileandthenitselfstops.Whichwasthetotaldisplacementofthesuitcaseonthefloor?

Amanmeasureshisweightinalift,whichisatrest,usingaspringandbalance,andfindsittobe700N.Withtheliftmoving,herepeatsthemeasurementandfindsittobe500N.Whatcanhedetermineabouttheliftacceleration?Andaboutitsvelocity?

Apersonsitsinachairstandingontheplatformofamerry-go-round,whichisturning.Thepersonholdsaplumb.Drawseparateforcediagramsfortheplumb,thewire,theperson,thechairandtheplatform.Describeeachoftheforcesinwords.Identifytheactionreactionpairs,bothforaframestationaryonearthandforonestationaryontheplatform.Inthelattercase,specifywhichoftheforcesareinertial.

Akidsitsonamerry-go-roundthatturnsatangularvelocityω,whilehisfriendisontheground.Theresultantoftheforcesonthelatteriszero.(a)Whatisthemotionofthesecondkidseenbythefirst?(b)Whatishisacceleration?(c)Whataretheforcescausingit?

Anoldvinyldiskrotatesat33turnsperminute.Itsradiusisr=15cm.Aninsectwalksfromthecentertowardstheborder.Willitbeabletoreachitifthestaticfrictioncoefficientisµs=0.1?

Atennisplayerat45˚latitudeisimpartingtotheballaspeedof100km/s,whichweassumetobeinitiallyhorizontal.Willingtohitgroundata

distanceof50m,shouldhetakeintoaccounttheCoriolisforce?

(1)

©SpringerInternationalPublishingSwitzerland2016AlessandroBettini,ACourseinClassicalPhysics1—Mechanics,UndergraduateLectureNotesinPhysics,DOI10.1007/978-3-319-29257-1_6

6.Relativity

AlessandroBettini1

DipartimentodiFisicaeAstronomia,UniversitàdiPadova,Padova,Italy

AlessandroBettiniEmail:alessandro.bettini@pd.infn.it

IntheprecedingchapterswehaveseenhowtheGalilei-Newtonmechanicsisabletodescribewithsimplelawsanenormousnumberofphenomenabothattheeverydayscaleandatcosmiclevel.Newtonianmechanicsisoneofthemajorconceptualconstructionsofhumangenius.However,thevalidityofthetheoryislimitedontwosides,onthesideofhighvelocitiesandonthesideofsmalldimensions.

Newtonianmechanicsisnolongervalidfornotverysmallvelocitiescomparedtospeedoflight.Thelatterisverylarge,about3×108m/s.Thevelocitiesofalltheobjectswehavetodealwithonearth,ofplanetsandofthemajorityofheavenlybodiesareverysmallincomparison.Thevelocityoftheearthandtheplanetsaroundthesun,forexample,areoftheorderofoneintenthousandofthespeedoflight.Consequently,Newtonianmechanicsgivesanextremelygoodapproximationforthesephenomena.Inthischapterweshallseehowthetheoryneedstobeextendedatspeedscomparablewiththespeedoflight,inrelativisticmechanics.

Classicalmechanics,asarecalledboththeNewtonianandrelativisticmechanics,ceasetoapplyforobjectsatmolecularorsmallerscales,oftheorder,say,ofnanometers.Theseareonethousandtimessmallerthanmicrobes.Thecorrecttheory,validatallordersofmagnitudeisquantumphysics.Classicalmechanicsisthelimitofquantumphysicsforsufficientlylargedimensions.Inthisbookweshallonlywarnthereaderofthelimitsofclassicalmechanicswhen

needed.Aswehavejuststated,inthischapterweshallstudythefundamental

principlesofrelativisticmechanicsandofthehighvelocityphenomenaitdescribes.InthelastpartoftheXIXcenturyMaxwellformulatedasetofequationsthatcompletelydescribewithgreataccuracyalltheelectromagneticphenomena.However,theseequationsseemedtobeincontrastwiththerelativityprinciple.Ifitwassoitwouldhavebeenpossibletoexperimentallyfindanabsolutereferenceframe.Astronomicalobservationsandaccurateexperiments,culminatedintheexperimentbyMichelsonandMorleyin1887(describedinSect.6.2),showedthatwasimpossible.Anindepthcriticismofthefundamentalconcepts,inparticularonthemeasurementoftime,followed,leadingtodevelopmentofspecialrelativitymainlybyA.Lorentz,H.PoincaréandA.Einstein,whichwassubstantiallycompletedin1905.

Therelativityprinciplewasfoundtobeuniversallyvalid,butnewtransformationsofcoordinatesandtime,validbetweentwoinertialframes,theLorentztransformationswereestablished.TheLorentzcovariance,firstestablishedforelectromagnetism,mustbevalidforallphysicslaws.Weshallemphasizethispointafterhavingfollowedthehistoricalpath.

InthesectionsfromSects.6.4to6.6weshallstudyhowtheconceptsofsimultaneity,timeintervalanddistancemustberevised.InSect.6.7weshallfindthelawofadditionofvelocities,whichshows,inparticular,thatthespeedoflightisthelargestpossibleone.Spaceandtimebecomecompletelycorrelatedconceptsandshouldbeconsideredasasinglefour-dimensionalmanifold,space-time,whichweshallstudyinSect.6.8.

InSects.6.9and6.10,weshalldiscussrelativisticdynamicsandseehowthefundamentalconceptsofmass,linearmomentumandenergychange.InSect.6.11weshallfindtherelativisticformofthesecondNewtonlaw.

Asalreadymentioned,allthephysicslawsshouldberelativisticallycovariant.Thisinfactisthecase.WeshallgiveafewhintsonthatinSect.6.12.

Finally,inSect.6.13weshallgiveasummaryofthedifferencesandequalitiesbetweenNewtonianandrelativisticmechanics.

6.1 DoesanAbsoluteReferenceFrameExist?InChap.5westudiedthecovarianceofphysicallawsundertransformationbetweentworeferenceframes.Wehaveseenthatinertialframesexist,whicharebydefinitiontheframesinwhichtheinertialawisvalid.InsuchframesalsothesecondandthirdNewtonlawshold.Inthischapterweshallconsideronlyinertialframes.Formally,wecanstatethatthephysicslawshavethesameform

1.

2.

3.

intwoinertialframesinthefollowingcases.Thetwoframeshavenorelativemotionbutdifferforatranslationorarotationoftheaxes,orthetwoframesareinarelativeuniformtranslationmotion.Inotherwordswehavenomeansforexperimentallyobservingdifferencesbetweenonelocationoranother(invarianceundertranslations,thespaceishomogeneous),orbetweenonedirectionoranother(invarianceunderrotations,thespaceisisotropic),or,toestablishwhetheraframeismovinginauniformtranslationmotionornotrelativetoanotherinertialsystem.ThelatterpropertyistherelativityprincipleestablishedbyGalilei.

Weobservenowthatthesetofalltranslationsofthereferenceframes,alltheirrotation,andallthetransformationsbetweentwoinertialframesinrelativeuniformtranslationmotionhavetheimportantmathematicalcharacteristicsofbeingagroup.Letusdefinewhatthismeans.

ConsiderasetoftransformationsA,B,C…forwhichaproductoperation,⊗,isdefined.Thesetisagroupifthefollowingconditionsaresatisfied:

ForanypairoftransformationsAandBoftheset,theproductC=A⊗Bisalsoatransformationoftheset.Theproductisassociative,namelyA⊗(B⊗C)=(A⊗B)⊗C.

ThesetincludestheidentitytransformationE,suchasA⊗E=A.

ForeverytransformationAoftheset,theinversetransformation,calledA−1,exists,suchasA⊗A−1=E

WithproductB⊗AwemeanthatwefirstapplytransformationAandthen,

ontheresultofthat,transformationB.Tobetterunderstandthat,considertheexampleofthestatictranslationsor

displacements,whichwetakeintwodimensionsforsimplicity.SupposethatthetransformationAisthedisplacementainthexdirectionfromthecoordinatesS(x,y)tothecoordinatesSʹ(xʹ,yʹ),asshowninFig.6.1.

Fig.6.1 Twotranslationsandtheirproduct

ThetransformationAis

(6.1)

LetthestatictransformationBbethedisplacementbintheyʹdirectionoftheresultofA,whichisSʹ(xʹ,yʹ),toSʺ(xʺ,yʺ),namely

(6.2)

TheproductofthetwoisthetransformationfromS(x,y)toSʺ(xʺ,yʺ).Isitatranslation?Theserelationsare

(6.3)

whichistheexpressionofatranslationtoo.Itisalsoeasytoseethattheassociativepropertyholds.Property2forbeingagroupissatisfied.

Theothertwopropertiesarealsosatisfied.Theidentityisthetranslationofnulldisplacement(donothing).Givenatranslationbyacertaindisplacement,thestatictranslationoftheoppositeoneisalsosuchatranslation.Doingoneaftertheotherleadstotheidentity.Inconclusion,statictranslationsformagroup.

Particularlyimportantaretherotations.Werecallthatthecovarianceofthelawsunderrotationsoftheaxescorrespondtothefactthatthequantitiesappearingintheequationsthatexpressthelaws(positionvector,velocity,acceleration,force,energy,etc.)musthavewell-definedtransformationpropertiesunderrotations.Theyshouldbescalar,pseudoscalar,vectorsorpseudovectors,andbothsidesoftheequationmustsharetheproperty.

Considernowthetime.InNewtonianmechanicstimeisthesameinallreferencesystems.Weneedtolookatthatmorecarefully.Thetimeinterval,asallthephysicalquantities,mustbeoperationallydefined.Itisnotobviousthattheoperationstomeasurethetimeintervalbetweentwoeventsisthesameforanobserveratrestrelativetotheeventsandonemovingrelativetothem.Asweshallsee,thisisnottrueathighenoughvelocities,inthedomainofrelativisticphysics.

Westateimmediatelythatthecovariancepropertiesofphysicslawsrelativetotranslationsandrotationsremainequaltothoseweknow,inrelativisticphysics.Thechangesareinthecovariancepropertiesbetweentwoframesinrelativeuniformtranslationmotion.Letusconsidertwo(inertial)referenceframes.Thefirstonehasthecoordinatesx,y,zandtimet.WecallitS(x,y,z,t).Thesecondframe,Sʹ(xʹ,yʹ,zʹ,tʹ),hasaxesparalleltothefirstone.Therelativevelocityisalongthe,overlapping,xandxʹaxes.TheconstantvelocityofSʹ,orofitsorigin,isvOʹ,isinthepositivedirectionofx.WechoosetheoriginsofthetimesinbothframesintheinstantinwhichOʹandOcoincide.Figure6.2showsthesituation.

Fig.6.2 Tworeferenceframesinrelativemotion

Thecovarianceofthelawsundertransformationsbetweentwoinertialframesisdefinedoncethetransformationequationsaredefined,namelytherelationsbetweencoordinateandtimeinSʹandinS.Thetransformationequationsweknow,includingtherelationbetweenthetimes,are

(6.4)

Moregenerally,foragenericdirectionofthevelocityvO,ofthepointsofSʹthetransformationsare

(6.5)ThesearecalledGalileitransformations.Animportantpropertyofthe

Galileitransformationisthattheyformagroup.Thetimeintervals,inparticular,areequalinthetwoframes.Inotherwords,

timeisabsolute,independentofthemotionoftheobserver.ThisimpliesthatitispossibletosynchronizetheclocksinSwiththeonesinSʹindependentlyoftherelativevelocityofthetwoframes.Asistime,simultaneityisabsolute.IftwoeventsaresimultaneousinStheyaresimultaneousinSʹtooandinanyother(inertial)frame,whateveritsvelocity.

TheGalileiandNewtonlawsofmechanicsthatwehavestudiedwereestablishedwhenonlyoneofthefundamentalinteractions,gravitation,wasknown.Threemorefundamentalinteractionswerediscoveredinthefollowingcenturies.Thefirstonewaselectromagnetism,includingelectricandmagneticphenomenaandwillbetreatedinthethirdvolumeofthiscourse.Theothertwoarethestronginteractionbetweenquarksinthenucleonsandtheweakinteractionresponsible,inparticular,forbetadecay.Bothofthemactatthenuclearandsubnuclearscalesandarequantumphenomena.Dotheyobeytheinvarianceprincipleswehavediscussed,inparticulartherelativityprinciple?Letussee.

ThestudyofelectromagneticphenomenawasdevelopedinthesecondhalfoftheXVIIand,mainly,intheXIXCentury.In1820HansChristianØrsted(1777–1851)discoveredthatelectriccurrentsgeneratemagneticfields,linkingforthefirsttimeelectricityandmagnetism.Between1820and1826AndréMarieAmpère(1775–1836)completelyclarifiedtherelationbetweenelectriccurrentsandmagnetismwithaseriesofexperiments.In1831MichaelFaraday(1791–1867)discoveredtheelectromagneticphenomena:magneticfieldsvaryingintimegiveorigintoelectricfields.Theprogressbecamerapidandin1865JamesClerkMaxwell(1831–1879)developedthecompletetheoryofelectromagnetism.Alltheelectricandmagneticphenomenaaredescribedbyasetofdifferentialequations,calledtheMaxwellequations.Inaddition,thetheorypredictedanewphenomenon.Electricchargesinaccelerationproduceelectromagneticwaves,whichpropagatewithawell-definedvelocity.Thisvelocitycanbeexpressedintermsoftwoquantitiesthatmeasurethestrengthoftheforcebetweentwoelectricchargesatrestandbetweentwostationaryelectriccurrentsrespectively.Maxwellhimselfaccuratelymeasuredthesequantitiesandfoundtheresultingvalueofthevelocitytobe,inroundfigures,

(6.6)Thisisjustequaltothevelocityoflight.AndMaxwellnoticedthat

theonlyusemadeoflightintheexperimentwastoseetheinstruments.

Heconcluded

thatlightisanelectromagneticdisturbancepropagatedthroughthefieldaccordingtoelectromagneticlaws.

Thedirectexperimentalconfirmationoftheexistenceandoftheforeseenpropertiesoftheelectromagneticwaveswasverydifficult.HeinrichRudolfHertzfinallysucceededinthatin1887.

TheMaxwellequationsledtotheunificationofelectric,magneticandopticalphenomena.However,soontheyshowedanunexpectedbehavior.TheirformchangesbetweentwoinertialframeswhencoordinatesandtimearetransformedaccordingtotheGalileitransformationsEq.(6.5).ItlookedlikeMaxwellequationsdidviolaterelativityprinciple.Ifitwereso,itshouldhavebeenpossibletodesignandperformelectromagneticandopticexperimentsabletoestablishanabsolutereferenceframe.

SupposeforexamplewehavealightsourceemittingalightpulseinthepositivexdirectioninFig.6.2.LetcbeitsvelocityinS.Noticethatlightisawavephenomenon,similartosoundorseawaves.Consequently,itspropagationvelocityisindependentofthevelocityofthesourcerelativetotheobserver.However,differentlyfromtheothermentionedcases,lightpropagatesinavacuumtoo.Indeeditcomestousfromverydistantstars.Consequentlythereisnosubstanceperturbedbythewaveandsupportingitsmotion.Thisfact,whichiscleartoeverybodynow,wasnotsoattheendoftheXIXCentury,whentheexistenceofasubstancepervadingallspacewasassumed,theluminifer(lightsupporting)ether.Theetherhypothesishasbeenaseriousprobleminthedevelopmentofelectromagneticphysics.

Anyway,asthespeedoflightisindependentofthemotionofthesource,itshouldtransformasanyothervelocity,aswehavebyderivationofEq.(6.4)inthecaseofourexample

(6.7)Asanyothervelocity,thespeedoflightshouldbedifferentfortheobserver

inSandinSʹ.IfwethenmeasurethespeedoflightinSʹandfinditdifferentfromcwewouldestablishthatSistheabsolutereferenceframe,namelytheonlyone,amongstalltheinertialframesinwhichthevelocityoflighthasthevalueofEq.(6.6).

Morespecifically,thenon-covarianceofMaxwell’sequationsundertheGalileitransformationsrequiresustoestablishwhichofthefollowing

(1)

(2)

(3)

alternativesistherightone.

TherelativityprincipleisvalidfortheNewtonlawsofmechanicsbutnotfortheMaxwelllawsofelectromagnetism.TheGalileitransformationsarecorrect.Thisimpliestheexistenceofanabsolutereferenceframe,whichshouldbeexperimentallyfound.

TherelativityprincipleisvalidforboththeNewtonlawsandelectromagnetism.TheGalileitransformationsarecorrect,buttheMaxwellequationsarewrong.InthiscaseweshouldfindmodificationstotheMaxwellequationsthatarenecessarytohavethemcovariantunderGalileitransformationsandthenexperimentallycontrolwhetherthepredictionsofthesemodificationsexistornot.

Therelativityprincipleisvalidformechanicsandelectromagnetism.TheMaxwellequationsarecorrect,butthetransformationequationsbetweenreferenceframesarenottheGalileitransformations.Inthiscasewemustfindnewtransformations,differentfromtheGalileionesandsuchastoinsurethecovarianceoftheMaxwellequations.Inaddition,theNewtonlawswouldnolongerbeanymorecovariantunderthenewtransformations.Weshouldfindthemodificationsneededtoguaranteethecovariancealsoofmechanicallawsandexperimentallyverifywhethertheconsequencesofthemodificationswemadearecorrect.Thehistoricalprocessleadingtotheclarificationoftheproblemwasnotstraight,butratheralongwindingpaths.AftertheimportantcontributionsofHendrikAntoonLorentz(1853–1928),in1905twofundamentalarticleswereseparatelypublished,thefirstbyHenryPoincaré(1854–1912),thesecondafewweekslaterbyAlbertEinstein,thatlaiddownthecompletetheory.Itbecameknownasspecialrelativity.

Thecrucialexperimenttochoosebetweentheabovestatedalternativesisthe

measureofthespeedoflightininertialframesinrelativemotion,allowingustoverifywhetheritisthesameornot.Theexpectedeffectshowever,areextremelysmallandverydifficulttodetect.TheexperimentwasdonebyAlbertAbraham

Michelson(1852–1931)in1881and,inamuchmoresensitiveversion,togetherwithEdwardWilliamMorley(1838–1923)in1887.Weshalldescribethe1887experimentinthenextsection.Weshallseehowitshowedthatthespeedoflightisthesameinallreferenceframes,soexcludingalternatives(1)and(2).

6.2 TheMichelsonandMorleyExperimentWestartwithabitofhistory.In1879Maxwellstudiedthepossibilityofestablishingtheabsolutemotionofearthrelativetothereferenceinwhichthespeedoflightisc,namelytheabsolutereferenceframe,onthebasisofastronomicaldata.Theabsolutereferenceistheframeinwhichtheether,whichwasthoughttoexist,isstill.Ifthisframeexists,itshouldbeatrestrelativetothefixedstars,accordingtoastronomicalobservations.Wedonotknowthevelocityoftheearthrelativetothishypotheticalframe,butweknowthatitshouldbeatleastthevelocityoftheearthinitsorbitalmotionaroundthesun.ThisisaboutυE≈30km/sinmagnitudeandvariesindirectionthroughouttheyear.Letusassumethisvelocitytobe,inorderofmagnitude,whatwehavetodetect.Itsratiotothespeedoflightis

(6.8)whichisaverysmallvalue.MaxwellestablishedthatonlyinastronomicalphenomenacouldoneexpecteffectsofthefirstorderinβE.Inlaboratoryexperiments,inwhichthelightleavesfromapoint,movestoacertaindistanceandcomesbacktothestartingpoint,orclosetoit,onlyeffectsofthesecondorderwereexpected,namelyoftheorderof10−8.Thisisreallyaverysmallnumber.Maxwell’sargumentisthefollowing.

Supposethatinourlaboratory,namelyinareferenceinwhichtheearthmoveswithspeedυE,weplaceabaroflengthlinthedirectionofthemotion.Atoneendofthebarwehaveasourceemittingflashesoflightandadetectoroflightnearby.Attheotherendthereisamirrorsendingthelightpulsesbacktothedetector.Thelightpulsetravelsthedistancelfromthesourcetothemirroratvelocityc+υEandwhengoingbackfromthemirrortothedetectoratvelocityc–υE.Thetotaltimeisthen

(6.9)

Now,2l/cwouldbetheround-triptimeifthebarwerenotmoving.Thisisa

veryshorttime.Butthetimetomeasureis ofit.Maxwellconcludedthatsuchanexperimentwasimpossible.

Theyoung,25yearsold,officeroftheUSAnavyAlbertAbrahamMichelson,whohadalreadyperformedanaccuratemeasurementofthespeedoflight,didnotacceptasobvioustheimpossibilityofalaboratoryexperimentsensitivetothesecondorder.Ratherheworkedontheproblemandin2yearsfoundasolution.In1881,hehadalreadyafirstresult.Thesensitivityofthisexperimentwasenoughtodetecttheeffectdowntoonehalfoftheprediction.Theresultwasnull.However,theconclusionwassoimportantthataconfirmationwasneeded.Michelson,nowwithMorley,designedandperformedin1887asecondexperimentsensitivetoeffects40timessmallerthanthepredictions.Againtheresultwasnull.

TheMichelson-MorleyexperimentisbasedontheemploymentoftheinterferometershowninFig.6.3,whichhadbeendevelopedbyMichelsonhimself(Michelsoninterferometer).

Fig.6.3 TheMichelsoninterferometer

ThesourceLemitsamonochromaticline.Thismeansthatthewaveisasinusoid.Thedistancebetweentwoconsecutivemaximaisthewavelength(λ=0.6µm).EachpointonthewavemovesupanddownperiodicallywithaperiodT.Inanequivalentmannerwecansaythatifwelookedatthewavepassingonafixedpoint,thetimeintervalbetweenthepassageoftwomaximawouldbeT.

Consequentlytheratiobetweenwavelengthandperiodisthespeedofthewave.Ifthisiscwehave

(6.10)IntheMichelsoninterferometer,thelightbeamisdividedintwobya

semitransparentmirrorMat45°withtheincidentbeamdirection.OneofthetwobeamsafterthismirrorreachesthetotallyreflectingmirrorM1,isreflectedback,reachesagainM,andisreflectedtowardsthetelescopeC.Theotherbeamonthearm2isreflectedbackbyM2and,afterM,whichpartiallytransmitsit,rejoinswiththefirstbeam.Thelengthsofthetwoarmsaremadeasequalaspossible.ThetwolightwavesareinphasewhentheyleaveMforthefirsttimeandarealsoinphasewhentheyrecombine,namelyinthetelescope,providedthatthetimes,callthemt1andt2,areidentical,ordifferexactlybyanintegernumberofperiods.ThisisthesituationdraftedinFig.6.4a.Inthissituation,thesignaltheyoriginatewhentheyrecombineisamaximum(constructiveinterference).

Fig.6.4 awavesinphase,bwavesinphaseopposition

Ifthetravellingtimest1andt2differbyhalfaperiod,oranoddnumberofhalfperiods,asinFig.6.4b,thetwowavesareinphaseoppositionandcanceleachothergivingazerosignal(destructiveinterference).Intheintermediatecases,theintensityisintermediatetoo.Iftheseweretheconditionsofthefieldseenbytheobserverthroughthetelescope,itwouldbeclearinconstructive,darkindestructiveinterference.Inpracticehowever,theplanesofthetwomirrorsareneverexactlyat90°.Consequently,theconditionsofconstructiveanddestructiveinterferencealternatethroughthewidthofthebeaminthevisualfield.Theobserverseesaseriesofclearanddarkbands,calledinterferencefringes.Onecouldchangetheplanesofthemirrorsbyadjustingscrewsinordertohavethefringeshorizontal,asinFig.6.5.Areferencewireintheeyepiecewasusedtomeasurethepositionofthefringes.

Fig.6.5 Theinterferencefringesbeforeandaftertherotationof90°oftheapparatus

Wenowevaluatethedifferencebetweenthetimest1andt2.Itisduetotwocauses.Thefirstoneisinstrumentalandduetothefactthatthelengths,sayl1andl2,oftwoarmsareneverexactlyequal.Noticethathereexactlymeanstobesowithinasmallfractionofthewavelength,namelyafewdozensofnanometers.Theothercauseiswhatwewanttomeasure,namelyadifferenceinthelightspeed,relativetotheinstrumentbetweenthetwoarmsduetothemotionoftheearth.

Supposewehavealignedthearm1paralleltoitstransportationvelocityandevaluatet1.InthepathfromMtoM1thespeedoflightisc+υEandinthepathbackfromM1toMisc–υE.Wehavealreadycalculatedtheround-triptime,Eq.(6.9).Wecanwrite

(6.11)Wenowcalculatethetimet2.IfearthmoveswithvelocityυErelativetothe

absoluteframe,inthetimet2isdisplacedbyυEt2asshowninFig.6.6.Lookingatthefigurewewrite

Fig.6.6 Pathofthelightinarm2

andhence

(6.12)

Noticethatwehavejustcalculatedt1intheframefixedtoearthandt2inthesupposedabsoluteframe.ThiswasallowedbecausewehaveassumedtheGalileitransformationstobevalid,inparticularthetimetobeabsolute.Noticealsothat,asanticipated,theeffectisofthesecondorder,namelyas .

Thedifferencebetweenthetwotimesisthen

(6.13)

Asweanticipated,thetwotimesdifferbythesearchedforeffect,i.e.thetermin ,andforthedifferencebetweenthearmlengths,2(l2−l1)/c.Togetridofthesecondeffect,Michelsonemployedameasurementmethodbycomparison.Thecomparisonwasbetweenameasurementinthejustdescribedconditionsandoneafterrotatingthewholeapparatusby90°.Thetimedifference,say∆tʹ,isEq.(6.9)withinvertedl1andl2,namely

(6.14)

Wetakethedifferencebetweenthetwodifferencesandobtain

(6.15)Ifthedifferencebetweenthedifferencesiszero,thepositionofthefringes

seenbytheobserverremainsfixedrelativetothereferencewirewhenwerotatetheapparatus.Ifitisequaltooneperiodthefringepatternmovesbyonefringe.Ingeneral,thenumber∆n(notintegeringeneral)offringescrossingthereferencewireduringtherotation,isgivenby

(6.16)where,inthelastmemberwehaveusedEq.(6.15)andintroducedthemeanvaluelofthelengthsofthetwoarms.

Inthe1881experimentthelengthofthearmswasl=1.2m,correspondingtoanexpectedshiftofΔn=0.04fringes.Michelsonwasabletoappreciateashiftof0.02fringes.Hedidnotobserveanyandconcludedthat:

Theconsequenceofastationaryetherresultsthereforecontradictedbythefactsanditmustbeconcludedthatthehypothesisoftheetherisfalse.

ThesecondexperimentisshowninFig.6.7.Theopticalpath,namelythepathofthebeams,isincreasedtol=11m,havingthebeamgoingbackandforthonitsarmeighttimeswithasetofmirrors(Fig.6.7b).The90°rotationwasanextremelydelicateoperation.Anyvibrationevenbyasmallfractionofawavelengthhadtobeavoided.Everythinghadtobestableatthislevel.MichelsonandMorleymountedtheinterferometeronamassivegranitebench,whichwasfloatingonamercurybath.TheshiftexpectedintheetherhypothesiswasnowΔn=0.40fringes.Figure6.4reproducesthefringesbeforeandaftertherotation.Noshiftcanbeseen.Thesensitivitywasonehundredthofafringe,correspondingtoadistanceof6nm.Figure6.8showstheresultofthemeasurements,whicharethefulllines.Thedottedcurvesare1/8th(toenhancethevisibilityofapossibledifference)oftheexpectationsassumingtheGalileitransformations.Noeffectwasdetectedrepeatingtheexperimentinday-timeandduringthenight,tocheckforanyeffectoftherotationvelocityoftheearth.Theconclusionwasdefinitive:theexperimentcannotestablishthemovementofearth.Thisisanexampleofhowanullresultcangiveextremelyimportantinformation.

Fig.6.7 TheMichelsonMorleyexperimentMichelsonMorleyexperiment

Fig.6.8 Observedshifts(continuouslines)and1/8oftheexpectations(dottedcurves—tokeeptheminsidethediagram)forGalileitransformations

Afirstattempttoexplaintheresultwasdonein1889byGeorgeFitzGerald(1851–1901)andindependentlyin1992byH.A.Lorentz.Theyadvancedthehypothesisthattheobjects,wheninmotion,contract,onlyinthedirectionofthemotionandnotintheperpendicularones.Thecontractionwasabletocanceltheeffectexpectedintheetherhypothesis.Itwasanadhoc,andwrong,hypothesisbutanimportantsteptowardsrelativitytheory.

InthefollowingyearstheMichelsonexperimentwasrepeatedwithincreasingprecision,alwayswithanullresult.Otherexperimentssensitivetotheabsolutevelocityweredone,againwithnullresult.In1904H.Poincaré,afteracarefulanalysisoftheexperimentalevidence,drewtheconclusionthattherelativityprinciple(sohenameditforthefirsttime)holdsforallphysicallaws.Hiswords,similartothoseofGalileithreecenturiesbeforehim,are:

AccordingtheRelativityPrinciplethelawsofthephysicalphenomenamustbethesame,whetheranobserverisfixed,orforanobservermovinginanuniformtranslationmotion:sothatwehavenomeans,andcouldnothaveany,ofdiscoveringifareorarenotcarriedalonginsuchamotion.

Hissecondconclusionwasthatthespeedoflightisthesameinallinertialreferenceframes,i.e.,thespeedoflightisinvariant.

Fromourside,weconcedethatonlythethirdalternativeofthoseconsideredintheprevioussectioncanbevalid.Wemustnow,firstofall,findnewtransformationlaws,inplaceoftheGalileitransformations.

6.3 TheLorentzTransformationsWeneednowtofindnewtransformationlawsofcoordinatesandoftimebetweentwoinertialframesinrelativeuniformtranslationmotion.Theymustbesuch,inordertoguaranteetherelativityprinciple,thattheMaxwellequations

arecovariant,namelymaintaintheirform,undersuchtransformations.Theinvarianceofthespeedoflightisanimmediateconsequenceofthat.ThesearetheLorentztransformations.Afterhavingrecalledtheimportanthistoricalelements,weshallgivetheresultwithoutdemonstrationandshalldiscussit.FinallyweshallstatewhicharethebasicassumptionsunderwhichtheGalileiandLorentztransformationsarevalid.

TheLorentztransformationswere,foundastheresultofadifficulttheoreticaleffortinseveralsubsequentstepsofimprovingprecision,byHendrikLorentzbetween1895and1904and,withafurthersmallcorrection,infinalform,byHenriPoincaréin1905,whopublishedtheresultonthe5thofJune1905.AlbertEinsteinreachedthesameresultonthe30thJune,whenhisfundamentalarticlewassentforpublication.

ConsideroncemorethetwoinertialreferenceframesSandSʹrepresentedinFig.6.2.WehaveinbothframesrulersalongtheCartesianaxes,tomeasurethecoordinates,and,ineverypointofthespace,wehaveidenticalclockstomeasurethetime.Alltheclocksineachframearesynchronizedwithoneanother.Weshalldiscussinthenextsectionhowthiscrucialoperationcanbedone.Wechoosetheoriginsofthetimesinbothframes,t=0andtʹ=0,asthetimeatwhichthetwoframesoverlap.

Weshallcallsomethinghappeninginadefinitepositionanevent,asdefinedbythethreeCartesiancoordinatesmeasuredbytherulersintheconsideredframe,atadefiniteinstantoftime,asmeasuredbytheclockinthatpositionintheconsideredframe.Therearetworelevantparameters,whichalwaysarepresentinrelativisticformulas.TheyarepurenumbersandarefunctionsofthevelocityυOʹofSʹrelativetoS.Thefirstoneistheratioofthisvelocityandthelightspeed

(6.17)andthesecondis

(6.18)TheLorentztransformationsare

(6.19)

WeimmediatelyseethattheyareageneralizationoftheGalileitransformations,tendingtothemfor ,namelyforvelocitiesmuch

smallerthanthelightspeed, .Theinversetransformations,togofromSʹtoS,canbefoundbyinvertingthe

systemofequationsor,inasimplerway,bychangingthesignofthevelocity.Hence

(6.20)

TheLorentztransformationsshowverystrangelookingaspects.Theymix,sotosay,spaceandtime.Weshallseetheconsequencesinthenextsections.Hereweshalllookatthemfromageometricalpointofview.Indeed,Eq.(6.20)aresimilartothetransformationsbetweenthecoordinatesintwoframesdifferingforarotationoftheaxes.Iftherotationis,forexample,aroundthecommonzaxis,thatwecancalltheheight,thetransformationsare

(6.21)

Alsointhiscase,thequantitiesinthesecondframearemixtures,betterlinearcombinations,ofthequantitiesinthefirst.Ifwelookatanobjectwerefertooneofitsdimensionsaswidth,anotherasthickness.Ifwenowrotateourpointofviewbyananglearoundaverticalaxis,thenewwidth,namelytheangleunderwhichweseetheobjectinthehorizontalplane,containsapartofwhatwecalleddepthbeforetherotation,andviceversa.Itfollowsthatdepthandwidtharenotabsoluteproperties,rathertheydependonthepointofview,namelytheyarerelativetothereferenceframe.TheLorentztransformationsareanalogous.Theytellusthatthelengthmeasurementsmadebyapersoncontainsomeofthetimemeasuredbyanotherpersonmovingrelativetothefirstone.Whenspeedsarehigh,closetothespeedoflight,theobjectsaremixturesofspaceandtime,asusuallytheyareofwidthanddepth.Whenweturnaroundanobjectandweseeitfromdifferentangles,ourbrainautomaticallyrecalculatesdepthandwidth,becauseitdevelopedundertheseconditions.Ifwewerelivingathighspeedwemighthaveabrainabletocalculatethenewmixtureofspaceandtimeeverytimewechangespeed.Wedonothavethisautomatichabitandmustunderstandthesituationbycarefullyreasoning.

Aswewellknow,thenormofavectorinourthreedimensionalspaceisthesumofthesquaresofitsCartesiancomponents.Inparticularthenormofthepositionvectoris

(6.22)Ifweconsiderforsimplicityaplane,wehave ,whichisthe

Pythagoreantheorem.Noticethatthesameisnottrue,forexample,onaspherical,ratherthanplane,surface.ThePythagoreantheoremisvalidifthetwodimensionalspaceisflat.Thesameistrueinthreedimensions.AspaceinwhichthesquaresofthedistancesaregivenbyEq.(6.22)issaidtobeanEuclideanspace.

Wealsoknowthatapropertyoftherotationoftheaxesistoleavethenormofthevectorsinvariant.WecanseethereasonforthatwritingEq.(6.21)asaproductofmatrices

(6.23)

Weseethatthesquarematrixinthetransformationisorthogonal.DotheLorentztransformationshavethesameproperty?

ConsiderthefollowingtwoeventsinS.ThefirstoneisthestartofalightpulsefromitsoriginOattheinstantt=0,thesecondisthearrivalofthepulseinthepoint(x,y,z)attimet.Weexpressthefactthatthespeedoflightiscbywriting

(6.24)InSʹtoothelightpropagateswiththesamevelocitycandwecanwrite

(6.25)Thequantitiesintheleft-handsideareverysimilartothenormofavectorin

fourdimensions.Theyarecalledintervals.Thedifferenceistheminussigninfrontofthetemporalterm.Technically,thefourdimensionalspace—timeissaidtobeapseudo-Euclideanspace.Anotherwaytocopewiththeissueistodefineanimaginarytimecoordinate,ict.Tosimplifytheexpressionsweshallusethesymbols

(6.26)Aneventisapointinspace-time.Analogoustothepositionvectorinthree

dimensionsisthefourdimensionalvectorofcoordinatesgivenbyEq.(6.26).Weshallcallthesevectors,four-vectors,todistinguishthemfromthevectorsinthreedimensions(three-vectors).TheLorentztransformationswrittenasproductsofmatricesare

(6.27)

AsfirstestablishedbyPoincaréin1905,theLorentztransformationsjoinedwiththespacerotations,formingagroupwhichhenamedtheLorentzgroup.Thematrixcorrespondingtotheproductoftwotransformationsistheproductoftheircorrespondingmatrices.

Equations(6.24)and(6.25)tellsustwothings.Firstly,twoeventsconnectedbyalightsignalareseparatedbyanullinterval.Thisdoesnotmeanthattheycoincidebutthatthenormoftheintervalbetweenthemiszero.Thispossibilityisaconsequenceoftheminussigninthetemporalterm.Thenormofafour-vectorcanbepositive,zero,ornegativeinspace-time.Secondly,ifanintervalisnullinSitisnullinSʹtoo.Thisisaformalwaytostatethatthespeedoflightisinvariant.Asamatteroffactwecanstatemore.ThesquarematrixinEq.(6.27)isorthogonal.Theconsequenceisthatalltheintervals,evenmorethenormsofallthefour-vectors,areinvariantunderLorentztransformations.

Inthree-dimensionalspacewedealtwithvectorsinthreedimensions,whichwenowcallthree-vectors.Asthereaderremembers,athree-vectorisanorderedtripletofrealnumbersthattransformsunderrotationoftheaxesasthepositionvector.

Inasimilarmannerinrelativisticphysicswedealwithfour-vectors.Afour-vectorisaquadrupleofnumbers,realthefirstthree,imaginarythefourth,whichtransformfromaninertialreferencetoanother,inrelativeuniformtranslationmotion,asthecoordinatesdo.Thenormsofallthefour-vectorsareconsequentlyinvariantunderLorentztransformations,inotherwordstheyarefour-scalars.Assuch,theyplayveryimportantrolesinrelativisticphysics.Weshallseeexampleslater.

InthenextsectionsweshalldiscussthedeepconsequencesoftheLorentztransformationsonthebasicconceptsofspaceandtime.Herewenoticethefollowing.

Historically,theLorentztransformationswerefound,asmentioned,bythethreemainauthors,intemporalorder,Lorentz,thenPoincaréandthenEinstein.Eachofthemstartedfromsomewhatdifferenthypothesesandfollowedadifferentlogicalpath.Thepathwehavefollowedhereistostartfromtheexperimentaldiscoveryoftheinvarianceofthespeedoflight.Thiswasindeedarevolutionarydiscovery.Thiswasalsooneoftheaxioms,togetherwiththerelativityprinciple,assumedbyEinstein.Fromthelogicalpointofview,however,thissecondaxiomisnotnecessary.Indeed,therelativityprinciple

1.

2.

3.

imposesthecovarianceoftheMaxwellequations.Oncethisisestablished,withtheLorentztransformations,theinvarianceofthespeedoflightisanimmediateconsequence.

However,thehistoricalapproachwehavefollowed,asthevastmajorityoftextbooksdo,tendstohidethelogicalstructureofspecialrelativityandtooveremphasizetheroleofelectromagnetisminthefoundationsofthetheory.Aftermorethanonecenturyfromitscreationweknowthatallthefundamentalinteractions,notonlytheelectromagneticone,butalsothegravitational,thestrongandtheweakinteractionsobeytherelativityprinciple.AllthelawsthatgovernthemarecovariantunderLorentztransformations.Thefieldsofthefundamentalinteractions,whichareanalogoustothegravitationalfieldwestudiedinChap.4,inquantummechanics,aremediatedby“quanta”.Thequantumoflightisthephoton.Itsvelocityisthevelocityoflight.AsweshallseeinSect.6.10thisimpliesthatthemassofthephotoniszero.However,itwouldbelogicallypossiblethatthephotonwouldbemassive.Inthiscase,theLorentztransformationswouldnotchange,buttheparametercappearingintheequationswouldnotbethespeedoflightandthelatterwouldnotbeinvariant.Indeed,thisisthecaseoftheweakinteraction,thequantaofwhich,calledZ0andW±bosons,havemassanddonotmoveatthespeedoflight.Ifthatwasthecase,thedemonstrationbasedontheinvarianceofthespeedoflightwouldnothold.Butthefinalresultwouldstillbevalid.

Fromthelogicalpointofview,wemustaskourselvesthefollowingquestions.Canweestablishtherelativitytheoryindependentlyonelectromagnetism?Whataretheassumptionsneededforthat?Theanswerisyes;onlyafewhypothesesonthebasicstructureofthespace-timeareneeded.Thesearethefollowing:

Space-timeisisotropicandhomogeneous.

Aclassofinertialreferenceframesexist,namelyframesinwhichtheinertialawholds.

Therelativityprincipleisvalid,namelythereisnoprivilegedreferenceframe.

4.

5.

Thetransformationsformagroup.

Aclassofeventsexistsforwhichthecausalityprincipleholds.Inthisclassthesignofthetimedifferencesbetweenevents,thatisthenatureofapossiblecausalrelation,isthesameinalltheinertialframes.

Itcanbedemonstratedthatonlytwotransformationsexistunderthese

hypotheses,theGalileiandtheLorentztransformations.1Thequantitycinthelatterhasthedimensionofavelocityandenjoysthepropertiesofbeinginvariantandbeingthemaximumpossiblevelocity.GalileitransformationsarethelimitoftheLorentzonesfor .Thereisnoneedtorelyonelectromagnetism.Theelectromagnetismentersthegameonlytogivetocthephysicalmeaningofspeedoflight.

6.4 CriticismofSimultaneityThemostimportantdifferencebetweenGalileiandLorentztransformationisontimemeasurements.IntheformertheresultofthemeasurementofatimeintervalisthesameinSandinSʹ.Timeisabsolute,independentofthereferenceframe,intheGalileiandNewtonphysics.Onthecontrary,thelastEq.(6.19)states,inparticular,thattheinstantatwhichaneventhappensinSʹdependsnotonlyonthetimeinwhichithappensinS(asexpected)butalsoonitspositioninS,asnotexpected.Hence,twoeventshappeningintwodifferentpointsthatappeartobesimultaneoustoanobserverinSdonotappeartobesotoanobserverinSʹ.Thesimultaneityoftwoeventsisnotanabsoluteconcept,butratheritisframedependent.Thein-depthcriticismofthesimultaneityconceptandofthetimeintervalsmeasurementwasmadebyH.Poincaréin1898.WeshallexplaintheargumentconsideringtheidealexperimentrepresentedinFig.6.9.

Fig.6.9 Alightflasherandtwodetectorsatequaldistances

WesupposetohavefixedintheSʹframearigidbarparalleltothexʹaxis.Inthemiddlepointofthebarwehaveinstalledalightsource,whichemitsalightflashatacertaininstant.Theflashpropagatesinalldirections,inparticulartowardstwodetectorsR1andR2atthetwoextremesofthebar.TheobserverinSʹconsidersthetwoeventsofarrivaloftheflashatthetwodetectorsassimultaneous.Noticethatthisconclusioncanbereachedonlyassumingthatlightpropagateswiththesamevelocityinbothdirections,namelythatspaceisisotropic.Noticethattheassumptionisdifferentfromtheinvarianceofthespeedoflight.

FortheobserverinSthetwoeventsarenotsimultaneous.SupposethatthevelocityvofthebarinShasthedirectionfromR1toR2.OneflashtravelstowardsR1thatisapproaching,theothertowardsR2thatisreceding.Theformerwillthentakeashortertimethanthelattertoreachitsdetector.Thetwoeventsarenotsimultaneous.

Thefactthatthesimultaneityoftwoeventshappeningintwodifferentpointsisnotabsoluteisaconsequenceoftheexistenceofamaximumvelocityforthepropagationofthesignals.Thisinturnhasdeepconsequencesonthemeasurementoftime.Wehavedefinedaneventasthesetofthethreespatialcoordinatesandthetemporalonethatcharacterizeaphenomenonhappeningatacertaintimeinacertainpoint.Togiveaphysicalmeaningtothisdefinition,weneedtodefinethesetsofoperationstobedonetomeasurethespaceandtimecoordinates.Inparticular,tomeasurethetimeoftheeventsweneedtohaveidenticalclocksinallthepointsofthereferenceframe.Alltheclocksmustbesynchronized.Thismeansthatthearmsofalltheclocksmustreachthesamepositionsimultaneously.Assimultaneityisframedependent,anobservermovingrelativetoaframe,theclocksofwhichhavebeensynchronizedbytheobserveratrestinthatframe,seesthoseclocksasnotsynchronized.Theconsequenceoftheframedependenceofsimultaneityistheframedependenceofthetimemeasurements.Letusseethatinthedetails.

6.5 DilationofTimeIntervalsConsidertwoeventshappeninginthesamepointx1oftheframeSintwodifferentinstantst1andt2.Intheseconditionswecanmeasurethetimewithasingleclockinx1.Inotherwords,wehavenoneedtosynchronizeclocksindifferentpositions.Thetwoeventshavethespaceandtimecoordinates(x1,0,0,t1)and(x1,0,0,t2).Theyareseparatedbythetimeinterval

wherethesubscript0istorecallthatthetimeintervalismeasuredintheframeinwhichtheobjectisatrest.Suchintervalsaresaidtobeofpropertime.TheobserverinSʹobviouslydoesnotseethetwoeventsinthesamepointofhisframe,but,say,inx1ʹandx2ʹ.Ifhewantstomeasurethetimest1ʹandt2ʹ,inwhichtheeventshappenheneedstwoclocks,oneinx1ʹandoneinx2ʹ,whichmustbesynchronized.Equation(6.19)tellusthat

ThetimeintervalinSʹisthen

or

(6.28)Considerforexampleaclockproducingperiodicticks.Theperiod,namely

thetimeintervalbetweentwoconsecutiveticks,intheframeinwhichtheclockisatrest,is,say∆t0.AnobservermovingwithvelocityυOʹtheclockappearsemittingtickswiththeperiod

(6.29)

whichislongerthanthepropertime∆t0.Itisusefultoshowthisresultalsowithaphysicsargument.Supposethatthe

observerinSandinSʹhavetwoidenticalclocks,builtasinFig.6.10a.ThelightsourceLemitsaflashatacertaininstant,whichreachesthemirrorRatthedistancelandisreflectedbacktothedetectorR.WhenthelightpulsereachesRatickisemittedandthesourceLemitsantherflash,andsoon.Letusseenowhowthetwoobserversseetheperiodoftheclock.Forbothobserverstheperiod

oftheirclocksisthetimetogotwicethroughthedistancel,asinFig.6.10a,namelythepropertime .

Fig.6.10 Aclockin,aseeninitsrestframe,bseenfromamovingobserver

Also,tobothobserverstheclockoftheotheroneappearstomovewithvelocityυOʹ=υO.Supposethatbothclocksareorientedperpendicularlytotherelativemotion.Intheseconditions,thepathoflightthattheobserverSʹseesintheclockinSisasrepresentedinFig.6.10b,andreciprocally.LighttakeshalfaperiodΔtʹ/2togofromLtoM,andtheotherhalfaperiodtogofromMtoL.Thedistancetravelledbytheflashinhalfaperiodisthen(Δtʹ/2)c.Inthesametimeintervaltheclockhasmovedadistanceof(Δtʹ/2)υOʹ.Hence(seefigure)

fromwhich

whichisEq.(6.29).Inthejustmadeargumentweimplicitlyassumedthatthelengthlofthe

clockisindependentofitsmotion,namelythesameforbothobservers.Asweshallproveinthenextsection,thisistruebecauseitisperpendiculartotherelativemotion.Wecanreachthesameconclusionbyobservingthatitisaconsequenceoftherelativityprinciple.Indeed,bothobserversmayagreetocuttwonotchesinthepositionsoftheextremesofthemovingclock,respectivelyontheyandyʹaxis,whenthispassesby.Suchnotchesmustresultinthesame

valuesofyandyʹ,otherwisetheresultswouldbeabletodistinguishwhichismovingandwhichisstill.Noticealsothatthehypothesisoftheindependenceoflightspeedofthedirectionisoncemorenecessary.

Thephenomenonoftimedilationisobservedeverydayinelementaryparticleslaboratories.Protonsandelectronsareacceleratedinacceleratorstospeedveryclosetothespeedoflight.Iftimedilationwerenottakenintoaccount,thesemachineswouldnotwork.

Considerasanotherexampleanaturalphenomenon,thecosmicrays.Theseareparticles,mainlyprotonsandatomicnuclei,acceleratedinthegalaxy,andaboveit,tospeedclosetothatoflightandconstantlyenteringtheearth’satmosphere.Intheatmosphere,soonerorlater,oneoftheseparticleshitsanucleusoftheairproducingashowerofsecondaryparticles.Someofthemareunstable.Amongthemaretheµleptons,ormuons,whichareverysimilartoelectrons,ifnotforthemassthatisabout200timeslarger.Theirlifetimeis2.2µs.Inthedecay,amuonproducesanelectron,aneutrinoandanantineutrino.

Thefollowingexperimentwasdonewithdidacticaims.Chargedparticlescanbedetectedusingablockoftransparentplasticmaterial,dopedwithsubstancesthatemitaflashoflightwhenachargedparticlegoesthrough.Thesmallflashoflightisconvertedintoanelectriccurrentpulse,whichissenttoanelectroniccircuit.Whenacosmicrayenterstheblockapulseisobserved.Ifitisamuonandifitstopsintheblock,afteratimeoftheorderofthelifetimeitdiesandthenewbornelectrongivesasecondpulse.Thissignatureallowsustodiscriminatethestoppingmuonsfromothereventsinducedbycosmicradiation.TheapparatuswasfirstusedonMountWashington(NewHampshire)at1800mheight.In1h568stoppingmuonswerecounted.

Howmuchtimeisneededforthemuonstotravelfrom1800mheighttosealevel?Obviouslythatdependsontheirspeed.However,alowerlimitisgivenbyassumingtheymovewiththespeedoflight.Thislowerlimitis6.3µs.Theexperimentersthencountedhowmanymuonshadlivedmorethan6.3µsofthe568detectedontheMountWashington.Theyfound27ofthem.Theythenmovedtheirdetectortosealevel.Inabsenceoftimedilation,about27eventshadtobedetected.Theyfound412.ThisnumberagreeswithEq.(6.29)iftheaveragemuonspeedisβ=0.99.

Nowconsideranobserversittingonamuon.Inthisframethelifetimeisnotdilatedandthemuonsurvivesonlyafewmicroseconds.Howcansomanyreachsealevel?Thereasonisthat,asweshallseeinthenextsection,thedistancefromthetopofMountWashingtontosealeveldoesnotappeartothemuontobe1800m,ratheritiscontractedbythesamefactor,theLorentzγparameter,asthe

timedilation.Forβ=0.99,wehaveγ=6.1andthedistancetotravelisonly257m.

6.6 ContractionofDistancesAsecondconsequenceofLorentztransformationsisthecontractionoflengths.Westartbyobservingthattheoperationaldefinitionsofthelengthofanobjectatrestandofanobjectinmotionarenotthesame.Theoperationstobedoneinthetwocasesareindeedcompletelydifferent.Consequently,thereisreallynoapriorireasonforwhichthetwolengthsshouldbeequal.Itisjustevery-dayexperiencewithobjectsmovingatrelativelylowvelocitiesthatmakesusbelieveinthisequality.Thelengthsareequal,asiseasilyseen,fortheGalileitransformations,not,asweshallnowsee,fortheLorentztransformations.Accordingtothelatter,whenabodymoveswithvelocityvrelativetotheobserver,itsdimensionparalleltovappearscontractedbyafactor1/γrelativetoitsvaluemeasuredatrest.Thetransversedirectionsareequalforbothobservers.

TodemonstratethesestatementsweimaginearulefixedtotheframeSlyingonitsx-axis.TheobserverinthemovingframeSʹdeterminesthelengthoftherulerbymeasuringthecoordinatesofitsextremesx1ʹandx2ʹatthesameinstanttʹ.Thetwocorrespondingeventshavethespace-timecoordinates(x1ʹ,0,0,tʹ)and(x2ʹ,0,0,tʹ).Thelengthfoundbytheobserverislʹ=x2ʹ–x1ʹ.ThecoordinatesinSofthetwoeventsare

andtheirdistanceinSis

Inconclusion,therelationbetweenthelengthparalleltotherelativevelocityofanobjectatrestandmovingwithvelocityυOʹis

(6.30)wherethesubscript0recallsthatthisisthelengthatrest.Thisiscalledtheproperlength.Inanyothermovingframethelengthappearscontractedbythefactor1/γ.

Asforthedimensionoftheruler,oranyobject,alongyandz,perpendiculartothemotion,thefactthattheydonotvaryfollowsimmediatelyfromthesecondandthirdEq.(6.19).

Inthiscasetoo,letusdemonstratetheresultalsowithaphysicsargument.Thiswillshowthatthecontractionofthelengthisalogicalconsequenceofthe

timedilation.Westillconsidertherulerfixedalongthex-axisofS.TheobserverinS

measuresthelengthl,andestablishesthattheobserverinSʹ,whichistravellingatspeedυOʹ,crossesthedistancelinthetimeintervalΔt=l/υOʹ.Thistimeisnotapropertime,becauseitisbetweentwoeventshappeningindifferentlocations,thepassageofthemobileobserveratoneextremeandattheother.Assuchitismeasuredwithtwodifferentclocks.Ontheotherhand,fortheobserverinSʹthetwoeventshappeninthesamepointandhecanmeasurethetimeinterval,∆tʹ,withthesameclock.∆tʹisapropertimeintervaland,forwhatwesawinthelastsection, ,and,as ,itis .ThemobileobserverseestherulemovingatthespeedυOʹandconsequentlyevaluatesitslengthtobe ,whichistheresultthathadtobedemonstrated.

6.7 AdditionofVelocitiesInthissectionweshallfindtheruleofadditionofvelocitiesinrelativisticphysics.WejustrecallthatfortheGalileitransformations,if,forexample,ashipmovesrelativetoshorewithvelocityuandontheshipapassengermoveswithvelocityvʹ,relativetotheship,thevelocityofthepassengerrelativetoshoreisv=u+vʹ.ThisistheGalileancompositionrulesofvelocities.WeshallnowfindthecorrespondingruleforLorentztransformation,stillintheparticularcaseinwhichthetwoframesSandSʹarethoseinFig.6.11.

Fig.6.11 Twoframesinrelativemotion

ThevelocityvʹofapointinSʹis andthecorrespondinginS

.Noticethateachderivativeineachframeiswithrespecttothe

timeinthatframe.WeshallnowusetheLorentztransformationsEq.(6.19)with

and .Wehave

Bydividingthefirstthreeequationsbythefourthwehave

Wethenwritetheconclusion

(6.31)

Noticethatnotonlythecomponentsparalleltotherelativemotion,butalsothenormalones,aredifferentinthetwoframes.Thecomplicatedbehaviorofthevelocitystemsfromthefactthatitscomponentsarenotthethreecomponentsofafour-vector.Thisisbecause,while(dx,dy,dz)aresuchcomponents,dtisnotafour-scalar.

ItiseasytoverifythattheEq.(6.31)tendtotheGalileanonefor .

ExampleE6.1Consideraparticlemovingwithvelocityυʹx=c/2relativetoSʹ,inthepositivedirectionofxʹ.ThereferenceSʹmovesrelativetoSatthespeedu=c/2inthesamedirection.NoticethatifthetransformationweretheGalileanonesthevelocityoftheparticlerelativetoSwouldhavebeenequaltoc.WiththeLorentztransformationwehave

ExampleE6.2ConsiderSʹtobea(veryfast)shipandshootingaballverticallyupwardswith

velocityυzʹ.Whichvelocityoftheballisseenfromshore?Withυxʹ=υyʹ=0Eq.(6.31)give

ConsidernowtheimportantcaseofalightsignalpropagatingalongthexʹaxisofSʹ.ItsvelocityrelativetoSis

(6.32)Namely,ithasthesamevalueinSʹandinS,whatevertheirrelativevelocity

canbe.ThisresultwasexpectedconsideringthatthespeedoflightisinvariantundertheLorentztransformations.

Acorollaryisthatcombingtovelocitiessmallerthanctheresultingvelocityisalwayssmallerthanc.Thespeedoflightisthemaximumpossiblevelocity.

6.8 Space-TimeWehaveseeninSect.6.3thattheLorentzare,fromthegeometricpointofview,rigidrotationsinthespace-time,ofcoordinates(x,y,z,ict).

Wecannotrepresentthefourdimensionsofthespace-timeonthetwodimensionsofapageofabook.However,wecanlearnalotconsideringaparticlemovinginjustonedimension,x.Thespace-timediagramhasthentwoaxes,thespacecoordinatexandthetime,or,bettertohavethesamephysicaldimensionsct,asshowninFig.6.12.

Fig.6.12 Thespace-timediagram

Apointonthisdiagramrepresentsaneventhappeninginthespacepointxattimet.Aparticleatrestintheframeis,inspace-time,asequenceofeventsatdifferenttimesthathaveallthesamecoordinates.Thisisalineparalleltothect

axis,asline1inthefigure.Suchaline,ingeneral,iscalledthelifelineoftheparticle.

Iftheparticlemoveswithaconstantspeedυitslifelineisastraightline,like2inthefigure,havingasloperelativetothectaxisequaltoυ/c.Noticethatthescalesoftheaxesaresuchthatthelifelinesofanyparticlemovingattheusualvelocitiesareveryneartobeingvertical(υ/c<<1).Ontheotherhand,thelifelinesofthelightsignalsarestraightlinesat+45°or−45°(dottedinFig.6.12)dependingonthedirectionofpropagationbeingthesameoroppositetothex-axis.Line3isthelifelineofaparticlethatisatrestatapositivevalueofxattime0,andthatlateronmovesinthepositivexdirectionofanacceleratedmotion,soonreachingspeedsclosetoc.Noticethatnolifelinecanhaveasloperelativetothectaxislargerthanone,namelyavelocitylargerthanc.

ConsidernowtheeventOintheoriginofthespacetimereferenceframe,namelytheinstantt=0inthepointx=0.Supposethiseventbeingthestartoflightsignalsinalldirections(thetwoofthexaxisinourcase).ThelifelinesofthesignalsarethebisectorsoftheaxesasshowninFig.6.13.Inthefourdimensionalspacetime,theselinesdrawahyperconewithvertexintheoriginandhalfvertexangleequal45°.Itiscalledthelightcone.Thepartofthelightconeonct<0correspondstoalightsignalreachingthepointx=0att=0.

Fig.6.13 Thelightcone

ItisnotdifficulttoseethataLorentztransformationtransformstheaxesasshowninthefiguresforxʹ,ctʹ.Therotationoftheaxesisdifferentfromrotationsinspacebecauseherethemetricispseudo-EuclideanratherthanEuclidean.Inthespacetimethexandctaxesrotateinoppositedirections,bythesameangle,approachingthelightcone.Therotationangleislargerforhigherrelativevelocityandtendsto45°whenthattendstoc.Obviously,thelightconesofthe

twoframescoincide,becausethelightvelocityisthesameinboth.Considernoweventsinsidethelightcone.Theintervalsbetweentheorigin

Oandeachofthem,likeAandB,arenegative.Suchintervalsaresaidtobetime-like,becausethepurelytimeintervalsarenegative.Theeventsoutsidethelightconeareseparatedfromtheoriginbypositiveintervals,calledspace-like.Theeventsonthelightconeareseparatedbynullintervalsandarecalledlight-like.Theintervalsbeinginvariant,thesepropertiesareindependentofthereferenceframe.

Twoeventsseparatedbyatime-likeintervalcanbejoinedbyasignaltravellingataspeedsmallerthanlight,iftheintervalislight-likeinterval,theycanbejoinedbyalightsignal,butifisspace-like,theycannotbejoinedbyanysignal.Suchasignalshouldtravelfasterthanlight.Consequently,nocauseandeffectrelationcanexistbetweentwoeventsataspace-likeinterval.Thisconclusionisconnectedwiththefactthattherelationpast-futureisnotanabsoluteoneforeventsoutsidethelightcone,suchastheeventCinthefigure.ThiseventisfuturerelativetoOin(ithast>0),whileitispastrelativetoO,thesameevent,inSʹ(tʹ<0)asisclearfromthefigure.

Theeventsseparatedfromtheoriginbytime-likeintervalsare,aswehaveseen,insidethelightcone.Wecandistinguishtwopartsofthecone.Intheupperhalfcone,witht>0,wehavetheeventsfuturerelativetoO.Inthelowerhalfcone,witht<0,wehavetheeventspastrelativetoO(asB).Considernow,forexample,theeventA.ItisseparatedfromObyanegativeinterval.AstheintervalsareinvariantthereisnoreferenceframeinwhichAiscontemporarytoO,becauseinthiscaseAwouldbeseparatedfromObyapositive,ornullinterval.WecanconcludethatthereisalsonoframeinwhichAispastrelativetoO,becauseinthiscase,forcontinuityreasons,aframewouldexistinwhichAandOaresimultaneous.Inconclusion,alltheeventsintheupperhalflightconearefuturetoOinanyreferenceframe(absolutefutureofO),thoseinthelowerhalfconearepastofOinanyframe(absolutepastofO).

6.9 Momentum,EnergyandMassAswehaveseen,theLorentztransformationsbetweeninertialframesaresuchastoguaranteethevalidityoftherelativityprinciplefortheMaxwellequations,theequationsthatgovernelectromagnetism.Theprinciplerequireshoweverthatallthephysicallawsshouldbecovariantunderthesetransformations.Consequently,wemustalsofindtheLorentzcovariantexpressionthatgeneralizesthesecondlawofNewton.Oncewehavefoundthenewlaw,itspredictionsshouldbecheckedagainstexperiments.

WestartwiththeobservationthatthenewlawshouldadmittheNewtononeasalimitforsmallvelocities.Wealsonoticethat,ifanequationhastobecovariant,allitstermsmusttransforminthesamewayunderLorentztransformations.Allofthemmustbefour-scalarorfour-vectors.

Wealreadyknowafour-vector,theonethatidentifiestheeventinspace-time,havingcomponents(x1,x2,x3,x4).Wehaveobtained“promoting”thespacethree-vectorr=(x1,x2,x3)withtheadditionofthefourthtimecomponent.Sucha“promotion”isnotalwayspossiblewitheverythree-vector.Aswehavealreadyseen,forexample,thethreecomponentsofthevelocitythree-vectorv=(dx/dt,dy/dt,dz/dt)arenotthethreespacecomponentsofafourvector,becausesuchare(dx,dy,dz),butdtisnotafour-scalar.

Thefirststeptowardsusingrelativisticdynamicsisfindingthecorrectexpressionoflinearmomentum.Aswewellknow,thelinearmomentumofaparticleofmassmandvelocitysmallrelativetocis

Wecansolvetheproblemofthenon-invarianceofdtbytakingthederivativerelativetothepropertimet0,thetimeinthereferenceframemovingwiththeparticle,ratherthanrelativetot.Recallingthatdt0=dt/γwehave

(6.33)

WecanimmediatelycheckthatthisexpressiontendstotheNewtonianoneforsmallvelocities,namelyfor .Asamatteroffact,γdoesnotdiffermuchfrom1evenatquitelargevelocities.Forexample,evenatυ=0.25c,γ=1.03,ithasincreasedbyonly3%.However,whenthevelocityapproachesc,theincreaseofγbecomesveryrapid,forexample,forυ=0.5c,γ=1.15,forυ=0.75c,γ=1.51,forυ=0.99c,γ=7.09,todivergeforυ→c.Ifwetrytoaccelerateaparticle,whenitsvelocityapproachesthespeedoflighttheworknecessarytoincreasethevelocityfurtherbecomeslargerandlarger.Theworkusesalargerandlargerfractionofforcetoincreasetheγfactorandlessandlesstoincreasethevelocity.Theworktoreachcwouldbeinfinite.

WehavenowfoundthespacevectorEq.(6.33)thatcanbepromotedtofour-vector,whichiscalledfour-momentum.Whatisitsfourthcomponent?Takingintoaccountthatdt/dt0=γitisclearly

(6.34)Thisveryimportantquantityis,asapartofaconstant,theenergyofafree

particle,aswillbecomeclearsoonafterhavingfoundthelawofmotion.Beforedoingthatweexpressthenormofthefour-momentum .As

allthenormsofthefour-vectors,thisisaLorentzinvariantquantity,afour-scalar.Itsexpressionisparticularlysimpleintherestframeoftheparticle,inwhichp=0,andwehave

(6.35)Thenormofthefour-momentumisproportionaltothemasssquaredofthe

particle.Wenowstatewithoutdemonstrationthat,oncetheexpressionofthe

momentumischangedaccordingtoEq.(6.33),theexpressionoftheNewtonlawdoesnotneedanyfurtherchange.However,therearenowtwotimedependentfactorsinthederivative,thevelocityandγ.Wehave

(6.36)Noticethatneithertheforcenorthetimederivativeofthemomentumarethe

spacecomponentsofafour-vector.However,suchareFdtanddp,andconsequentlyEq.(6.36)isLorentzcovariant.Historically,theequationwasfoundforthefirsttimeinJune1905byH.Poincaré,whodemonstrateditscovarianceand,inaddition,thatitistheuniqueexpressionenjoyingsuchaproperty.

Wearenowreadytoseethephysicalmeaningofthefourthcomponentsofthefour-momentumandofFdt,namelyof .Weshallproceedinawayquitesimilartowhatwedidforthekineticenergytheorem.LetF(r)betheresultantforceactingontheparticleatthepositionvectorr.WecalculateitsworkwhentheparticlemovesfromAtoBonacertaintrajectory,asshowninFig.6.14.

Fig.6.14 Thetrajectoryofaparticleandtheforceactingonit

Theelementarydisplacementdsinthetimeintervaldtisds=vdt.The

workdonebyFis .Toevaluatethelastdot

productwedifferentiateEq.(6.35),obtaining .Substitutingp=mγv,andsimplifying,wehave

(6.37)andtheelementaryworkisthen

(6.38)TheworkdonebytheforcewhentheparticlemovesfromAtoBis

(6.39)

ExactlyasinNewtonianphysics,theworkdonebytheresultantoftheforcesontheparticleisthedifferencebetweenthevaluesofafunctionofthevelocityonlyattheendandatthebeginningoftheconsideredtrajectory.Inthefollowingweshallconsideronlyfreeparticles,namelyinabsenceofpotentialenergy.Intheseconditions,wecansaythattheenergyoftheparticleis

(6.40)Weseethatthefourthcomponentofthefour-momentumisjusttheenergyof

theparticle,dividedbyc.Forthisreason,thefour-momentumisalsocalledanenergy-momentumvector.Itscomponentsare .Itsnorm,orbettertheoppositeofitsnormis

(6.41)Therelativisticenergyofafreeparticle,Eq.(6.40),isnotonlykinetic

energy.Indeed,theparticlehasenergyalsowhenitisatrest.Itiscalledrestenergyandweshallindicateitwith

(6.42)Wecansaythattherelativistickineticenergyofafreeparticleisitstotal

energylessitsrestenergy,namely

(6.43)Weseeimmediately,bydevelopinginseriesofβ2,thattherelativistic

kineticenergytendstothenon-relativisticoneatlowvelocities:

Ontheotherhand,atveryhighvelocities,Eq.(6.40)showsthattheenergy

oftheparticlegrowswithoutlimitswhenitsvelocityapproachesthespeedoflight.Aswehaveseenforthemomentum,thisisduetodivergenceoftheγfactor.Theparticle“accelerators”ofthelaboratoriesstudyingtheelementaryparticlesworkusuallywithprotonsorelectrons“accelerated”ataspeedveryclosetoc.Acceleratorsacttoincreasetheenergyoftheparticles,whiletheirvelocitymaychangeonlybyverysmallamounts.Theyshouldbemoreproperlycalled“energizers”.Indeed,particlesofnon-zeromasscanneverreachthespeedoflight.Theirenergyandmomentumwouldbeinfinite.Weshallcomebacktomasslessparticlessoon.

Thefundamentalmechanicalquantitiesofafreeparticleareitsmass,itsmomentumanditsenergy.Thesequantitiesarelinkedbytwofundamentalequations,Eq.(6.41)thatweshallnowwriteinabitdifferentform(multiplyingbyc2)andasomewhatdifferentexpressionofEq.(6.33).Theyare

(6.45)

(6.46)Wenowobservethatinnatureelementarymasslessparticlesexist.Suchare

thephotons,thequantaoflight,andalsothequantaofthestronginteractionbindingthequarksinaprotonandinanucleon,whicharecalledgluons.Whenm=0,theexpressionEq.(6.33)hasnomeaning,becauseitcontainstheratiobetweenanullandaninfinitequantity.ThemostgeneralexpressionoftherelativisticmomentumisEq.(6.46)thatisvalidbothformassiveandformasslessparticles.

LetushaveabetterlookatEq.(6.45)withthehelpofthe“cartoon”ofFig.6.15.Inthegeneralcase,Fig.6.15a,theenergyislikethehypotenuseofarighttrianglehavingmc2andpcassides.Itisgivenbythequadraticsumofthetwoquantities,namelyitisthesquarerootofthesumoftheirsquares.Oneofthem,mc2,isthemassenergy,theotherone,pc,istheenergyofitsmotion.

Fig.6.15 Relationbetweenenergy,momentumandmass.aGeneric,bparticleatrest,cmasslessparticle

Iftheparticleisatrest,itsenergyisonlymassenergy,orrestenergy

(6.47)Herewemustwarnthereaderthatthisequationisoftenwritteninthepress,

butalsointhescientificliterature,asE=mc2,whichisnottrue,because,aswesawingeneralitisE=mγc2,Eq.(6.40).Theconfusionisincreasedbywritingmγ“relativisticmass”andtalkingofmassvaryingwithvelocity.Thesearearchaicconceptsthatwereintroducedwhenrelativitytheorywasbeingdeveloped,butshouldbeavoided.Indeedthemassisaninvariantquantityanddoesnotvarywithvelocity.Thetermmγisapartfromafactorc2notelsethantheenergy,whichisthefourthcomponentofafour-vector.

Equation(6.46)tellsusthatthemassenergyisenormous,duetothec2factor.Howevermatterandenergyarenotequivalent.Indeed,matterhasexistedsincetheoriginoftheuniverseanddoesnotconvertintoenergy.Thereasonisthatthematterparticleshavecharges,theelectric,theweakandthestrongones.Thesechargesareconserved.Wecannotdestroy,forexample,anelectronandgetenergyfromitsmass.Wecanhowever,annihilateanelectronwithitsantiparticle,thepositronthathasoppositecharge.However,thequantityofantimatterintheuniverseisverysmall.Weshallcomebacktothemassandtoenergytransformationsinthenextsection.

Figure6.15cshowsthecaseofamasslessparticle,sayaphoton.ForEq.(6.45),beingmasslessmeansthat

(6.48)andfromEq.(6.46),forphotons

(6.49)afreemasslessparticlecanmoveatonlyonespeed,thespeedoflight.

6.10 Mass,MomentumandEnergyforaSystemofParticlesWenowconsiderasystemoffreeparticles,namelytherearenoforces,externalorinternal,actingonthem.Asinnon-relativisticphysics,thetotalmomentumandthetotalenergyofthesystemarethesumofthehomologuesquantitiesofthesingleparticles,namely

(6.50)

Thesituationismorecomplexiftheparticlesinteractwithinternalforces.In

particular,Eq.(6.50)arenotvalid.Wedonothavethetimetodiscusstheissuehere,butonlymentionthat,inadditiontothemechanicalonesoftheparticles,therearebothenergyandmomentumdistributedinthefieldsofforces.

Comingbacktothesystemofrelativisticnon-interactingparticles,weshallnowlookatitstotalmass.Asforthesingleparticle,thetotalmomentumandthetotalenergyofasystemare(takingintoaccountthecfactors)thefourcomponentsofafour-vector,ofwhichMc2isthenorm.

(6.51)

Weseehereafundamentaldifferencefromthenon-relativisticcase:themassofthesystemisnotthesumofthemassesofitsconstituents.

Considernowseveralexamples.

ExampleE6.3FindtheexpressionsforthemassofthesystemoftwophotonsofthesameenergyE,iftheymoveinequaloroppositedirections.

Forthephotonthathaszeromass,pc=E.ConsequentlythetotalenergyEtot=2E.

Ifthephotonshavethesamedirection,thenthetotalmomentumisptot=2E/candthereforethemassism=0.

Ifthevelocitiesofthephotonsareopposite,itisstillEtot=2E,butptot=0,andhencem=2E/c2.

Ingeneral,ifθistheanglebetweenthevelocities,

andhence

ExampleE6.4Considertwoparticleswiththesamemassmmovingwiththesameinitialvelocityυofoppositedirection.Thetwoparticlescollideandsticktogether.Thefinalkineticenergyiszero.Macroscopicallywecallthecollisioncompletelyinelastic.However,thetotalenergydidnotvary,becausetherestenergyhasincreasedbythesameamount.Inrelativisticmechanicstheinelasticcollisionsdonotexist.Energyisalwaysconserved

Inotherwords,themassofthefinalbodyisnotM=2m,but,whichislargerthan2m.Themassincreaseisextremely

smallatlowvelocities.Asanexample,supposethatυ=300m/s,whichisquitelargeforeverydaylife,butverysmallcomparedtoc,beingthatβ=υ/c=10−6.Developingtheaboveexpressioninserieswehave

whichdiffersfrommby,inorderofmagnitude,10−12.Thisissosmallthatitcannotbemeasured.Inotherwords,therestenergyissolargethatitsincreasecorrespondingtothedecreaseinkineticenergyisundetectable.Thedecreaseofkineticenergybetweeninitialandfinalstateisonthecontraryevident.Itlookslikeenergyisnotconserved.But,whatappearstohavebeenlostisratherhiddeninthemassenergy.

ExampleE6.5Themostmassivenuclei,assomeoftheUraniumisotopes,areoftenunstable.Theycanbreakupinfragmentsspontaneously,ormakethemabsorbaneutron.Supposethefragmentstobetwoandm1andm2theirmasses,whileMisthemassofthemothernucleus.Westatethatm1+m2<M.Indeed,theenergyconservationrequiresthat

ThefinalkineticenergyEK1+EK2istheenergyproducedforexampleinapowerstation.Theremainingenergydifference maycorrespondtoasmallmassdifference,butthecorrespondingenergycanbelargeduetothefactorc2.Letusseeanumericalexample.

Weprofitfromanexampletointroduceameasurementunitofmassthatiswidelyusedinatomicandsubatomicphysics.Aswehaveseen,energycanbemeasuredinelectronvolt,Eq.(3.78).Asthemassisequaltotherestenergydividedbyc2,weshallmeasureitineV/c2.

Thesimplestnucleus,thehydrogenone,issimplyaproton,themassofwhichis .Themassoftheneutronisabitlarger,

.Themassoftheelectronisabout2000timessmaller,

.ThemostmassivenucleihavemassesofhundredsofGeV/c2.Inaheavynuclearfission,namelyabreakup,thereleasedenergyisofseveralMeV.Inotherwords,themassdifferencebetweentheinitialandthefinalstateis,inrelativevalue,ofafewpartsinhundredthousandths.Thesevaluesaresmall,butcanbemeasured,andthepredictionsofthetheorycanbechecked.

Inthelightestnucleitheoppositeprocesscanhappen.Thatprocessisfusion.Forexample,twoneutronsandtwoprotonscanjointogethertoproduceaHenucleus.Thisisbecausethemassofthelatter, ,issmallerthanthesumoftheinitialmasses.Letuscalculatethemassdefect,namely

Themassdefectcorrespondstothebindingenergy,namelytoseparatethefourcomponentsofaHenucleuswemustgiveitanenergyof28.3MeV.

ExampleE6.6Considernowthehydrogenatom,whichismadeofaprotonandanelectron.Itsbindingenergy,namelytheenergytoseparatetheelectronfromtheprotonis∆E=13.6eV.Themassdifferenceinrelativevaluesis

whichisaverysmallfraction.Theatomicenergyscaleismuchsmallerthanthenuclearone.

ExampleE6.7WhenenergyismeasuredineV,themomentaaremeasuredineV/c.Letussee,forexample,thevalueinSIofa1meV/cmomentum.Itis

6.11 ForceandaccelerationAswehaveseen,therelativisticlawofmotionofaparticleofmassmundertheactionoftheforceFstatesthattheforceisequaltotherateofchangeofmomentum.Thisisp=mγ(υ)v.Itcontainstheproductoftwofunctionsoftime.Consequently,thederivativeisthesumoftwoterms

(6.52)

Takingthederivativeofγ(υ),weobtain

WesubstitutethisexpressioninEq.(6.52)takingintoaccountthatdυ/dtisthecomponentoftheaccelerationinthedirectionofthevelocity,namelythat

,whereuυistheunitvectorofvelocity,obtaining

(6.52)whereβisthevectorv/c.

Weseethattheforceisthesumoftwoterms,oneparalleltotheaccelerationandoneparalleltothevelocity.Therefore,wecannotdefineany‘mass’astheratiobetweenforceandacceleration.Athighspeeds,themassisnottheinertiatomotion.

TosolvefortheaccelerationwetakethescalarproductofthetwosidesofEq.(6.52)with .Weobtain

Hence

(6.53)and,bysubstitutioninto(6.52)

(6.54)Theaccelerationisthesumoftwoterms,oneparalleltotheforce,andone

paralleltothespeed.Equation(6.52)anditsequivalentEq.(6.54)havebeentheobjectofalarge

numberofexperimentalcontrolswithhighenergychargedparticleslikeprotons,nucleiandelectronsunderelectricandmagneticforcesindifferentconfigurations.Theengineersdesigningtheacceleratorsatrelativisticenergiesusetheseformulasintheireverydaywork.

Wenoticethatforceandaccelerationhavethesamedirectionintwocasesonly:1.forceandvelocityareparallel:F=mγ3a;2.forceandvelocityareperpendicular:F=mγa.Theproportionalityconstantsaredifferent.Considerforexampleaparticlemovingwith95%oflightspeed,thatisβ=0.95andγ=3.2.Iftheparticletravelsonacircle,thecentripetalforceshouldbe3.2timeslargerthanwhatwasforeseenbyNewtonianmechanics.However,ifitisinarectilinearacceleratedmotiontheforcenecessarytogiveitthesameaccelerationisγ3=32.8timeslargerthaninNewtonianmechanics.Weseethat,eveninthesespecialcases,wecannotconsidermassastheinertiatomotion.

6.12 LorentzCovarianceofthePhysicsLawsWehaveseenhowtherelativityprinciple,originallyestablishedbyG.GalileiintheXVIIcentury,wasfoundtoholdforelectromagneticinteractions,providedthatthetransformationsofcoordinatesandtimebetweentwoinertialreferenceframesareLorentztransformations.Thisledtospecialrelativity.Thetheory,however,canworkonlyifallthephysicslawsturnouttobeLorentzcovariant.Indeed,wehavealreadydiscussedthatforthesecondNewtonlaw.

WehavealreadyfirmlystatedthattheLorentztransformations,whiletheyhistoricallydiscoveredaguaranteefortherelativityprincipleofaspecificinteraction,canbedemonstratedindependentlyofelectromagnetism,onthebasisofverygeneralassumptionsaswesawattheendofSect.6.4.

Itremainstobeseen,however,whethertheotherforces,orbetterinteractions,ofnaturesatisfytherelativityprinciple,namelyiftheequationsthatrulethembehaveinaLorentzcovariantform.Theanswerisyes,butwecangivehereonlyafewhints.

TheNewtonlawofthegravitationalforce,

(6.55)isclearlynotLorentzinvariant.Indeed,thisexpressionimpliesinstantaneouspropagationoftheeffectsoveranydistance.If,forexample,oursunwouldsuddenlydisappear,thegravitationalforceonearthwouldgotozeroimmediately.ButLorentzinvariancerequiresthatallthefundamentalinteractionspropagatewithaspeednotlargerthanc,whichistheparameterinaLorentztransformation.Consequentlywewouldbesafestillfor8min,thetimetakenbythegravitationalwaveresultingfromtheexplosiontoreachus.Therelativistictheoryofgravityiscalledgeneralrelativity,aswehavealreadymentioned.Theequationsweresentforpublicationattheendof1915independentlybyDavidHilbert(1862–1943)andA.Einstein.Wehavenowanenormousquantityofexperimentalproofsofitsvalidity.Weonlymention,asanexample,thatthedataoftheglobalpositionsystem,theGPS,whichisbasedonaconstellationofartificialsatellites,wouldgivewronginformationonourpositionifnotelaboratedwithgeneralrelativity.

AlltheotherforceswestudiedinChap.3,theelasticforce,theforcesoftheconstraints,theforcebetweenmolecules,etc.are,atafundamentallevel,duetoelectromagneticinteraction.Assuch,thelawsbywhichtheyaregovernedareLorentzinvariant.

Theothertwofundamentalinteractions,theweakinteractionandstrong

1.

2.

3.

4.

5.

6.

7.

8.

interaction,werediscoveredaftertheestablishmentofspecialrelativityandtheirequations,whicharequantumtheories,werewritteninaLorentzcovariantformsincethestart.Theirvalidityhasbeenprovenwithamyriadofveryhighprecisionexperimentsonhighenergyparticlesbothfromnaturalsources,liketheradioactivedecaysandcosmicrays,and,mainly,intheacceleratorlaboratories.

6.13 WhatIsEqualandWhatIsDifferentWesummarizeheretheconceptsthataredifferentinrelativisticmechanics(r.m.)fromNewtonianmechanics(n.m)andthosethatremainunaltered.

Therelativityprincipleisvalidbothinn.m.andinr.m.

Thecoordinatetransformationsaredifferent,Galileiinn.m.,Lorentzinr.m.

Timeandsimultaneityareabsoluteinn.m.,relativeinr.m.

Thelawforsummingvelocityisdifferent.

Inn.m.,velocitiescanhaveanyvalue;inr.m.theycannotbelargerthanc.

Theexpressionsofmomentumaredifferent.

Theforceshavethesameexpressions.

Theforceisequaltothetimederivativeofthemomentuminboth.

9.

10.

11.

12.

13.

14.

15.

Thetotalmomentum(andthetotalangularmomentum)ofanisolatedsystemareconservedinbothcases.

Theenergyhasdifferentexpressions.Thekineticenergyisdirectlyproportionaltothesquareofvelocityinn.m.,notinr.m.Therestenergydoesnotexistinn.m.

Theenergyofanisolatedsystemisconservedonlyifalltheforcesareconservativeinn.m.,alwaysinr.m.

Thetotalmomentumofasystemofnon-interactingparticlesisthesumofthemomentaofthesingleparticles.Thesameistrueforenergies.Thisbothinn.m.andinr.m.Inr.m.thesameisnottrueforthesystemsofinteractingparticles.Wecanonlyhintatthereasonforthathere.Itlaysinthefactthatthefieldoftheinteractionforcecontainsbothenergyandmomentum.

Themassofacompositebodyisthesumofthemassesofitscomponentsinn.m.itisnotinr.m.

Inn.m.,forceandaccelerationareparallel;theyarenotso,ingeneral,inr.m.

Inn.m.theproportionalityconstantbetweenforceandaccelerationisthemass,whichactsasinertiatothemotion.Inr.m.accelerationisnotproportionaltotheforce,thereisno“inertial”mass.

16.

6.1.

6.2.

6.3.

6.4.

6.5.

6.6.

ThemassisinvariantbothundertheGalileiandtheLorentztransformations.

6.14 ProblemsConsidertworeferenceframes,S,whichwecallfixed,andSʹ,whichwecallmobileasinFig.6.2.InthetwoframesthereareclocksasthoseinFig.6.5.DeveloptheargumentanalogoustothatofSect.6.5ifthearmsoftheclocksareinthedirectionofthexaxis,namelyoftherelativevelocity.

Amuonisproducedbycosmicraysintheatmosphere.Ittravelsatυ=0.99cfor4kmandthendecays.(a)Howlongdoesitliveinourreference?(b)andinitsframe?(c)Howmuchisthethicknessoftheatmosphereitcrossedinitsreference?

Aparticleofmassmmovesinastraightmotionalongthexaxiswith

.Finditslimitvelocityfort→∞.Findtheexpressionof

theforceactingonthepoint.

Aparticleofmassmmovingwiththespeedυ=(4/5)c,hitsaparticleatrestwiththesamemass.AfterthecollisionthetwoparticlesformauniquebodyofmassM.FindMandthevelocityofthisbody.

Thecosmicrayscontainprotonswith1010GeVenergy.FindthetimeinthereferenceframeofsuchaprotontocrosstheGalaxy.

Finditsmomentum(inMeV/c)ofanelectronof1meVkineticenergy.

6.7.

6.8.

6.9.

6.10.

6.11.

6.12.

1

Findthemomentum,inMeV/cofanelectrontravellingatc/2.

Findtheenergyofanelectrontravellingat80%ofthespeedoflight.

Aparticlecalledρhavingmass770meV/c2decaysatrestintwoparticlescalledπ,whichhavemassm=140meV/c2.Findtheirvelocity.

IntheLEPacceleratoratCERN,electronswereaccelerateduptoanenergyof50GeV.Findtherelativedifferencebetweenthevelocityoftheelectronsandlight.

Aparticlecalledtauhasalifetimeof0.3ps.Findthevelocityitshouldhavetotravel1mminalifetime.

AZ°(mass91.2GeV/c2)particledecaysatrestinanelectronandapositron(theyhaveequalmasses).Findtheenergyandthemomentumoftheelectrons.Howmuchdoes,inrelativeterms,thevelocityoftheelectrondifferfromc?

FootnotesForanelementaryproofofthisresult,seeJ-M.Lévy-Leblond“OnemorederivationoftheLorentztransformation”AmericanJournalofPhysics44(1976)271andA.PelissettoandM.Testa“GettingLorentztransformationswithoutrequiringaninvariantspeed”AmericanJournalofPhysics83(2015)338.

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©SpringerInternationalPublishingSwitzerland2016AlessandroBettini,ACourseinClassicalPhysics1—Mechanics,UndergraduateLectureNotesinPhysics,DOI10.1007/978-3-319-29257-1_7

7.ExtendedSystems

AlessandroBettini1

DipartimentodiFisicaeAstronomia,UniversitàdiPadova,Padova,Italy

AlessandroBettiniEmail:alessandro.bettini@pd.infn.it

Inthischapterweshalldiscussthemechanicsofextendedsystems,namelyofmechanicalsystemscomposedofmorethanoneparticleorbybodiesoffiniteextension.Asamatteroffact,eveninthesimplestcaseofapoint-likebodyundertheactionofaforce,atleastanotherbody,givingorigintotheforce,mustexist.Everyactionisalwaysaccompaniedbyareaction.Inotherwords,thesimplestmechanicalsystemconsistsoftwointeractingparticles.Wehaveconsidered,forexample,themotionofearthorofaplanetaroundthesun.Wehadignoredthesun.Wecoulddothatwithoutmucherrorbecauseitsmassisenormouslybigger.Thesehoweverareparticularcases.

Inthefirstthreesectionsweshallstudytwo-bodysystems.Weshallsee(Sect.7.1)howthepotentialenergy,correspondingtotheforcethatonebodyexertsontheotheris,infact,relativetothepair.Inotherwords,itisaninteraction.WeshallthenintroduceinSect.7.2theconceptsofcenterofmassandofreducedmass.InSects.7.3and7.4weshalldiscusstwoexamplesofatwo-bodysystem,thedoublestarsandthetides,aphenomenoninanothertwo-bodysystem,earthandmoon.

Theexperimentalstudyofcollisionsbetweentwobodieshad,andstillhas,anenormousimportanceinthedevelopmentofphysics.InthesectionsfromSects.7.5to7.7weshallseethecollisionexperimentsbetweentwopendulumsthatledNewtontoestablishtheprincipleofconservationoflinearmomentum.Thisisoneofthefundamentalprinciplesinphysics,strictlyconnectedwiththe

action-reactionlaw.Weshallthenmovetosystemsofmanyparticles,introducingtheconceptsof

totallinearmomentum(orquantityofmotion)andtotalangularmomentumofasystem.Weshallfindthefundamentallawsgivingtheirrateofchange,andstudythepropertiesofaprivilegedpoint,thecenterofmassofthesystem.

Inthelasttwosectionsweshallcomebacktothestudyofcollisionsbetweenextendedbodies.

7.1 InteractionEnergyInourdiscussionsonpotentialenergyintheprecedingchapters,wehaveanalyzedtheproblemsasifonlyonebodyexisted,onwhichgivenforceswereacting.Forexample,wesaidthatthepotentialenergyoftheweightofabodyofmassmattheheighthismgh.Thisisaperfectlycorrectstatementwhenthemassofthebodyunderconsideration,anappleforexample,ismuchsmallerthanthebodywithwhichitinteracts,theearthintheexample.Inthissituation,areferenceframeunitedwiththelargerobjectcanbeconsideredatrest.Asamatteroffact,whentheapplefallstowardstheearth,alsotheearthfallstowardstheappleinanacceleratedmotion.Inpractice,bothearthvelocityandaccelerationarecompletelynegligible.Rigorouslyspeakinghowever,wearedealingwithatwo-bodysystem,theappleandtheearthandmghisthevariationofpotentialenergyoftheearth-applesystem,whenthedistancebetweenthecenteroftheappleandthecenteroftheearthincreasebyh.Inotherwords,thepotentialenergyisapropertyofthecoupleofobjectstogether;itcannotbeassociatedtooneortheotherindividually.Indeed,ifthetwointeractingbodieshavecomparablemasses,bothofthemaccelerateconsiderablyundertheactionofinteractionforces.Thekineticenergyofeachofthemwillvaryatthevariationoftheinteractionpotentialenergy.Letusnowstudytheissue.

WestartwithasimpleexampleinFig.7.1.Itismadeoftwosmallspheresofmassm1andm2joinedbyaspring.Intheupperpartofthefigurethesystemisinitsconfigurationatrest.Wenowmovebothspheresandcallx1andx2thetwodisplacements,invalueandsign,measuredeachfromitsequilibriumposition,asinthelowerpartofthefigure.Bothforcesnowact,F21onsphere1andF12onsphere2.Theforces,anactionreactionpair,areequalandopposite.Theseareelasticforces,whichareproportionaltothestretchthatis∆x=x1–x2(N.B.x1ispositive,x2isnegative).

Fig.7.1 Twomasseslinkedbyaspring

Theelasticpotentialenergy,Eq.(3.2),is

(7.1)wherekisthespringconstant.Noticethatthisenergydoesnotbelongtooneortheothersphere,buttothewholesystem,inotherwordsistheinteraction(throughthespring)energybetweenthespheres.

Thepotentialenergyofanysysteminagivenstateisalwaystheworkthatmustbedoneagainsttheforcesthatthesystemdevelopstochangeitsstatefromthe(arbitrarily)definedzeroenergystatetothegivenstate.Inourcasethezeroenergystateiswhenthespringisnotdeformed.Intheabovestatement,alltheworkmustgointoachangeofthepotentialenergy,namelyitmustbedoneatconstantkineticenergy(zeroinparticular).Letuscheckwithadirectcalculationthatourstatementsarecorrect.

Supposewestartfromtheequilibriumposition.Wefirstmovesphere1,keeping2atrest.Callxthedisplacement(withsign)ofsphere1fromitsequilibriumposition.Wearemovingitfromx=0tox=x1.Duringthedisplacementthestretchofthespringisjustx.ThexcomponentoftheforceisconsequentlyF21x=–kxandtheworktobedoneisagainstit,

Wenowmovesphere2,keeping1steady.Wenowcallxthedisplacementofsphere2fromitsequilibriumposition.Wearemovingitfromx=0tox=x2.Thestretchofthespringisnowx1–xandthexcomponentoftheforceF12x=k(x1–x).Theworktobedoneagainstitis

Finallythetotalworkis

whichisclearlyEq.(7.1).Considernowasecondexample:thepotentialenergyofthegravitational

force.Considerapoint-likebodyofmassmonthesurfaceoftheearth(massM),atthedistanceREfromitscenter.Asweknow,thepotentialenergyis

(7.2)RecallingtheargumentsofSect.2.14oneeasilyseesthatthisistheworkto

bedoneagainstthegravitationalforcetomovethemassm,sayanapple,atzerokineticenergy,frominfinitedistance(thestatewehavedefinedtohavezeropotentialenergy)tothesurfaceofearth.Theenergyisnegativebecause,fromoutsideofthesystem,wemustworkagainstanattractiveforce.Inotherwords,theworkweareconsideringistheoppositeoftheworkofthegravitationalforce.Wealsoseethattheenergyisnotintheapplealonebutintheearthandapplesystem.

Asthelastexampleweconsidertheweightforce.Thepotentialenergyofabodyofmassmatheighthoverthelevelwehavedecidedforthepotentialenergytobezero,saytheground,is

(7.3)WeknowthatthisenergyisjustEq.(7.2),apartfromanadditiveconstant.

Indeed,inthetwocaseswemadeadifferentchoiceofthezeropotentialenergystate.Atfirstsightthetwoequationslookquitedifferent.However,considerthatEq.(7.3)isanapproximateexpression,validforsmallleveldifferencesrelativetotheearth’sradius,h«RE.WethenstartfromEq.(7.2)expandingitinseriesofh/REstoppingatthefirstorder.Weget

Nowconsiderthat issimplythegravityaccelerationontheearth’ssurfaceg.Equation(7.3)isvalidwhentakingthepotentialenergyontheearth’ssurfaceequaltozero, AndthelastequationbecomesEq.(7.3).Inconclusion,theenergyoftheweightforcemghisnotofthebodybutofthesystembodyandearth.

7.2 CentreofMassandReducedMassWenowcomebacktothesimplemechanicalsystemoftwospheresjoinedbyaspring(Fig.7.2)andconsideritsmotions.Weareinterestedinthemotionofoneofthem,saysphere1.ThesphereissubjecttotheelasticforceF21.InSect.3.2wehavealreadydiscussedthemotionofamaterialpointundertheactionofanelasticforceandfoundittobeharmonic.Inthatcase,however,theotherendofthespringwasfixedtoawallanddidnotmove.Wecanthinkofthewallasanalogousinthatcasetosphere2inthiscase.Inbothcases,theforceF12actsonthesecondbody.Butthemassofthewallissolargethatitsaccelerationiscompletelynegligible.Inthepresentcase,onthecontrary,sphere2willaccelerate.

Fig.7.2 Twospheresconnectedbyaspring

Theproblemwehavenowisthatbothpointsmove.Asweshallseeinthischapterhowever,foreverymaterialsystemaprivilegedpoint,calledcenterofmassofthesystem,exists.Itisageometricalpoint,notaphysicalone.Inthepresenceofonlyinternalforces,asinthecaseunderdiscussion,theaccelerationofitscenterofmass,inaninertialreferenceframe,iszero.Weshallprofitfromthatanddescribethemotioninareferenceframemovingwiththecenterofmassandwithitsorigininit,calledthecenterofmassframe,forabriefCMframe.Thecenterofmassofatwopoint-likebodiessystemisthepointonthesegmentjoiningthetwopointsthatdivideitinpartsinverselyproportionaltothemassesatthecorrespondingextremes.

WeshallcallCthecenterofmass,ξ1andξ2,thedistancesofthetwomassesfromitandrthecoordinateofpoint1measuredfrompoint2.Bydefinitionofcenterofmass

(7.4)Consideringthatr=ξ1+ξ2isthecoordinateofpoint1,themotionof

whichwewanttostudyis

(7.5)TheforceF21actingonpoint1willgiveittheaccelerationa1accordingto

theNewtonlaw

Wehavesofound,inthelastsideofthisequation,animportantquantity,calledthereducedmassofthesystem

(7.6)Wecanthewritetheequationofmotionofpoint1as

(7.7)whichisaverysimpleexpressionindeed.Theequationofmotionofpoint1isidenticaltoitsequationofmotionvalidwhenpoint2isfixed,providedthatweareintheCMframeandwesubstituteforthemassofpoint1thereducedmassofthesystem.

LetuscheckiftheargumentswemadeinSect.3.2agree.First,weobservethatwhenm2becomesverylargecomparedtom1,thereducedmasstendstothesmallerofthetwomasses,m1.Toseethat,justwriteEq.(7.6)as

,fromwhichimmediately for Clearly,whatwesaidinSect.3.2isthelimitcaseofwhatwearediscussinghere.

Wenowcomebacktotheproblemofthemotionofpoint1.Wecallr0thelengthatrestofthespringandsitsstretch.Hencer=r0+sandF21=–ks.But

andEq.(7.7)becomes

(7.8)whichwerecognizeastheharmonicoscillatorequation.Wealreadyknowitssolution

(7.9)whereAandϕdependontheinitialconditionand

(7.10)IntheCMframethemotionofpoint1isaharmonicoscillation.The

differencewiththecasewhenpoint2isatrestisthatinplaceofthemassoftheoscillatingbodywehavethereducedmassofthesystem.Clearly,point2moves

withaharmonicmotionofthesamefrequencybecausethereducedmassisthesameinbothcases.

InSect.3.11wehaveconsidered,asanexampleofmechanicalresonance,adiatomicmolecule,inparticularHCl.Thetwonucleiaresmallenoughtobeconsideredpoint-likeparticlesinaverygoodapproximation.Callr0theirequilibriumdistance.Whenthedistancerisdifferentfromr0,theelectroncloudthatinthemoleculesurroundsthenucleiexertsaforce,which,inafirstapproximation,isproportionaltothedisplacements=r–r0.Theforceisthenelasticandthesystemisquitesimilartotheonewejustdiscussed.Asamatteroffact,theinternalmotionsofmoleculesarecorrectlydescribedbyquantummechanics.Ourdiscussionshouldbeconsideredafirstapproximation.

Thepotentialenergyoftheinteractionbetweenthetwonuclei,whichwehavealreadyconsideredinSect.3.11,isshowninFig.7.3.Thedottedparabolaaroundtheminimumisanapproximationofthepotentialenergycorrespondingtotheelasticforce.Inthisapproximationthepotentialenergyis

Fig.7.3 Theenergypotentialofadiatomicmolecule

(7.11)TheequationoftheparabolaiswritteninFig.7.3ineVunitsofenergyand

nanometerunitsoflength.Expressingtheminjouleandmetersrespectivelyweobtain andthe“springconstant”equivalentis

.Wecalculatenowthereducedmass.Inatomicmassunits

(u=1.66×10−27kg)themassesofhydrogenandchlorineare(approximately)equalto1uand35u.Inthesameunits ,whichisclosetothesmallerhydrogenmass.

Finally,theproperoscillationfrequencyis ,whichisthevalueweusedinSect.3.11.

Asasecondexample,consideramoleculeofcarbonoxide(CO).ThepotentialenergyisquitesimilartoHCl,alsoquantitatively.Wethentakethesamevalueofthe“elasticconstant”.

Asforthereducedmasswemustconsiderthatthemassesof12Cand16Oarerespectively12uand16u.Thereducedmassisthen

Noticethat,thistime,thetwomassesaresimilarandthereducedmassissubstantiallydifferentfrom,andsmallerthan,eachofthem.Thereducedmassofasystemoftwoequalmassesisonehalfofeachofthem.

Concludingourcalculation,wefindtheoscillationfrequency ,whichisnottoodifferent,consideringourapproximations,fromthemeasuredvalue

7.3 DoubleStarsInthissectionweshallconsideratwo-bodysystemmovingintwodimensionsratherthanonedimensionasthediatomicmolecules.Itwillalsobeamuchlargerastrophysicalsystem.InChap.4wediscussedthemotionofaplanet,ofmassm,aboutthesun,ofmassMorofasatellitearounditsplanet,assumingthesuninthefirstcase,theplanetinthesecond,tobeatrest.Fromthediscussionofthelastsectionsoneclearlyunderstandsthattheassumptionisnotrigorouslytrue.Indeed,bothbodiesmovearoundtheircenterofmass.However,inthosecasesthemassofthecentralbodyismuchlargerthantheoneoftheorbitingbody,andtheapproximationisquitegood.Weshallnowconsideranastronomicalsysteminwhichthetwomasses,saym1andm2aresimilar.

Weknowtodaythatalargefractionofthestarsareinfactdouble,or,inseveralcases,evenmultiple.Toestablishthesefacts,theimageofthestarsystemneedstoberesolvedinthoseofitscomponents.Telescopesofadequateresolvingpowerareneeded.

Thefirstdoublestarsystemwasdiscoveredin1780bySirWilliamHerschel(1738–1822)intheUrsaMajorconstellation.ItiscalledXiUrsaeMajoris.MoredoublestarswerediscoveredbySirWilliamandhissonJohn(1792–1871)inthefollowingyears.ThestudyofdoublestarsgivesafurtheropportunitytochecktheNewtontheory.

Figure7.4showstheapparentpositions,namelytheanglesunderwhichthe

objectsareseenfromearthoftheXi,asmeasuredformorethanacentury.ThemotionmustbestudiedintheCMframe,asinFig.7.5.Cisthecenterofmassofthesystem,r1andr2thepositionvectorsofthetwostars,whicharepoint-likeinagoodapproximation,andm1andm2theirmasses.Letrbethevectorfromm2andm1.

Fig.7.4 Theapparentpositionsofonestarrelativetotheother(thedotinsidethecurve)fortheXiUrsaeMajorisdoublestar

Fig.7.5 aDiagramforthemotionofadoublestarsystem;bcaseofcircularorbits

Fromthedefinitionofcenterofmass and

(7.12)Weknowthattheforce,callitF(r),actingonm1istheattractionofm2and

consequentlythatitisdirectedasr.Theaccelerationis andtheNewtonlaw

(7.13)Inthiscasetoo,asinonedimension,wehavefoundthatthemotionofa

bodyofmassm1aroundanotherbodyofmassm2whenbotharemovingisthesameaswhenm2isatrestif,(a)wesubstituteform1thereducedmassofthesystem,(b)weworkintheCMtakingintoaccountthatthecenteroftheforcesisthecenterofmass.

Figure7.4showsthattheorbitshapeisanellipse.However,oneofthestarsdoesnotlooktobeinafocusoftheellipse.Thisisanopticaleffectduetothefactthatwearenotlookingnormallyattheorbitplane,butatacertainangle.

Aninterestingfeatureofbinarysystemsisthattheirperioddependsonlyonthesumofthemassesandnotontheirratio.Thisistrueingeneral,but,forsimplicity,werestrictourselvestothecircularones,asshowninFig.7.5b.Thetwostarsrotatearoundthecenterofmasswiththecommonangularvelocityω.Themotionofoneofthem,m1forexample,isgivenbytheNewtonequation

andhence and,forEq.(7.12)

(7.14)

BymeasuringtheperiodTandthedistancerbetweenthestarswecandeterminethesumoftheirmasses.

7.4 TidesThelevelofwatercontainedbytheseasandoceansvariesduringtheday.Thelevelgrows(flux)tillitreachesamaximumlevel(hightide)andthendecreases(reflux)toaminimum(lowtide)andsoon.Thephenomenonisperiodicwithaperiod(forexamplebetweenconsecutivehightides)of12h25′,whichisexactlyequaltoonehalfthetimetakenbythemoontocomebacktothesamepositionrelativetoearth,namelyitsrevolutionperiod.Consequently,sinceancienttimestideswerethoughttobeduetothemoon.TheexplanationofthephenomenonhoweverisnotatallsimpleandhadtowaitforNewton.

Consideringthatweobservethephenomenononearth,weshalldescribeitinareferenceframefixedonher.Thefirstideacomingtomindisthatthemoonattractsthepartsoftheoceansnearesttoitmorestrongly,causingtheirrise.Butitdoesnotwork,becauseafterhalfaperiod,whenthemoonisinitsfarthestposition,weobserveanotherriseratherthanalowering.Theexplanationmustbedifferent.

Wecannotconsiderheretheearthaspoint-like.Wemusttakeintoaccountthatthegravitationalfieldofthemoonisdifferentindifferentpointsoftheearth’ssurface,thathavedifferentdistancesfromthemoon.Weshallworkinareferenceframewiththeorigininthecenteroftheearth.Noticethatitcannotbeconsideredinertialinthepresentdiscussion.Ifthegravitationalforcewasequalinallthepointsofearth,itwouldbeexactlybalancedbytheinertialforce(centrifugal)duetotheacceleratedmotionofthecenteroftheearth,aswehaveseeninSect.5.7.Actually,thegravitationalforceisexactlybalancedbythecentrifugaloneonlyintheearth’scenter.Onthepartofthesurfacenearertothemoon,themoongravitationalforceislargerthanthecentrifugalone.Ontheoppositepartthecentrifugalforceislargerthanthegravitationalone.

Weunderlinethattheinertiaforceweareconsideringisduetotheaccelerationoftheoriginofthereferenceframe(thecenteroftheearth)thatisrotatingduringthedayaroundthecenterofmassoftheearth-moonsystem.Wealsoobservethatweareneglectingtheactionofthesunonearth,whichismuchmoreintensethanthatofthemoon.Wecandothat,inafirstapproximation,becausewhatmattershereisnotthegravitationalfielditselfbutitsdifferencesinthedifferentpointsoftheearth.Asaconsequenceofthemuchlargerdistance(400times)ofthesunthanthemoon,itsfield,evenifstronger,ismuchmorehomogeneous.However,thesundoeshaveaninfluence.Weshallcomebacktothatattheendofthesection.

Tosimplifytheproblem,weshallconsidertheearthasasolidspherewithalayerofwaterofconstantdepthonthesurface.WealsoassumethemoonmovingintheplaneoftheEquator.Figure7.6showsaviewinthisplane.Theearthandthemoon,atwo-bodysystem,rotateabouttheircommoncenterofmass.Theaccelerationsofbotharedirectedtowardsthecenter.Wecanalsothinkthatbotharecontinuouslyfallingtowardsthecenterofmass.

Fig.7.6 aThegeometryoftheproblem,bthetideforceindifferentpointsoftheearthsurface

InthepointA,inwhichthemoonisatthezenith,itsgravitationalattractionislargerthaninO,becauseAisclosertoit.Asaconsequence,thewaterparticlesinAfalltowardsthecenterofmass,andtowardsthemoontoo,withalargeraccelerationthantheearth’scenterO.Onthecontrary,inthepointBinwhichthemoonisonnadir,thegravitationalattractionofthemoonissmallerthaninOandthewaterparticlestherefalltowardsthecenterofmass,andthemoon,withanaccelerationsmallerthanO.

WehavefollowedtheargumentofNewtontillhere.However,atthispoint,Newtonmadeamistake(followedbyseveralauthors).Theerroristoextendwhatwasestablishedfortheaccelerationsofwaterparticlestotheirdisplacements.Ifwecoulddoso,wewouldsaythatthewaterparticlesinAmovetowardsthemoonmorethanthecenterOandthesearises,whilethoseinBmovetowardsthemoonlessthanO.Theseamovesawayfromthemoon,andrisesheretoo.ThesituationisshowninFig.7.7a.Theoceanpresentstwobumps,diametricallyopposed,onthelinejoiningthemoonwiththeearth’scenter.Thebumpsmoveinphasewiththemoon.Wethenexpecthightidestotakeplacejustwhenthemoonpassesatthezenithandatthenadir,thelowtidesinquadrature,i.e.ataquarteroftheperiodrelativetothosepositions.

Fig.7.7 Schematicviewoftheearthandofthetides.aPhaseasforeseenbyNewton.bPhaseasactuallyobserved(approximately)

Theobservations,however,donotconfirmthesepredictions.Rather,hightideshappenwhenthemoonisaboutinquadrature,thelowtideswhensheisatthezenithandatthenadirasshowninFig.7.7b.Thepresenceofthecontinentsandothereffectsmakethesituationmorecomplex.Inanycase,however,thedelaybetweenthepassageofthemoononthezenithandnadirandthehightideisalwaysofseveralhours.Thedisagreementisaconsequenceoftheabove-

mentionedmistake.Thedisplacementofawaterparticleatacertaintimeisnotparalleltotheaccelerationinthatinstant.

Thecorrecttreatmentofthetidescanbedividedintwoparts.Inthefirstpartwecalculatethetide-generatingforceasafunctionofapointontheearth’ssurface.Thesecondpartisthecalculationoftheforcedoscillationoftheoceansundertheactionofthatforce.Thiscalculationiscomplicatedbythepresenceofcontinents.Weshallshowthebasicpointsoftheargument.

Letusstartwiththeforce.Boththegravitationalandtheinertiaforceactingonawaterparticleareproportionaltoitsmass.Wecanthenconsidertheforceperunitmass.Theweightperunitmass,g,hasnoinfluenceonthetides,becauseit,evenifdifferentfrompointtopoint,isconstantintimeineachpoint.Ifthemoondidnotexist,thesurfacesoftheseawouldbeinanypointperpendiculartog.Thetide-generatingforceperunitmass,thatweshallcallf,is,aswealreadysaid,theresultantofthegravitationalattractionofthemoonandofthecentrifugalforceduetotherotationoftheearth’scenteraroundthecenterofmassofthemoon-earthsystem.

Weshallnotperformthecalculationoff(whichisnotdifficult).WeshowtheresultinFig.7.6b.Weshallhowever,evaluatetheorderofmagnitudeoff,calculatingitinA,whereitisparticularlyeasy.InthecenterOthegravitationalattractionofthemoonandthecentrifugalforceareequal.InAthecentrifugalforceisthesameandislargerthaninO.Inthispointtheyhaveequalandoppositedirections.ThemagnitudeofthesumofthegravitationalattractionandtheinertiaforceinAisconsequentlythedifferencebetweenthegravitationalattractioninAandthegravitationalattractioninO(becausethelatterisequaltotheinertiaforcebothinOandinA).Inconclusion,withrEMtheearthmoondistance,REtheearthradiusandMMthemoonmass,wehave

Consideringthattheradiusoftheearthismuchsmallerthantheearth-moondistance,RE/rEM~1/60,wecanexpandthisexpressioninseriesofthisquantityandstopatthefirstterm.Wehave

(7.15)Thisisthetide-generatingforceperunitmassinthepointA,whichhasthe

dimensionsofanacceleration.Letuscompareitwiththeweightperunitmass,,whereMEistheearthmass.Wehave,intheright-handside,withand ,

(7.16)

Firstweobservethatthetide-generatingforceisinverselyproportionaltothecubeoftheearth-moondistance.Infactitdependsonthedifferencesbetweenthegravitationalforceindifferentpoints,namelythederivativeofthegravitationalforce.Thelattervariesinverselyasthesquare,itsderivativeasthecube.

Weobservethatthetide-generatingforceisverysmall,butstillenoughtobeacauseofsuchimportantphenomena.Asamatteroffact,theheightofthetideisoftheorderofafewtoseveralmeters,correspondingtoafractionof10−7oftheearthdiameter.

Calculationsshowthatthemagnitudeofthetide-generatingforceisthesameeverywhere,henceisequaltowhatwecalculated.Itsdirection,asshowninFig.7.6bvariesasafunctionofthepoint.

Tobeprecise,wenoticethatthemoon’sorbitiselliptic.Itsdistancefromearthvariesbetween57and63.7earthradii.Consequentlyf/gvariesfrom1.33×10−7to0.96×10−7.

Wenowpasstothesecondpartofthetheory.Letuslookatthesituationinapointoftheearth’ssurface.Aswehavesaid,themagnitudeofthetidegeneratingforceisconstantintime,butitsdirectionvaries.Itsvariationisarotationatconstantangularvelocity.Inotherwords,thecomponentsoftheforce,saythehorizontalandverticalones,varyperiodicallyintime.Whentheformerisamaximumthelatterisnullandviceversa.Theocean,whichwestillimaginetocovertheentiresurface,issubjecttoaperiodicforce,varyingintimeasacircularfunction.Evenifthesystemismuchmorecomplexthanapendulum,itbehavesasaforcedoscillator.

Considerforexampleadropofwaterintheairofaspaceship.Itsnaturalshapeisspherical.Ifwedeformitabitandthenweletitgo,itwilltendtogobacktoitsnaturalshape.Butitcannotdothatdirectly.Rather,likeapendulum,itwilloscillatebetweendifferentshapesandalternatebetweenoblateandprolate.Theoscillationshaveaproperperiod,whichdependsonthephysicalcharacteristicsofthedrop,and,ifdissipativeforcesarepresent,aredamped.Thesamewouldhappenif,inabsenceofthemoon,wewoulddeformthesurfaceoftheoceanaroundtheearthandabandonit.Thesystemwouldoscillateatitsproperoscillationfrequencyor,inotherwords,withtheperiod,callitT0,ofthefreeoscillationsofthesystem.CalculatingT0isextremelydifficultduetothecomplicatedshapeofthecontinentsandoftheseabottom.Calculationsonsimplifiedmodelsleadhowever,tovaluesofT0=20–30h.

Wecanimaginetheoceanasanoscillator,withproperoscillationperiodT0.TheoscillatorisforcedbyaperiodicforceofperiodT=12h25′,whichismuchsmallerthanT0.Inotherwords,itisanoscillatorforcedatafrequencysubstantiallylargerthantheresonancefrequency.Intheseconditions,asweknow(seeFig.3.21b),displacementandforceareinphaseopposition.Consequently,thecorrectshapeisthatofFig.7.7b,notthatofFig.7.7a,insubstantialagreementwithobservations.

Wenowcomebacktotheactionofthesun.Thereasoningisexactlythesameasforthemoon,andtheresultanalogoustoEq.(7.16)isreached,obviouslywiththemassandthedistanceofthesunintheplaceofthoseofthemoon.Itissofoundthatthemagnitudeofthetide-generatingforceduetothesunisabouthalfthanthatduetothemoon.Thetwoforcesmustbeobviouslysummedasvectors.Thetwoforcesreinforceoneanotherwhenthesunandthemoonareaboutonthesameline(newandfullmoon).Thetidesarethenparticularlyample(aconditioncalledasyzygy),aboutoneandahalflargerthanthevalueforthemoononly.Onthecontrary,whenthemoonisatthefirstorlastquarter,at90°withthesun,thetwoforcespartiallycanceleachotherandthetideshavesmallamplitude(quadraturetides),aboutonehalfasforthemoonalone.

Inpractice,theheightofthetidesdependsonseveralotherfactors,liketheshapeoftheshoresofthecontinentsandtheislands,theshapeoftheseabottom,theoceaniccurrents,thewinds,etc.Neartheoceanicislandstheheightofthetidesistypicallyonemeterandnearthecontinentalshoresitisabouttwicelarger.However,insomesitesthetidesreachthreemetersandinafewevensixmeters.Particularlygreattidesareobservedindeepgulfsorfiordsfacingtheopensea.ThegreatesttidesareintheBayofFundy,inNovaScotia,Canada.Theiramplitudeis4matthebayentrance,toreach14matitsendandevenmoreatthesyzygy.

7.5 ImpulseandMomentumConsideramaterialpointofmassminaninertialreferenceframe.LetFbetheresultantforceactingonthepoint.ThesecondNewtonlawcanbewrittenintheform

(7.17)Inwords,theeffectofaforceinthetimeintervaldtisavariationof

momentumequaltotheproductoftheforceandofthetimeinterval.ThevectorquantityFdtiscalledelementary(meaninginfinitesimal)impulseoftheforcein

dt.Theimpulseofaforceinanon-infinitesimaltimeintervalfromt1tot2isdefinedtobe

(7.18)

IntegratinginthattimeintervalEq.(7.17)weimmediatelyhave

(7.19)

Thisequationexpressestheimpulse-momentumtheorem:themomentumchangeofamaterialpointundertheactionoftheforceFinthetimeintervalfromt1tot2isequaltothecorrespondingimpulse,whateveristhetimevariationoftheforceandwhateveristhelengthoftheinterval.

Theimpulse-momentumtheoremisusefulwhentheforceactsforashorttime,likeinthecollisions,strokes,explosions,etc.Inthesecasestheforceisinitiallynull,thenitquicklygrowsandasquicklygoesbacktozero.Inthesecaseswedonotusuallyknowtheinstantaneousvaluesoftheforce,butonlyitsaveragevalue.Theaveragevalueofaquantityinagiventimeintervalis,bydefinition,theintegralofthequantityoverthattimeintervaldividedbythelengthoftheinterval.Fortheforce,asshowninFig.7.8,

Fig.7.8 Animpulsiveforceanditsaveragevalue

ExampleE7.1Thehammerisaninstrumentusedsinceancienttimestoamplifythemuscular

force.Initially,attimet1,ahammerofmassm,isatrest.Withourarmweapplytoitaforceofaveragevalue tilltheinstantt2inwhichthehammerstrikestheheadofthenail.Inaccordancewiththeimpulse-momentumtheorem,inthisinstantthemomentumofthehammeris Afterthat,thehammerslowsdownandstops(itsmomentumbecomeszero)attimet3.Forthesametheorem,theaverageforceonthenailintheintervalfromt2tot3is

Inconclusion, Clearly,t3–t2ismuchsmallerthant2–t1sothatweobtainalargeamplificationoftheforce,byfactorsthancanwellbethreeordersofmagnitude.

7.6 TheAction-ReactionLawConsideragainatwo-bodysystem,madeoftwomaterialpoints,whichwecall1and2,andlater,tofollowNewton,AandB,ofmassesm1andm2.Thetwopointsinteract,point1actingon2withtheforceF12andpoint2actingon1withtheforceF21.Thetwoforcesareanactionandreactionpair.ThethirdNewtonlawstatesthattheyareequalandopposite

(7.20)Weshallassumethatnoexternalforceexistsor,ifsomedo,theirresultantis

zero.Wedealwithanisolatedsystem.F21beingtheonlyforceactingonpoint1,itisequaltotherateofchangeof

itslinearmomentum,orquantityofmotion,p1,andsimilarlyF12isequaltotherateofchangeofp2.Equation(7.20)immediatelygives

(7.21)andalso

(7.22)wherewehaveputP=p1+p2.Thisistotallinearmomentum(ortotalquantityofmotion)ofthesystem.Equation(7.22)impliesthat

(7.23)Thisequationexpressestheprincipleofconservationoflinearmomentumin

thecaseofatwo-particlesystem.Theprinciplestatesthattotalmomentumofanisolatedsystemisconstant.Weshallproveitsgeneralvaliditylaterinthischapter.Inthissectionweshalluseitinanexperimentalproofofthethirdlaw,

asNewtonhimselfdid.Indeed,wehavejustseenthat,foratwo-bodysystem,theprincipleisaconsequenceoftheaction-reactionlaw.Itisalsotruethat,ifthetotallinearmomentumofanisolatedsystemisconstant,theinternalforcesmustbepairsofequalandoppositeones.Indeed,themostaccurateverificationsoftheaction-reactionlaware,infact,verificationsofconservationofthetotalmomentum.Weobserve,however,thatinthiswayweverifytheinteractionforcestobeequalandopposite,notthattheyhavethesameapplicationline.Weshallcomebacklaterontothispoint.

Historically,thefirstexperimentalchecksoftheaction-reactionlawweredonebyNewtonandhiscontemporariesChristopherWren(1632–1723),ChristiaanHuygens(1629–1695)andJohnWallis(1616–1703).Theirexperimentsareveryaccurate,conceptuallysimpleandelegant.Theexperimentsstudythecollisionsbetweentwospheresofdifferentsizes,measurethemomentabeforethecollision,sayp1andp2,andafter,sayp1′andp2′,asaccuratelyaspossibleandcheckiftherelationissatisfiedornot.Theexperimentsweredonebyattachingthetwospherestotwowiresofequallengths,

(7.24)thusmakingtwopendulumsofthesameperiod.WhenatrestthetwospherestoucheachotherasinFig.7.9a.Wemovethespheresfromequilibrium,eachatacertaindistance,whichwemeasure.Ifweletbothspheresgoatthesameinstantfromrest,theywillaccelerate,collidewitheachotherintheirlowestpoints,separateandmovebacktogether.

Fig.7.9 Thetwo-pendulumexperimenttoverifythemomentumconservation.aPositionatrest,baninitialconfiguration

Theexperimentprofitsfromtwopropertiesofthependulum.Thefirstpropertyistheisochronismofthe(small)oscillations.Havingthesamelengths,

theperiodsofthetwopendulumsareequal,independentlyofthemassesofthespheresandoftheirinitialpositions(amplitude).Consequently,alsothetimestakentoreachtheequilibriumpositionareequal(aquarterofaperiod)andtheywillalwayscollidethere,ifabandonedatthesametimewithnullvelocity.Thesecondpropertyis:thevelocityofthependulumwhenreachingtheequilibriumpositionstartingfromacertaindistancewithnullvelocityisproportionaltothatdistance.Letusshowthisproperty.

Letmbethemassandlthelengthofthependulum.Letusremoveitfromtheequilibriumpositionbyx0asinFig.7.10andletitgowithnullvelocity.Inthisposition,thependulumisatacertainheight,sayh,abovethehorizontalthroughtheequilibriumposition.Forsmalldisplacementangleswecanuseforhtheapproximateexpression,Eq.(4.14)

Fig.7.10 Geometryofthestartingconfigurationofthependulum

(7.25)

Ifυ0isthevelocityofthependuluminanequilibriumposition,theenergyconservationlawstatesthat Hence,usingEq.(7.25),

andalsoif istheperiodofthependulum

(7.26)Weconcludethatthevelocityυ0atacollisionwillbeknownifwemeasure

theperiodonceandtheinitialpositionx0foreachandeveryexperiment.

WearenowreadytoreadhowNewtondescribeshisexperimentsinthePrincipia.Hedoesthatjustafterhavingstatedthethirdlawtoproveexperimentallyitsvalidity.Newtonbuilttwopendulumseach10ft(about3.25m)long,attachingtwospheresAandBofthematerialstotest,andfixingthetwowiresinCandDasinFig.7.9a.Wecallm1andm2themassesofAandBrespectivelyandx1andx2theirdisplacement,measuredforeachpendulumfromitsequilibriumposition(thepositionofitscentertobeprecise).Weremovebothspherestox10andx20respectivelyandaccuratelymeasurethesedistances.Noticethatx10andx20canbeonoppositesides,bothononesideorbothontheotherofO.Ifweletthemgoattheverysameinstantwithnullvelocities,theywillcollideinOwithvelocities

(7.27)Letυ1′andυ2′bethevelocitiesimmediatelyafterthecollision.Wecan

determinethembymeasuringthemaximumdistances,x10′andx20′reached(contemporarily)intheirswingback.Indeed,wehave

(7.28)Thetwoparticlesinteractonlyduringtheinstantofthecollision.The

externalforcesactingonthem,theweightandthetensionofthewire,havezeroresultant.However,thesystemisnotexactlyisolatedbecauseairresistanceexistsandisanexternalforce.Thisissmall,butitmustbetakenintoaccountinprecisionmeasurements.Newtondidthatasfollows.Hestartedoperatingwithonependulumonly.Heremoveditfromequilibriumateachofthedistancesthathewasgoingtouseinthefollowingexperiments.Heletitgowithzerovelocityandobservedthepositionreachedafteroneperiod,whichdidnotcoincideexactlywiththeoriginalone.Hemeasuredthemiss.Aquarterofthatiswhatislostinaquarterofaperiodduetotheairresistance.

Hemadeanumberofexperimentswithspheresofdifferentsubstances.Foreachofthem,hetrieddifferentpairsofstartingpositionsx10andx20,measuredthosereachedafterthecollisionx10′andx20′andappliedthejustdescribedcorrection.Eachofthemcorrespondstoavalueofthequantityofmotion;hecallsthatsimply“motion”,beforethecollision.Thelinearmomentumconservationlaw(whichisequivalenttothethirdlaw)thatweneedtoverifyis

(7.29)Hewrites(inparenthesissomeexplanations):

Thustryingthethingwithpendulumsoftenfeet(3.25m)inunequalas

wellasequalbodies,andmakingthebodiestoconcurafteradescentthroughlargespaces,asof8,12,or16feet(2.6,3.9,5.2m),Ifoundalways,withoutanerrorof3inches(8cm),thatwhenthebodiesconcurredtogetherdirectly(inastraightline),equalchangestowardsthecontrarypartswereproducedintheir(quantitiesof)motions,and,ofconsequence,thattheactionandreactionwerealwaysequal.

Hecontinuesgivingnumericalexamplesofhisresults.Theinitialandfinalmomentaaregivenin“partsofmotion”,namelyinanarbitraryunit.Theunitisclearlyirrelevant.Forclarity,weshallwritethevaluesofthetwosidesofEq.(7.29)foreachquotedresultatthebeginningofeachexperiment.Foreachexperiment,hementionsalsothechangesofthemomentumofeachbody.

InthefirstexperimentBisinitiallyatrest(9+0=2+7).

ifthebodyAimpingeduponthebodyBatrestwith9partsofmotion,andlosing7,proceededafterreflectionwith2,thebodyBwascarriedbackwardswiththose7parts.

Inthesecondexperimenttheinitialvelocitieshaveoppositedirections(12–6=–14+8).

Ifthebodiesconcurredwithcontrarymotions,Awithtwelvepartsofmotion,andBwithsix,thenifAreceded(initsmotionafterthecollision)with14,Brecededwith8;namely,withadeductionof14partsofmotiononeachside.ForfromthemotionofAsubtractingtwelveparts,nothingwillremain;butsubtracting2partsmore,amotionwillbegeneratedof2partstowardsthecontraryway;andso,fromthemotionofthebodyBof6parts,subtracting14parts,amotionisgeneratedof8partstowardsthecontraryway.

Inthethirdexperimentthetwoinitialdisplacementsareinthesamedirection(14+5=5+14).

Butifthebodiesweremadebothtomovetowardsthesameway,A,theswifter,with14partsofmotion,B,theslower,with5,andafterreflectionAwentonwith5,Blikewisewentonwith14parts;9partsbeingtransferredfromAtoB.Andsoinothercases.

Newtonthendiscussesthecausesoftheerrorsinthemeasurementsofthedistancesand,aswehavereadabove,evaluatesthemlessthan3in.,8cm.The

distancesbeingseveralmeters;thisisabout2–3%error.Therelativeerroronthemomentawassimilar(massesandperiodsbeingknownwithamuchbetteraccuracy).

ItwasnoteasytoletgothetwopendulumssoexactlytogetherthatthebodiesshouldimpingeoneupontheotherinthelowermostplaceAB;nortomarktheplacess,andk,towhichthebodiesascendedaftercongress.Nay,andsomeerrors,too,mighthavehappenedfromtheunequaldensityofthepartsofthependulousbodiesthemselves,andfromtheirregularityofthetextureproceedingfromothercauses.

He,andwewithhim,thenobservethatthetotalmomentumisconservedbothforelasticandnon-elasticcollisions.Acollisioniscalledelasticifenergyisconserved.Thisisanidealization;inpracticeperfectlyelasticcollisionsdonotexist.However,thecollisionbetweentwosteelspheresisclosetobeingso,betweentwowaxonesisnot.InanelasticcollisionthetwoforcesF12andF21areconservative.Elasticcollisionsconservemechanicalenergy,inelasticonesdonot,butinbothcasesthetotalmomentumisconserved.LetusgobacktoNewton.

Buttopreventanobjectionthatmayperhapsbeallegedagainsttherule(theactionandreactionlaw),fortheproofofwhichthisexperimentwasmade,asifthisruledidsupposethatthebodieswereeitherabsolutelyhard,oratleastperfectlyelastic(whereasnosuchbodiesaretobefoundinNature),Imustaddthattheexperimentswehavebeendescribing,bynomeansdependinguponthatqualityofhardness,dosucceedaswellinsoftasinhardbodies.

Obviously,therelativevelocityofthebodiesafteracollisionissmallerfortheinelasticthanforelasticcollisionswiththesameinitialconditions.Itmayevenbenull;thetwobodiesremainattached.Thetotalmomentumhoweverisalwaysequaltotheinitialone.

ThisItriedinballsofwool,madeuptightly,andstronglycompressed.

Hecomparedtheresultsobtainedwithballsofsteel,glassandcork.TheNewtonconclusionisthat

AndthusthethirdLaw,sofarasitregardspercussionsandreflections,is

provedbyatheoryexactlyagreeingwithexperience

Incollisionexperimentstheinteractionforcesactforaveryshorttime,duringwhichtheyareveryintense.Wetalkofimpulsiveforces.Thejustdescribedexperimentsestablishthatthetotalmomentumisconservedinanisolatedsysteminwhichtheinternalforcesareimpulsive.Andiftheforcesarenotimpulsive?ToanswerthisquestionNewtondidthefollowingexperiment.Hefixedamagnetonapieceofwoodandapieceofirononanotherone.Heleanedbothofthemonthesurfaceofthewaterinacontainer,carefullycontrollingthemtobeperfectlyatrest.Heletthetwobodiesgo.Thetwobodiesmovedonetowardstheother,undertheattractionofthemagnet,attachedthemselvestoeachotherandremainedstill.Theimportantobservationisthatthefinalbody,ironplusmagnet,doesnotmoveonwater,evenifthereisnoimpedimenttodoso.Thetotalfinalmomentumiszero,astheinitialonewas.Inthisexperimenttoothesystemisisolated.Indeed,theexternalforces,weightandArchimedesforceequilibrateeachother.

Theconservationoflinearmomentuminanisolatedsystemisafundamentallawofuniversalvalidity.

7.7 Action,ReactionandLinearMomentumConservationTheconclusionsfromexperimentswehavedescribedandmanyotheronescanbesummarizedasfollows.Webuildanisolatedtwo-bodysystem.Theresultantexternalforceiszero.TheinternalforcesareF21=–F12.Westartfromaninitialstateiandmeasurethetwomomentapi1andpi2.Weletthesystemspontaneouslyevolveundertheactionoftheinternalforces.Whenthesystemhasreachedthestate,whichwecallfinal,f,wemeasureagainthemomenta,pf1andpf2.Wealwaysfindoutthat

(7.30)Thelinearmomentumisconserved.Westatedthatthisprovestheactionand

reactionlaw(equalityoftheapplicationlinesapart).Letuslookatthatmoreclosely.Equivalently,wecanwrite

(7.31)Inwords,thechangesofthelinearmomentumofthetwobodiesareequal

andopposite.Fortheimpulse-momentumtheoremwehave

TheexperimentalverificationofEq.(7.31)isthenaverificationof

(7.32)

Inconclusion,theseexperimentsverifythatthetimeoftheforcebody1exertsonbody2isequalandoppositetothetimeintegraloftheforcebody2exertsonbody1.Inabsenceofanycontraryevidence,weassumetheinstantaneousvaluesofF21andF12tobeequalandoppositetoo.

Figure7.11showsthetimeevolutionofinternalforcesintheexampleofahypotheticalcollision.Rigorouslyspeaking,weknowfromtheexperimentonlythatthetwoareasareequalandassumethatthecurveshave,inaddition,mirrorshapes,namelythattheforcesareequalandoppositeinanyinstant.

Fig.7.11 Thetimeevolutionoftheinternalforcesduringacollision

Thisassumptionisbasicallyapostulate.Moreover,thepostulate,namelytheaction-reactionlaw,isnottrueineverycircumstance.Thereisnoproblemwhenthetwobodiesinteractthroughcontactforces,asinacollision.Problemsarisewhenthetwobodiesareseparatedbyadistance.AswehaveseeninChap.6,noeffectcanpropagateoveradistanceinstantaneously.Consequently,whenthepropagationtimeiscomparablewiththetimeinwhichthechangeinmotiontakesplace,theconceptofinstantaneousequalityofactionandreactionlosesvalidity.

Figure7.12showsasimplemechanicalmodelofa“delayed”interaction.Thetwobodiesaretwotrolleysmovingwithnegligiblefrictiononstraightrails.Trolley1carriesagun,trolley2ablockofmaterialattheheightofthegun.Atacertaininstantthegunshootsabunchofprojectiles.Supposetheprojectilestobeinvisible,havemassesmuchsmallerthanthetrolleysbutveryhighspeeds.Consequently,thebunchcarriesanappreciablemomentump.Weobservethe

systemandsee,intheinstantoftheshot,trolley1torecoilwithamomentum–pwhiletrolley2remainsstill.Thetotalmomentumofthetwotrolleysischanged.

Fig.7.12 Mechanicalmodelofanactionatadistance

IfυisthevelocityofthebulletsandLthedistancebetweenthetrolleys,thebulletswillreachtrolley2inatimeL/υandsticktotheblock.Weobservetrolley2acquiringamomentump.Thetotalmomentumofthetwoblocksisnownull,asitwasinitially.Momentumconservationisrestored.

Theexamplelooksabitstupid.Themomentumseemsnottobeconservedduringthetimethebulletsareinflight,justbecausewedidnotincludetheirmomentuminthetotal,assumingthemtobe“invisible”.Ifweincludethat,asweshould,thetotalmomentumisconservedineveryinstant.However,thingsarenotverydifferentinthecasesofactionsatadistanceasthegravitationalandelectromagneticones.Light,inparticular,isanelectromagneticphenomenon.Consideragaintwotrolleys,nowverylight,againwithnegligiblefriction.Thefirsttrolleycarriesalampthatemitsalightflashatacertaininstant.Now,lightcarriesmomentum,evenifinaverysmallamount.Consequently,thefirsttrolleyrecoilswithanoppositemomentum(p),whilethesecondisstillatrest.Supposethesecondtrolleycarriesablackscreen,whichabsorbsthelightpulsecompletely,acquiringthemomentum–p.Thesituationisquitesimilartothe“stupid”mechanicalexample.Howevernowduringthetimeofflightofthelightthetotalmechanicalmomentumisnotconserved.Themissingmomentumis,duringthistime,intheelectromagneticfield.Weshallstudythisinthe3rdvolumeofthiscourse.WeonlynoticeherethatthisisbasicallythereasonforwhichEq.(6.50)thatwefoundindiscussingrelativityisnotvalidfornon-interactingparticles.Inquantummechanicstheanalogyisevencloser;lightismadeof“invisible”particles,thephotons.Aquitesimilarsituationexistsforthegravitationalinteraction.Inthiscasealsothegravitationalfieldcarriesmomentum.Thisisdescribedbygeneralrelativity.

7.8 SystemsofParticles

Weshallnowstartourstudyofsystemsofseveral,sayN,ofmaterialpoints.TherelevantphysicalquantitiesareshowninFig.7.13,inaninertialreferenceframe.LetribethepositionvectorofthegenericpointPiinagenericinstant.Wecallthesetofpositionsofitsconstituentmaterialpointsaconfigurationofthesystem.

Fig.7.13 Asystemofmaterialpoints

LetmibethemassofPi,viitsvelocityandpi=mivIitsmomentum.Theforcesactingoneachpointcanbeusefullydividedininternal,duetotheotherpointsofthesystem,andexternal,duetoagentsexternaltothesystem.

ConsiderforexamplethesystemofJupiteranditssatellites.Theforcesononeofthem,Ganymedeforexample,aretheinternalonesduetoJupiterandtotheothersatellites,Io,Europa,Callisto,andtheexternalonesduetothesunandtheotherplanets.Obviously,beinganinternalorexternalforcedependsonthesystemunderconsideration.Ifthesystemisthesolarsystem,allthementionedforcesareinternal.

Wecall theresultantexternalforceand theresultantinternalforceactingonPi.Alltheforces,bothexternalandinternal,accordingtotheNewtonlaw,determinethemotionofPi,

(7.33)ThemotionofasystemofNpointsisdescribedbyNindependentEq.(7.33).

Theirsolutionisingeneralquitedifficult.Indeed,justthinkofthefactthattheforceactingonacertainpointatacertaintimedependsnotonlyonitsposition,butonthoseofalltheotherpointstoo.Theproblemissocomplicatedthateven

(1)

(2)

(3)

inthesimplestcaseN=3cannotbeingeneralsolvedanalytically.Numericalmethodsaretodayavailabletosolvetheproblemwiththehelpofpowerfulcomputers.

Weshallnotanalyzethemotionsofsinglepoints,butratherconsiderquantitiesrelativetothewholesystem.WeindicatewithΩageometricpointthatwechooseasthepoleofthelinearmomentaandofthemomentsoftheforces.Thispointisnotnecessarilyatrestinthereferenceframe,ratheritmoveswithavelocitythatisafunctionoftime,vΩ.TheangularmomentumofpointPiaboutΩis

(7.34)Letf1,i,f2,i,….betheforcesactingonthepointPiandFi=f1,i+f2,i

+….theirresultant.Alltheseforcesareappliedtothesamepointand,consequently,theirtotalmomentisequaltothemomentoftheirresultant.TheexternalmomentactingonPiisthen

andtheinternalmoment

Theglobalquantitiesofthesystemthatweshallneedarethefollowing:

Thetotallinearmomentumofthesystem,whichisthevectorsumofthelinearmomentaoftheconstituentpoints

(7.35)

thetotalangularmomentum

(7.36)

thetotalkineticenergy

(4)

(5)

(7.37)

theresultantforce

(7.38)

wherethevectorsinthelastsidearetheresultantsofinternalandexternalforcesactingonthesystem.

Wenowmakeaveryimportantobservationthatwillgreatlysimplifyseveralproblems.Theinternalforcescomeinpairs;theforceexertedonpointPibyanotherpointPjisequalandoppositetotheforcethatPjexertsonPiandtheirsumisnull.Consequentlytheresultantinternalforceiszero,

,andEq.(7.38)becomes

(7.39)

ThetotalmomentaboutthepoleΩis

(7.40)

wherethevectorsinthelastsidearethetotalmomentoftheinternalandoftheexternalforcesrespectively.

Noticethatwecancalculatethetotalmomentoftheforcesactingonasingle

pointPiorcalculatefirstthemomentsofthedifferentforcesandthensumthem,orsumtheforcesandthencalculatethemomentoftheresultant.Onthecontrary,tocalculatethetotalmomentactingonthesystemwemustfirstcalculatethemomentsoftheforcesonthesinglepointsandthensumthosemoments.Indeed,inthiscasetheforcesareappliedindifferentpoints.

Asecondimportantobservationisthefollowing.Theinternalforcescomeinpairsthat,fortheaction-reactionlaw,notonlyarecouples,butalsozeroarm

couples.Consequently,themomentofeachcoupleisnull,whateveristhepole.Thetotalinternalmomentiszero, andwecanwrite

(7.41)

7.9 TheCenterofMassWecontinuewiththesystemofNmaterialpoints.Figure7.14representsthesituation.

Fig.7.14 Amaterialsystemanditscenterofmass

Wedefineasthecenterofmassofthesystemthegeometricpoint(itisnotamaterialpoint)definedbythepositionvector

(7.42)

whereMisthetotalmassofthesystem.Thecoordinatesofthecenterofmassare,clearly

(7.43)

Itcanbeshown,butweshallnotdoso,thatthepositionofthecenterof

massisindependentofthechoiceofthereferenceframe.However,obviously,itscoordinatesdependonthat.Wealreadymetthecenterofmassintheparticularcaseofatwo-pointsystem.Inthiscasethecenterofmassisthepointofthesegmentjoiningthetwopointsatdistancesfromtheminverselyproportionaltothemasses.Itcanbeshownthatthetwodefinitionsagreeinthisparticularcase.

Wenowconsiderthemotionofpointsofthesystem.WecallvithevelocityofPi(whichisafunctionoftime).ByderivingEq.(7.42)wefindthatthevelocityofthecenterofmassis

(7.44)

Weobservethatthesumintheright-handsideofthisequationisjustthesumofthelinearmomentaofthepoints,namelyisthetotalmomentumofthesystem

(7.45)

WecanwriteEq.(7.44)as

(7.46)whichisaveryimportantequation.Itstatesthatthetotalmomentumofthesystemisequaltothemomentumofthecenterofmass,ifconsideredasamaterialpointinwhichallthemassofthesystemisconcentrated.

Considernowhowthetotalmomentumvariesintime.Weworkinaninertialreferenceframe.TakingthederivativeofEq.(7.45)wehave

(7.47)

but,asweareinaninertialframe,miaiisequaltotheresultantforce,bothexternalandinternal,actingonPi.

(7.48)SubstitutingthisinEq.(7.47)wehave

but,asweknow,theresultantinternalforceiszero,afactthatenormouslysimplifiestheequation.Itbecomes

(7.49)Thisfundamentalequationstatesthattherateofchangeofthetotal

momentumofamechanicalsystemisequaltotheresultantexternalforceactingonthesystem.Thefactthattheinternalforcesdonotcontributetothevariationofthetotalmomentumsimplifiesmanyproblems.

WenowgobacktoEq.(7.46)andimmediatelyseethat

(7.50)whichiscalledthetheoremofthecenterofmassmotion:thecenterofmassmovesasamaterialpointinwhichallthemassofthesystemisconcentratedandacteduponbytheresultantexternalforce.Noticethatwhilethemotionofthecenterofmassisdeterminedbytheexternalforcesonly,themotionofeachpointofthesystemdependsonbothexternalandinternalforces.

Asanexample,supposewetakeinourhandthehandleofahammer,andwelaunchitintheair.Themotionofthehammerwillbeacomplicatedcombinationofrotationsanddisplacements.Themotionofitscenterofmass,onthecontrary,willbesimplyaparabola,withthehammerrotatingaboutit(neglectingairresistance).Forthatthebodydoesnotneedtoberigid.Ifwelaunchachainintheair,itscenterofmasswilldescribeaparabolatoo.Inasimilarway,considerthebulletshotbyacannon.Itdescribesaparabola.Ifatacertainmomentthebulletexplodes,itspieceswilldescribecomplicatedtrajectories,buttheircenterofmasswillcontinueonthesameparabola,aslongasthefirstpiecehitstheground.Whenthishappensanewexternalforce,duetotheactionofground,startsactingonthesystem.

Thecenterofmass,aswehaveseen,isnotamaterialpointbutbehavesassuch.

7.10 LinearMomentumConservationThelaw(orprinciple)ofconservationoflinearmomentumstatesthat:if,inaninertialframe,resultantexternalforceonasystemiszero,thetotallinearmomentumisconstantintime.ThepropertyisimmediatelyobtainedfromEq.(7.50)

(7.51)Wecanalsosaythat,underthesamehypotheses

(7.52)Iftheresultantexternalforceiszeroinaninertialframethecenterofmass

remainsstillifinitiallystillorcontinuesonitsrectilinearuniformmotion.InSects.7.5and7.6,wehavealreadyusedthecenterofmasspropertiesand

thelinearmomentumconservationprincipleintheparticularcaseoftwo-bodysystemsanddiscussedtherelationswiththeaction-reactionlaw.

7.11 ContinuousSystemsThemechanicalsystemswehaveconsideredsofararediscrete,namelycomposedofanumberofpoint-likeparticles.Weshallnowconsidercontinuousmechanicalsystems.Sucharethesolidbodieswhentheirphysicaldimensionscannotbeneglected.Figure7.15representsacontinuousbodyofmassMandvolumeV.

Fig.7.15 Acontinuousbodyandaninfinitesimalvolumeelement

WecandividethebodyintosmallvolumesdV,whichwetakeascubeswithsidesparalleltothecoordinateaxes.LetrbethepositionvectorofthegenericdVandΔmitsmass.Wedefinethedensityρ(r)ofthebodyinthepositionrtobetheratiobetweenthemassandthevolumeoftheelementinthelimitinwhichthevolumebecomesverysmall,namely

(7.53)Thedensitycanvaryfrompointtopoint.Thinkforexampleofthe

atmosphericdensitythatdecreaseswithaltitude.Abodyissaidtobehomogeneousifitsdensitydoesnotvaryfrompointtopoint.

Hereweneedtospecifythatthelimit shouldbeunderstoodasa

physicalratherthanmathematicallimit.Indeed,whenseenatamolecularscale,matterisnotcontinuous,butmadeofsmallparticles,themolecules,separatedonefromanother.Consequently,thelimitforvolumesgoingmathematicallytozeroisnotdefined.However,thegranularityofmatterissosmallcomparedtothemacroscopicsizesandwecansafelystatethatthelimitistakenforvolumesverysmallcomparedtomacroscopicdimensionsbutstilllargeenoughtocontainagreatnumberofmolecules.Indeed,wecansay,forvolumesphysicallytendingtozero.

ThedefinitionofcenterofmassforacontinuoussystemiscompletelyanalogoustothatwegaveinSect.7.9foradiscretesystem.WedividethesysteminNsmallvolumes∆Vi,thenuseEq.(7.42)todefinethecenterofmassandtakethelimitforthesmallvolumestendingphysicallytozero.Weobtain

Thepositionvectorofthecenterofmassisthen

(7.54)

or,itscoordinatesare

(7.55)

Inthischapterweshallcontinuethestudyofmaterialsystems.Forthesakeofsimplicity,weshallconsiderthemdiscrete.Thediscussionofcontinuoussystemsiscompletelysimilar,justchangingsumswithintegrals.Thelimitationtodiscretesystemsdoesnotsubtractanythingfromthephysicsconclusions.

Asexamples,weshallnowcalculatethepositionofthecenterofmassintwoexamplesofhomogeneousbodiesofsimplegeometricalshapes.

ExampleE7.2Figure7.16representsathinsheetintheformofanisoscelestriangleofheighthandbaseb.Itcanbeconsideredtwo-dimensionalandthevolumeintegral(7.54)becomesasurfaceintegral.Itisevident,forsymmetryreasons,thatthecenterofmassmustbeontheheightofthetriangle(thesamequantityofmassmustlayontherightandontheleft).Weneedonlytofinditsycoordinate.Itisconvenienttotakeassurfaceelementsstripsofheightdyrunningfromonesidetotheother.Indeedallpointsofsuchastriphavethesameyandequallycontributetotheintegral.Thelengthl(y)ofthestripatheightycanbefound

consideringtheproportionl(y):b=y:h.Hencewehave Theareaofthestripis and,ifσisthesurfacedensity,namelythemassperunitarea,itsmassis Wethencalculatetheintegral(Fig.7.17)

Fig.7.16 Calculatingthecenterofmassofahomogenousisoscelestriangle

Fig.7.17 Calculatingthecenterofmassofahomogenouscone

ThemassMofthebodyisσtimestheareahb/2andwehave

ExampleE7.3Figure7.17representsahomogeneousconeofheighthandbaseradiusR.Asevidentinthiscasetoo,thecenterofmassisontheaxis.Tocalculateitsheighty,wetakeasvolumeelementsthinsheetsparalleltothebase.Allthepointsofasheethavethesameheighty.ThevolumeofthesheetatyisdV=πR2(y)dy.Butr(y)=Ry/hand,ifρisthedensity

whichwemustdividebythemass,thatis ,obtaining .

7.12 AngularMomentumThefundamentalequationEq.(7.49),describestheevolutionintimeofthetotallinearmomentumofamechanicalsystem.Weshallnowseehowthetotalangularmomentumvariesintime.Figure7.18showsamechanicalsysteminareferenceframe,whichwechoosetobeinertial.WearbitrarilychooseageometricΩtobethepoleofthemomentsandangularmomenta.Thepoleisnotnecessarilystill,andwecallvΩitsvelocity.

Fig.7.18 Thematerialsystem

Thetotalangularmomentumaboutthepolewehavechosenis

(7.56)

Wetakethetimederivativeandobtain

(7.57)

Thevector isthedifferencebetweentwovectors, ,bothofwhichvaryintime.Consequentlyitstimederivativeis .

Inthesecondtermintheright-handsidewehavetheratesofchangeofthelinearmomentaofsinglepoints.Asweareinaninertialframe,therateofchangeofpiistheresultantforce,bothinternalandexternal,actingonthepointPi.Wecanwrite

Thefirsttermintheright-handsideiszero,beingthesumofcrossproductsofparallelvectors.ThesuminthesecondtermisthetotallinearmomentumPofthesystem.ThethirdtermisthetotalmomentoftheexternalforcesM(e).Thelasttermisthetotalinternalmoment,whichiszero.InconclusionEq.(7.57)becomes

(7.58)Theexpressionbecomesstillsimplerwithtwodifferentchoicesofthepole.Ifthepoleisfixedinthe(inertial)referenceframe,vΩ=0and

(7.59)Thisfundamentalequationreads:therateofchangeofthetotalangular

momentumofamechanicalsystemaboutapolefixedinaninertialframeisequaltothemomentoftheexternalforcesaboutthesamepole.

Ifthepolecoincideswiththecenterofmass,whichgenerallymoves,thesecondtermintheright-handsideofEq.(7.58)isagainzero.Itisthecrossproductoftwoparallelvectors,thevelocityofthecenterofmassandthetotallinearmomentum.Wecanwrite

(7.60)Inwords:Therateofchangeoftheangularmomentumofamechanical

systemaboutitscenterofmassasapoleisequaltothetotalexternalmoment(aboutthesamepole).

7.13 AngularMomentumConservationTheprincipleofconservationofangularmomentumstatesthatinanisolatedsystemthetotalangularmomentum,aboutanypolefixedinaninertialframeisconserved.

Indeed,inanisolatedsystemtheresultantexternalforceandthetotal

externalmomentarezero.IfthepolestandsstillinaninertialframeEq.(7.59)holds,wecanstatethatthetimederivativeofthetotalangularmomentumiszero.

SimilarlyforEq.(7.60)wecanstatealsothatinaninertialframethetotalangularmomentumofanyisolatedsystemaboutitscenterofmassisconstant.

Evenifinthecaseofnon-isolatedsystems,namelyinthepresenceofexternalforces,itissometimespossibletochooseafixedpole,suchthatthetotalexternalmomentaboutitiszero.Then,thetotalangularmomentumaboutthatpoleisconserved.Weshallseesomeexamplesinthefollowing.

Noticealsothatthetotalexternalmomentmaybezeroandtheirresultantdifferentfromzero,orviceversa.Consequently,thelinearmomentumandtheangularmomentumconservationsareingeneralindependentissues.Ifthesystemisisolatedhowever,bothquantitiesareconserved.

Finallyweobservethefollowing.Aswehaveseen,conservationofthetotallinearmomentumisaconsequenceofoneaspectoftheaction-reactionlaw:actionandreactionareequalandopposite.Thetotalangularmomentumconservationisaconsequenceofthesecondaspectofthethirdlaw:actionandreactionhavethesameapplicationline.Alltheexperimentalevidence,withoutexceptions,isinfavoroftheangularmomentumconservation.Consequently,alsothissecondaspectofthethirdlawmustbeconsideredexperimentallyproven.

Thelinearandangularmomentumconservationlawsarefundamentalprinciplesofphysics,notonlyofmechanics.Inadvancedtreatmentswecanshowthattheyareconsequencesrespectivelyofthehomogeneityofthespace(therearenoprivilegedpointsinspace)andofitsisotropy(therearenoprivilegeddirections).

7.14 EnergyofaMechanicalSystemWecontinuetoconsideramaterialsystemofNmaterialpointsPiinanrithepositionvectorofPi,miitsmassandviitsvelocity.ThegenericpointPihasthekineticenergy ,andthetotalkineticenergyofthesystemis

(7.61)

Duringthemotionofthesystemitskineticenergywill,ingeneral,vary,becausethesinglekineticenergiesofthepointsvaryundertheactionofthe

forces.Let and betheresultantsofexternalandinternalforcesactingonPirespectively.Inthegenericelementarytimeintervaldtthedisplacementofthepointisdri.Thecorrespondingelementaryworkoftheforcesis

ConsiderthepointPimovingonacertaintrajectoryfromaninitialpositionAinriAtoafinalpositionBinriB.Thevariationofitskineticenergyisgivenbythekineticenergytheorem

Inwords,thevariationofthetotalkineticenergyofasystemisequaltotheworksofboththeexternalandinternalforces.Differentlyfromthecasesofthetotallinearandangularmomenta,thecontributionofinternalforcesisnotzero.

Ifallforcesactingonthesystemareconservative,theworkcanalsobeexpressedasadifferenceofpotentialenergy.CallingUPthetotalpotentialenergy,whichisthesumofthepotentialenergiesofallpointsofthesystem,weimmediatelyfindthat

(7.62)WedefinethetotalenergyofthesystemUtotasthesumofitspotentialand

kineticenergyandweseethatithasthesamevaluesinAandinB.Consideringthatthesepointsarearbitrary,weconcludethatthetotalenergyisconstantduringmovementofthesystem

(7.63)Ifthesystemisisolated,therearenoexternalforcesandonlytheinternal

onesmakework.Thisdoesnotimplythatthetotalenergyisconserved.Forthattobethecasealloftheinternalforcesmustbeconservative.Asanexampleconsiderasystemmadebyablockandatrolleysupportingit.Thetrolleycanmoveonrailswithoutappreciablefriction,butthereisfrictionbetweentheplaneofthetrolleyandtheblock.Theblockmovesonthatplane.Theplaneexertsafrictionforceontheblockandsodoestheblockontheplane.Thetwoforcesareequalandoppositewiththesameapplicationline.Duringamotion,thetotalandangularmomentumareconserved,butnotkineticenergy.

7.15 CenterofMassReferenceFrame

Itisoftenusefultoconsiderthemotionofamechanicalsysteminthecenterofmassframe,CMforbrevity,evenifthatframeisnotusuallyinertial.Westartfromaninertialframeandwedefineascenterofmassframe,theframewithorigininthecenterofmassofthesystemandwithaxesparalleltotheaxesoftheinertialframe.

Figure7.19showsthetwojustmentionedreferenceframesandagenericpointofthesystemPi.TheCMframedoesnotrotaterelativetotheinertialframe,ittranslateswiththecenterofmassvelocity.Thismayvaryand,asaconsequence,theCMframeisnotingeneralinertial.Itissoiftheresultantexternalforceiszero(evenifthetotalexternalmomentisnot)becausethenthevelocityofthecenterofmassisconstant.

Fig.7.19 Theinertialframexyzandthecenterofmassframex*y*z*

WeshallindicatewithanasteriskthequantitiesintheCMframe.Therelationbetweenthepositionvectorsinthetwoframesis

(7.64)andtherelationbetweenthevelocitiesis

(7.65)andsimilarlyfortheaccelerations

(7.66)Obviously,thecenterofmasspositionvectorandthevelocityarenullinits

reference

(7.67)InSect.7.9wehavefoundthatineveryreferenceframe,bothinertialand

not,thetotallinearmomentumofasystemisequaltothemassofthesystemtimesthevelocityofthecenterofmass.ButthelatterisnullintheCMframe

andwehave

(7.68)

TheCMframeisalsotheframeinwhichthetotallinearmomentumiszero.Itissometimescalledthecenterofmomentaframe.

WeobtainanotherinterestingpropertybyexpressingEq.(7.42)intheCMframe.ForthefirstofEq.(7.67)thisbecomes

(7.69)

Wenowconsiderthetotalangularmomentum,whichhasanimportantroleinmechanics.WemightexpectittobedifferentinthetwoframesofFig.7.19,theinertialandthecenterofmassframes.Asamatteroffacttheyareequal.Indeedthetotalangularmomentumintheinertialsystemis

Thefirstterminthelastsideistheangularmomentumaboutthecenterofmass,asapole,intheCMframe,whilethesecondiszeroforEq.(7.69).Hence

(7.70)Weconcludethatthetotalangularmomentumofamaterialsystemaboutits

centerofmassisanintrinsiccharacteristicofthesystem,independentofthereferenceframe.

7.16 TheKönigTheoremsThetwoKönigtheoremsthatweshalldiscussinthissectiongivetworelations,onebetweenthekineticenergyintheinertialandCMframesUKandrespectively,andonebetweentheangularmomentaLandL*.Inbothcasesthequantityintheinertialframeisequaltothesumoftwoterms,oneofthesystemasawhole,theothercorrespondingtoitsmotionrelativetothecenterofmass.ThetheoremsarenamedafterJohannSamuelKönig(1712–1757)

Königkineticenergytheorem.Thekineticenergyofthesystemintheinertialframeis

UsingEq.(7.65)itbecomes

TheexpressioninparenthesisinthelasttermofthelastsideisthetotalmomentumintheCMframe,henceisnull.Andweobtain

(7.71)

Wereadthisexpressionas:thekineticenergyintheinertialframeisthesumoftwoterms.Onetermisthekineticenergy“ofthecenterofmass”,ifwethinkofitasbeingamaterialpointwithallthemassofthesystem.Thesecondtermisthekineticenergyinthecenterofmasssystem,namelyrelativetothemotionofthepartsofthesystemaboutthecenterofmass.

ExampleE7.4Achildissittingonawheelchairneartoawallwithhisfeetrestingonitwithfoldedlegs.Thechild,instretchinghislegs,pushesonthewallandacceleratesbackward.Afterhisfeetdetachfromthewallhecontinuestomoveatconstantvelocity(neglectingfrictions).Whatforceshavecausedtheacceleration?Whichforceisthevariationofkineticenergy?

Oursystemisthechildandthechair.Wecannotconsideritaspoint-like,becausethestretchingofthelegschangestheshapeofthesystem.Theresultantexternalforceisthenormalreactionofthewall,N.Thisistheforcecausingtheacceleration.IfmisthemassandaCMthecenterofmassacceleration,wehaveN=maCM.

TheworkoftheexternalforceNis,ontheotherhand,zero,becauseitsapplicationpointdoesnotmove.Whichisthecauseofthekineticenergyvariation?

Intheanalysisofthistypeofproblem,thefollowingmistakeisoftenmade.Itconsistsinapplicationofthekineticenergytheoremtothecenterofmass,inaformvalidforthematerialpoint.Indeed,thecenterofmassbehavesasamaterialpointfromseveralpointsofview,butnotfromthisone.Letuslookatthat.WecanwriteEq.(7.50),whichisvalidforthecenterofmass,as

whichisformallyidenticaltothelawandisvalidforthematerialpoint.We

trynowtogoaheadaswedidinSect.2.10toshowthekineticenergytheoremforamaterialpoint.WeindicatewithdsCMtheelementarydisplacementofthecenterofmassindt,inordertohave WetakethedotproductoftheaboveequationanddsCMobtaining

WeindicatewithΓthetrajectoryofthecenterofmassandweconsidertwopositions,AandBonΓ.AswedidforthematerialpointweintegratetheaboveexpressiononΓfromAtoBobtaining

(7.72)

whichhasthesameformas(2.36).Itsmeaningishoweverfundamentallydifferent.Whiletheright-handsideofEq.(7.72)isindeedthedifferenceofcenterofmasskineticenergy,theleft-handsideisnottheworkoftheresultantexternalforce.ThisisbecausedsCMisthedisplacementofthecenterofmass,notoftheapplicationpointoftheresultant.Thelattermaynotevenhavebeendefined.Itisdefinedonlyifalltheforcesareappliedtothesamepoint.Consequently,Eq.(7.72)isnotveryusefulinpractice.

Wecanconcludethattheworkoftheresultantexternalforcehasnothingtodowiththevariationofkineticenergy.Thelatterisduetoaninternalforce,theoneduetothemusclesofthelegsofthechild.

Similarly,whenacaraccelerates,theforceproducingaccelerationisthefrictionoftheroadonthetires.Theworkofthisforceisnull.Thekineticenergyvariationisequaltotheworkoftheinternalforcesduetotheengine.

ExampleE7.5Figure7.20showstwoblocksofmassesm1andm2supportedbyahorizontalplanewithnegligiblefriction.Aspring,initsnaturallength,isfixedtotheleft-handsideoftheblockontheright.Itselasticconstantisk.Thetwoblocksmovewithvelocitiesv1andv2inthesamedirectionandwithυ1>υ2.Block1reachesblock2andhitsit,compressingthespring.

Fig.7.20 Twolocksonahorizontalplaneofnegligiblefriction

Wethencalculatethemaximumspringcompression.Weshallsolvetheproblemintwoways,usingatrivialreasoningfirst,then

usingtheKönigkineticenergytheorem.Letυ1′andυ2′bethevelocitiesafterthecollision.Thelinearmomentum(onedimension)andenergyconservationgiveustwoequations

Fromthefirstequationweexpressυ2′asafunctionofυ1′.Then,withthesecondequation,weexpressthespringenergyasafunctionofυ1′.Wedenotebyxthecompressionofthespring

Themaximumspringdeformationcorrespondstothemaximumelasticenergy.Weobtainthelatterbytakingthederivativeofthelastside,puttingitequaltozeroandsolvingforυ1′:

Bysubstitutingthisintheexpressionoftheenergyjustfound,weseethatthevelocitycorrespondingtothemaximumcompressionisthecenterofmassvelocity.Consideringthesymmetryoftheproblem,weexpectυ2′tobeequal.Thisisimmediatelyfoundfromtheaboveequation,asthereadercanverify.

Thesecondapproachtosolvetheproblemismuchquicker.Moreover,itimmediatelyshowsthereasonforbothvelocitiesbeingequaltothecenterofmassvelocity.WewritetheenergyintheformgivenbytheKönigtheorem.

whereυCMisthecenterofmassvelocityand andarethevelocitiesrelativetothecenterofmass.Wethenhave

Thefirsttwotermsintheright-handsidedonotvaryduetotheenergyand

linearmomentumconservationrespectively.Theelasticenergyisthenamaximumwhenthetwolastterms,namelythekineticenergies,andconsequentlythevelocities,relativetothecenterofmassarezero.

TheangularmomentumKönigtheorem.WithreferencetotheinertialframeofFig.7.19,wechoosethepoleinthe

originO.Theangularmomentumis

which,usingEqs.(7.64)and(7.65)becomes

Thelastsidecontainsfourterms.Thefirsttermisthetotalangularmomentuminthecenterofmassaboutthecenterofmassasapole,say .ThesecondtermisthetotallinearmomentumintheCMframeandisnull.ThethirdtermiszeroforEq.(7.69).Thefourthtermisthecrossproductofthepositionvectorofthecenterofmassandthetotallinearmomentumintheinertialsystem,say .Wecanwrite

(7.73)Wecanstatethatthetotalangularmomentumintheinertialframeisequalto

thesumoftwoterms.Onetermistheangularmomentum“ofthecenterofmass”,whichistheangularmomentumthatthecenterofmasswouldhaveifitwereamaterialpointwiththetotalmassofthesystem.Thesecondtermistheangularmomentumrelativetothecenterofmass.

7.17 ElasticCollisionsInthischapterwehavealreadydiscussedcollisionexperiments.Fromtheobservationofthelinearmomentumconservationwehavededucedthevalidityoftheactionandreactionlaw.Weshallnowtaketheoppositepointofview.Assumingthemechanicslawstobevalid,weshalldiscussinsomedetailcollisionphenomena.Weshalllimitthediscussiontomaterialpoints.Thisisanidealization.However,wecanconsidertherealbodiesaspointscoincidingwiththeircentersofmass,inwhichalltheirmassisconcentrated,aslongasinthe

collisionthekineticenergiesofeachbodydonotvary.Ifwearedealing,forexample,withthecollisionoftworigidballs,theirmotionsshouldbetranslations,withnorotation.

Wespecifythatwhentalkingofcollisionoftwobodieswedonotnecessarilyimplythatthetwobodiescomeintocontact.ConsideringforexampletheNewtonexperimentsonthecollisionsbetweentwopendulumswemightsubstitutetheballswithtwobarmagnets,withtheirnorthpolesfacingeachother.Ifwetakethependulumsoutofequilibriumandletthemgo,thetwomagnetswillapproacheachothersubjecttotherepulsiveforcebetweenthemagnets.Thisforcewillslowthemdowntilltheystopandbunchback,withouttouchingeachother.Asanotherexample,considertwoionsofthesamechargemovingonetowardstheother.Whentheyarefarfromoneanothertheyfeelpracticallynoforceandmovewithconstantvelocities.Butwhentheybecomecloseenough,therepulsiveelectricforcewillcausebothtrajectoriestodeflect.Thetwoionswillmoveoncurvedtrajectories,approachingtoaminimumdistanceandthenseparatingagain.Whentheyarefarenoughapart,theionswillagainmovepracticallywithconstantvelocities.Thefinalvelocitiesareingeneraldifferentinmagnitudeanddirectionfromtheinitialones.

Inacollisionprocesswecandistinguishthreephases.Intheinitialphasethetwobodiesaredistantanddonotinteract,namelytheforceexertedbyoneontheotherisnegligible.Thesecondphaseisthephaseofthepropercollision,whichhasalimitedduration,sayΔt.Duringthistimethetwobodiesinteract.Theinteractionforcesareinternal,anactionandreactionpair.Theinternalforcesaremuchlargerthantheexternalforces,whichcanconsequentlybeneglected.Asamatteroffactwetalkofacollisionwhenthisconditionismet.Inthethirdphasethebodiesnolongerinteractbutmoveawayfromeachother.Noticethatduringthecollisionthetwobodiesmaychangetheirshape,theinternalenergycandecreaseorincrease,oneorbothofthemcanbreakintoanumberofpieces,ortheymayjoininasinglebody,etc.Inotherwords,thebodiesinthefinalstatemaybedifferent,alsoinnumber,fromthetwoinitialones.

Asduringthecollisiononlyinternalforcesarepresent,wecanstateincompletegeneralitythatthetotallinearmomentabeforeandafterthecollisionareequal.Assumeforsimplicitytohavetwobodiesbothintheinitialandfinalstate.Weindicatewiththesubscriptsiandfthequantitiesintheinitialandfinalstatesrespectivelyandwrite

(7.74)Thecollisionissaidtobeelasticifeachofthebodiesafterthecollisionis

thesameasbeforethecollision,itsinternalenergyincluded,andifthetotalenergiesafterandbeforethecollisionareequal.Astheexternalforcesarenegligibleandastheinternalforcesarezerointheinitialandfinalstates,theinitialandfinaltotalkineticenergiesareequalaswell.Ifm1andm2arethemassesofthetwobodies,wehave

(7.75)andEq.(7.75)canbewrittenas

(7.76)Equation(7.75)isonerelation,Eq.(7.76)arethreerelations,intotalfour,

betweentheinitialandfinalstates.Weshallnowconsiderafewimportantcases.Oftenoneoftheparticlesisatrest.Ifitisnotso,wecanalwayschangethe

referenceframebychoosingaframemovingwithoneparticle(thinkofanobserversittingontheparticle).Theframeinwhichoneparticlestandsstilliscalledalaboratoryframe.Theparticlethatisstill,sayparticle2,iscalledthetargetparticle.InthelaboratoryframeEqs.(7.75)and(7.76)become

(7.77)

(7.78)ThevelocityofthetargetparticleafterthecollisionisgivenbyEq.(7.78),

(7.79)Considerthecaseinwhichthemassofthetargetisverylarge,namely

.Weseethatthefinalvelocityofthetargetparticleisverysmall.Itsfinalkineticenergy,namelytheenergygainedinthecollision,isalsoverysmall.Inthelimitofinfinitetargetmass,thefinalvelocityandkineticenergyofthetargetarezero.Forexample,astandingrailcarhitbyaping-pongballdoesnotmove,neitherdoesabilliardtablewhenaballhitsoneofitssides.Asaconsequence,thekineticenergiesofalightparticlehittingaverymassivetargetparticlebeforeandaftercollisionareequal.

Wenowconsiderthecaseoftwoequalmassparticles,whichareatrestinthelaboratoryframe.Themassesbeingequal,wecaneliminateitfromEqs.(7.77)and(7.78)andwrite

(7.80)Thefirstoftheseequationstellsusthatthethreevelocityvectorscanbe

thoughtofasthesidesofatriangle,asshowninFig.7.21.Forthesecondequationwehavearighttriangle,thehypotenuseofwhichisvi1.Thefinal

velocitiesoftwoparticlesofequalmassesinthelaboratoryframearealwaysat90°fromoneanother.Thiscanbeobserved,forexampleinabilliardgame.

Fig.7.21 InitialandfinalvelocitiesinacollisionoftwoequalmassparticlesintheCMframe

ConsidernowFig.7.22,whichrepresentstheinitialstateofthecollisionbetweentwosphericalbodies.Oneisinitiallyatrest.Thedistancebetweenthelineonwhichthecenterofthemovingbodytravelsandthecenterofthetargetiscalledimpactparameter.Itisbinthefigure.Clearly,thefinalstatedependsonb.Suppose,forexample,thatthetwobodiesarerigidspheres.Whentheytouch,theyinteractwithaforceinthedirectionofthenormaltothecontactsurface,whichdependsonb.Thisisthedirectionalsoofthevariationofthemomenta.

Fig.7.22 Theimpactparameter

Thesimplestcaseiswhentheimpactparameteriszero.Thecollisionisthensaidtobecentral.Theincomingparticletravelsonalinepassingthroughthecenterofthetarget.Whentheparticlescollide,theactionandreactionforcesaredirectedonthatline,andsoareconsequentlythefinalmomenta.Afterthecollisionbothparticleswilltravelonthisline.ThemomentumconservationlawEq.(7.78)becomesasimplerelationbetweenmagnitudes

(7.81)TheenergyconservationequationEq.(7.77)becomes

(7.82)Weseektwofinalvelocitiesasfunctionsoftheinitialoneυ1i.AsEq.(7.82)

canbewrittenas ,wecanusefullydivideitby

Eq.(7.81)obtaining .Andfinally

(7.83)Letusdiscussthefirstequation.Ifthemassoftheincomingparticleis

smallerthanthemassofthetarget(m1<m2)itsfinalvelocityisnegative,meaningthatafterthecollisionitbouncesback.Onthecontrary,ifitsmassislargerthanthemassofthetarget(m1>m2),afterthecollisionitcontinuestomoveforward,evenifwithasmallervelocity.Aninterestingcaseiswhenthetwomassesareequal.Afterthecollisionthevelocitiesareυf1=0andυf2=υi1.Thetwoballsexchangetheirvelocities.Thephenomenoniseasilyseenhittingtwopendulumsofequalmass.

Finally,if ,thenυf1=–υi1andυf2=0.Thisisthecaseofanelasticcollisionofaball,forexampleatennisone,againstawall,showninFig.7.23.Herewesupposethewalltobesmooth.Inthiscasetheforceofthewallontheballisnormaltothesurface.Wedecomposethequantityofmotionoftheballincomponentsnormalandparalleltothewall.Thelatterisnotchangedbythecollision.Tothenormalcomponentwecanapplytheresultswefoundforthecentralcollisions.Particle1istheball,particle2isthewall,hence .Afterthecollisionthewallisstillatrestwhilethenormalcomponentoftheballvelocityhaschangeditssign.

Fig.7.23 Elasticcollisiononawall

Wenowanalyzethegeneralcaseoftheelasticcollisionbetweentwoparticles.Asduringthecollisionthesystemisisolated,thecenterofmassvelocityisconstantandtheCMframeisinertial.Recallingthatυi2=0,theCMvelocityinthelaboratoryframeis

(7.84)WeobtainthevelocitiesoftheparticlesintheCMframe,whichweindicate

withanasterisk,bysubtractingtheCMvelocityfromtheirvelocitiesinthelaboratoryframe

(7.85)IntheCMframethetotallinearmomentumiszerobothbeforeandafterthe

collision.Thismeansthatthemomentaofthetwoparticlesareequalandoppositebeforethecollisionandsimilarlyafterit.Thesequantitiesarecalledcenterofmassmomentumbeforeandafterthecollisionrespectively.If isthemomentumofparticle1beforethecollision,themomentumofparticletwois–.Similarly,afterthecollisionthemomentaare,say, and– .Wewritethe

kineticenergyconservationas

andalso

(7.86)Inwords,inanelasticcollisionintheCMframe,themagnitudeofthelinear

momentumofeachparticleisequalafterandbeforethecollision.Theonlyeffectofthecollisionistochangethecommondirectionofthemomentabyanangle,say,θ,asshowninFig.7.24.

Fig.7.24 ElasticcollisionintheCMframe

Theangleθiscalledascatteringangle.Itcannotbefoundonlyonthebasisoftheconservationlaws.Firstofall,itdependsontheimpactparameterb,whichintheCMframeisthedistancebetweenthelinesonwhichthecenterofmassofthetwobodiestravelintheinitialstate.

Thedependenceofthescatteringangleontheimpactparameter,givenbythefunctionθ(b),dependsonthestructureofthecollidingbodies.Suppose,forexample,thatoneofthem,theincomingoneinthelaboratoryframe,ispoint-like,whilethetargetbodyhasastructure.Wecanthinkofthefirstasanelectron,thesecondanatom.Weimaginetheatomasasphericalcloudofnegativeelectricchargewiththepositivelychargednucleusatthecenter.Thisis

verysmallandhard.Iftheimpactparameterislargerthantheatomicradius,theelectronisnotdeflectedinitsmotion,namelythescatteringangleisθ=0.Iftheimpactparameterissmallerthantheatomicradius,theelectronpenetratesinthechargedcloud,isdeflectedbytheelectricforceandexitsinadirectiondifferentfromtheincidentone.Thescatteringangleisnowθ≠0,whichisincreasingwithadecreasingimpactparameter.Inpracticehoweveritisneververylarge.Whentheimpactparameterissmallerthanthenuclearradius,thecollisioniswiththenucleus,andisviolent.Thescatteringangleislarge.Itcanevenreach180°,namelythedirectionofmotioncaninvertifthecollisioniscentral,b=0,becausethemassofthenucleusismuchlargerthanthatoftheelectron.

Thisexampleshowshowthemeasurementofthefunctionθ(b)inascatteringexperiment(aitiscalled)canbeextremelyusefultounderstandthestructureoftheobjectsthat,likeatoms,aretoosmalltobevisible.Asamatteroffact,theexamplewehavejustmadeisquitesimilartotheexperimentperformedin1911byHansWilhelmGeiger(1882–1945)andErnestMarsden(1889–1970)thatledLordErnestRutherford(1871–1937)todiscovertheatomicnucleus.GeigerandMarsdenusedenergeticαparticles(ratherthantheelectronintheexample)sendingthemonathingoldsheetandmeasuringhowmanyofthemwerescatteredatdifferentangles.Theyfound,inparticular,thatsometimestheyweredeflectedbackwards.Iftheatomsweresoftcloudsofcharges,asinthecurrentmodel,thiscouldnothappen.Rutherfordconcludedthatasmallhardnucleushadtobepresentinsidetheatom.Inthesamewaytheinternalstructureoftheatomicnucleiwasstudiedand,in1967,thepresenceofthequarksinprotonsandneutronswasdiscovered.

7.18 InelasticCollisionsAswehavestated,linearmomentumisalwaysconservedinacollision.Thisisnotthecaseforenergy.Whenthefinalenergyisdifferentfromtheinitialonethecollisionissaidtobeinelastic.Rigorouslyspeaking,inrealcollisionsbetweeneverydaysizeobjects,atleastasmallfractionofthemechanicalenergyislost.Forexample,ifwedropasteelballonarigidfloor,itwillbunchbackbutwillnotreachexactlytheinitialheight.Ifwedothesameexperimentwithawaxballweseethatitsticksonthefloor.Therealcollisionsareneverperfectlyelastic,buthaveasmallerorlargerdegreeofinelasticity.Weshallgiveaquantitativedefinitionofthisconcept.Beforedoingthat,letusconsiderthecaseofthecompletelyinelasticcollision(thecaseofthewaxballintheexample).

Considertwosphericalbodiesofmassesm1andm2andinitialvelocitiesv

i1andvi2.Thecollisioniscompletelyinelasticifthetwobodiessticktogether,namelyiftheirvelocitiesafterthecollisionareequalvf1=vf2.Wecanindicatesimplywithvfthefinalvelocityandwritethemomentumconservationas

(7.87)Thefinalvelocityisthen

(7.88)whichisthesameasthecenterofmassvelocity(thatdoesnotvaryinthecollision)asexpected,consideringthatinthefinalstatethereisonlyonebody.Wewritedowntheinitialkineticenergy,usingtheKönigtheorem

where isthekineticenergyintheCMreference.Thefinalkineticenergyis

Weseethatinthecompletelyinelasticcollisionallthekineticenergyrelativetothecenterofmass islostinthecollision.IfwewanttolookatthecollisionintheCMframewecantakeoveralltheconclusionsofthelastsection,withtheexceptionofequalityofthemagnitudesoftheinitialandfinalmomenta.Ifthecollisionisinelastic,thefinalcenterofmassmomentumissmallerthantheinitialone,nullifitiscompletelyinelastic.Figure7.25showsthesituation.InthecompletelyinelasticcollisionallthemomentumintheCMreferenceandallthekineticenergyrelativetothecenterofmassarelost.Inthelaboratoryframenotallthekineticenergygetslost,becausethevelocityofthecenterofmassmustbethesameafterandbeforethecollision,duetothemomentumconservation.Consequently,thekineticenergy“ofthecenterofmass”cannotbelost.Inthecompletelyinelasticcollisionalltheenergythatcanbelostislost,butthisisnotalltheenergy.

Fig.7.25 CollisionsintheCMframe.aElastic,binelastic,ccompletelyinelastic

Intheaboveexampleofthewaxballfallingonthefloor,theballlosesallitsenergy.However,inthiscasethemassofthetargetisenormous,practicallyinfiniteandconsequentlythevelocityofthecenterofmassiszero.

Anapplicationofacompletelyinelasticcollisionistheballisticpendulumusedtomeasurethevelocitiesofbullets.Figure7.26showsthedevice,whichismadeofasandbagofmassMsuspendedwithabartothepivotO.ThebulletPofmassmandvelocityυtobemeasuredhitsthependulum,penetratesthebagandsticks.Bymeasuringtheresultingoscillationamplitudewedeterminethevelocityofthebagυfafterthecollision.Equation(7.81),consideringthatv21=0,andthatM»m,becomes ,or .Thisgivesthebulletvelocity,thetwomassesbeingknown.

Fig.7.26 Ballisticpendulum

LookingatFig.7.25weunderstandimmediatelythatalltheintermediatecasesbetweenelasticandcompletelyinelasticcollisionsarepossible.Theparametercharacterizingthedegreeofelasticityiscalledthecoefficientofrestitutionandisdefinedastheratiobetweenthecenterofmassmomentumafterandbeforethecollision

(7.89)Bydefinition,thecoefficientisanon-negativenumber.Itisequaltoonein

theelasticcollision.Noticethatitcanbelargerthanone.Supposeforexamplethatthetargetbodycontainsaspring,whichiscompressedandblockedbyanail.Inthecollisionthenailisbroken,thespringexpandsandgivesenergytothecollidingbodies.Thefinalcenterofmassmomentumislargerthantheinitialones.Asanotherexample,energycanbegainedincollisionsbetweentwomolecules.Sucharetheexothermicchemicalreactions.

LetuswritedownthekineticenergyintheCMreferenceframe

whereµisthereducedmass.Similarlythefinalkineticenergyis

7.1.

7.2.

7.3.

7.4.

7.5.

andfinally,forEq.(7.89)

(7.90)namelytheratioofthefinalandinitialkineticenergiesrelativetothecenterofmassisequaltothesquareofthecoefficientofrestitution.

7.19 ProblemsWhatisthetotalmomentumPofasystemofparticlesintheCMframe?

Asystemofinteractingbodiesmovesintheneighborhoodoftheearth’ssurface.Neglectairresistance.Howdoesthecenterofmassmove?

Tworailcarsmoveoneagainsttheotheronarail.Thefirstonehasamassof1000kgandmovesatthespeedof2m/s.Thesecondonehastwicethemass.Afterthecollisionthetwocarsareatrest.Whatwastheinitialvelocityofthesecondcar?Didthekineticenergychange?

Arailcarof5tmassandspeed10m/sisstoppedbybumpersin0.5s.Findtheimpulseandtheaveragevalueoftheforce.

Twopendulumscollideelastically.Initially,oneofthetwo,ofmassm2standsstillintheequilibriumposition,theotherone,ofmassm1isabandonedatacertainheightabovethat.Afterthecollisionthetwovelocitiesareequalandopposite.(a)Whatistheratiooftheirmasses?(b)Whatistheratiobetweenthecenterofmassvelocityandthevelocityofpendulum1beforethecollision?

7.6.

7.7.

7.8.

7.9.

7.10.

7.11.

InProblem7.5,knowingthekineticenergyUKi(1)ofpendulum1immediatelybeforethecollision,find:(a)thetotalkineticenergyintheCMreference,(b)thekineticenergyUKf(1)ofthefirstpendulumimmediatelyafterthecollision.

Inafirstapproximation,themoonrevolvesaroundthecenteroftheearth.Moreprecisely,earthandmoonrevolvearoundtheircommoncenterofmass.Knowingthatthemassoftheearthisabout81timesthatofthemoonandthatthedistancebetweenthetwocentersisabout60earthradii,RE,calculatethepositionofthecenterofmass(inREunits).

AplanetofmassMhasasatelliteofmassm=M/10.ThedistancebetweentheircentersisR.(a)ExpresstherevolutionperiodasafunctionofRandM.(b)Findtheratiobetweenthe(revolution)kineticenergiesofthetwobodies.

WehavemeasuredtheperiodofTearthyearsofabinarysystemandthedistancebetweenthetwostarsinRastronomicunits.Findthesumofthetwomassesinsolarmass(MS)units.

Twopoint-likebodieshaveacompletelyinelasticcollision.Thefirstbodyhasamassm1=2kgandthevelocitybeforecollisionv1i=(3,2,–1)m/s.Thesecondbodyhasamassm2=3kgandthevelocitybeforecollisionv2i=(–2,2,4)m/s.(a)FindthevelocityVofthecompositebodyafterthecollision.(b)Findthetotalenergyandtheenergyrelativetothecenterofmassbeforethecollisionandcomparewiththekineticenergyafterthecollision.

Amaterialpointofmassm,movingwithvelocityv1icollideswitha

7.12.

7.13.

7.14.

7.15.

secondpoint,ofmass2m,thatisstanding.Wemeasurethevelocityafterthecollisionoftheparticleofmassmfindingitsdirectionat45°withtheincidentoneanditsmagnitudeonehalfoftheinitialvalue.(a)Findthemagnitudeanddirectionofthevelocityofaparticleofmass2m.(b)Wasthecollisionelastic?

TheforceF=(3,4,0)NisappliedonthepointPhavingcoordinates(8,6,0)m.Find(a)itsmomentabouttheorigin,(b)theleverarmboftheforce,namelythedistanceofitsapplicationlinefromthepole.(b)thecomponentFnoftheforceperpendiculartothepositionvectorr.

Aballfallsonthefloorfrom5m.Whataretheheightsitreacheswhenbouncingbackthefirst,thesecondandthethirdtimesifthecoefficientofrestitutionis0.8?Whatarethecorrespondingenergies?Neglectairresistance.

Anairguideisarailwithaseriesofsmallholesthroughwhichcompressedairisblown.Asledgecanrunontheguidepracticallywithoutfriction.Weputtwosuchsledgesontherail.Thefirstone,ofmassm1=2kgisstill.Onitsrightsideliesaspringofelasticconstantk=300N/mand1mlong,initsnaturallength.Thesecondsledge,ofmassm2=3kgislaunchedtowardsthefirstwithvelocity5m/s.Ithitsthefirstsledgeputtingitandthespringinmotion.WhatisthemaximumdeformationΔxofthespring?

Amaterialsystemismadeofaparticleofmassm1=0.1kginthepointofcoordinates(1,2,3)m,aparticleofmassm2=0.2kgatthecoordinates(2,3,1)mandaparticleofmassm13=0.3kgatthecoordinates(3,1,2)m.Findthecoordinatesofthecenterofmass.

7.16.

7.17.

Abodyofmassm=2kgisshotverticallyupwardswithinitialvelocityυ0=10m/sfromapointwithcoordinates(0,20,0)m.Thez-axisisverticalupwards.Findthedifference∆LOoftheangularmomentumofthebodyabouttheoriginbetweentheinstantwhenitisbackintheinitialpositionandtheinitialinstant.

Aparticleofmassmislaunchedwithinitialvelocityv0atanangleαwiththehorizontal.InthereferenceframeofFig.7.27,neglectingairresistance,findthetimedependenceof(a)themomentoftheforceabouttheoriginO,(b)theangularmomentumLOofthebodyaboutthesamepole.

Fig.7.27 Thetrajectoryofproblem7.17

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©SpringerInternationalPublishingSwitzerland2016AlessandroBettini,ACourseinClassicalPhysics1—Mechanics,UndergraduateLectureNotesinPhysics,DOI10.1007/978-3-319-29257-1_8

8.RigidBodies

AlessandroBettini1

DipartimentodiFisicaeAstronomia,UniversitàdiPadova,Padova,Italy

AlessandroBettiniEmail:alessandro.bettini@pd.infn.it

Inthischapter,weshalldiscussthemechanicsofanimportantclassofextendedsystems,therigidbodies.Inaperfectlyrigidbody,thedistancebetweenanypairofitspointsdoesnotvaryforanyactingforcesoranymotion.Clearly,thisisanidealization,but,inpractice,thesolidobjectsarerigidinagoodapproximation.

Themotionoftherigidbodiesisgovernedbytwodifferentialequations.Theknownmembersarethensoftheexternalforcesandoftheexternalmoments.Thesolutiontothesemayinvolveadvancedcalculus.Weshalllimitthediscussiontothesimplestsituations.

InSect.8.1,weshalldefinetherigidbodyanditsmotions,andthendothesameforthepropertiesofthesystemsofappliedforcesinSect.8.2.InSect.8.3,weshallconsidertheequilibriumconditions.

Wethenconsidertherotationsaboutafixedaxis,whicharethesimplestmotions.Weshallfindtheexpressionsofthekineticenergyandtheangularmomentum.Weshallintroduceanewkinematicquantity,themomentofinertia,andseehowthedynamicsoftherotationsaboutafixedaxishavesomesimilaritieswiththedynamicsofthematerialpoint.InSects.8.9and8.10,wediscusstwoimportantexamples,thetorsionbalanceandthecompoundpendulum.

Weshallthenmoveontomorecomplexmotions,thoseofarigidbodyaboutafixedpoint.Weshallfirstfindtheexpressionoftheangularmomentumaboutthatpoint,andofthekineticenergyinSects.8.12and8.13.Weshallseethat,in

general,thedirectionsoftheangularmomentumandtheangularvelocityaredifferent.Oneconsequenceofthatisthattheforcesdeveloponthesupporting,asdemonstratedinSect.8.14.InSects.8.15and8.16,westudythepurerollingmotionofcylindricalandsphericalrigidbodiesonaplane.

InSect.8.17,weconsiderthegyroscopes,whicharerigidbodiesmovingaboutafixedpoint,atopbeingagoodexample.

Finally,inSect.8.18,weshallstudythecollisionsbetweenrigidbodies.

8.1 RigidBodiesandTheirMovementsThesolidbodiesareapproximatelyrigid.Uponfirstapproximation,theirshapedoesnotchangeifwestretch,compressortorquethem.Clearly,thisisneverrigorouslytrue,becausesmalldeformationsalwaystakeplace.However,severaldynamicalpropertiesofthesebodiescanbestudiedconsideringthemasrigid.Wedefinethemasrigidifthedistancebetweenanypairoftheirpointsdoesnotvary.

Thespacelocationofarigidbodyiscalleditsconfiguration,whichisthesetofthepositionsofitspoints.TodefinetheconfigurationofagenericsystemofNpoints,weneed3Ncoordinates,butonlysixforarigidbody.Letusseewhy.

Westartbydefiningthepositionofonepoint,forexample,AinFig.8.1,namelyitsthreecoordinatesinthereferenceframewehavechosen.Inacertaininstant,itisinA1=(xA1,yA1,zA1).Wethenshallgivethecoordinatesofasecondpoint,likeB,whichisinB1=(xB1,yB1,zB1).But,waitamoment:wecannotdothatarbitrarily.Wecanchooseonlytwocoordinates,becausethedistancebetweenAandBisfixed,independentoftheconfiguration,namely

Fig.8.1 Twoconfigurationsofarigidbody

Withthis,wedonotyetknowtheconfiguration.Weneedthepositionofathirdpoint,likeC,whichisinC1intheconsideredinstant.Ofitsthreecoordinates,wecanarbitrarilychooseonlyone,theothertwobeingdefinedbythetwoconditionsthatthedistancesC1A1andC1B1arefixed.Now,thepositionsofallthepointsaredefined,hencetheconfigurationofthebodytoo.

Intotal,theconfigurationofarigidbodyisdefinedbysixcoordinates.Wesaythatthesystemhassixdegreesoffreedom.

Considernowtwoconfigurations.Thetransportfromoneconfigurationtoanothercanalwaysbeobtainedwithatranslationfollowedbyarotationaroundanaxis.Thisisgeometricalandnotnecessarilyfixed.InFig.8.1,thetranslationbringspointAfromA1toA2andthebodygoestothedottedconfiguration.Tobringtheotherpointstotheirfinalpositions,weneedarotationaboutanaxisthroughA2(thispointisalreadyinposition,andshouldnotmoveanymore).Theaxisshouldhavetherightdirectionandtherotationangletherightvalue.

Intheaboveargument,thechoiceofthepointAwasarbitrary,butitdeterminesthetranslation.Ifwehadchosen,forexample,B,thetranslationwouldhavebeendifferent.Consequently,thereareinfinitetranslation-rotationpairsthatproducethegivendisplacement.Itcanbedemonstrated,however,thatoncegiventheinitialandfinalconfigurations,therotationdirectionandtheanglearedetermined.

Obviously,thesameresultcanbeobtainedbyperformingtherotationfirstandthenthetranslation,orasequenceofrotationandtranslationpairs.Asamatteroffact,themotionoftherigidbodycanbethoughtofasacontinuousseriesofinfinitesimalroto-translations.Therotationaxis,ingeneral,variescontinuouslyduringthemotionandwesubsequentlytalkofaninstantaneousrotationaxis.

Whilethechoiceofthetranslatingpointis,aswesaid,arbitrary,itisinpracticeconvenienttochoosethecenterofmass,consideringitsprivilegedroleindynamics.WerecallherethedynamicalequationsgoverningeverymechanicalsysteminaninertialframethatwefoundinChap.7.WechooseapointΩfixedintheinertialframeasthepole.LetMΩbethetotalexternaltorqueaboutΩandLΩthetotalangularmomentumaboutthesamepole,FtheexternalresultantforceandPthetotallinearmomentum.Thetwodynamicalequationsare

(8.1)

(8.2)Wealsorecallthatthesecondequationissimilarlyvalidwhenwechoosea

particularpoint,evenifitismovingintheinertialframe,namelythecenterofmassofthesystem

(8.3)Thetwovectorequationsgivesixindependentconditions.Forany

mechanicalsystem,thesearenecessaryconditions,butingeneral,theyarenotsufficient.Theyare,however,sufficientforarigidbody,whichhassixdegreesoffreedom,asmanyastheconditions.Inotherwords,ifweknowtheexternalresultantforceandthetotalexternaltorque(ormoment)andtheinitialconditions,wecanknowthemotionofthebodysolvingtheabovedifferentialequations.

WenoticethatEq.(8.1)rulesthemotionofthecenterofmassofthebody.RememberingthatP=mvCM,wheremisthemassofthebodyandvCMthevelocityofitscenterofmass,wecanwriteEq.(8.1)intheequivalentform

(8.4)whereaCMisthecenterofmassacceleration.Themotionofthecenterofmassisexactlyinthesamewayasthemotionofamaterialpoint.

Equation(8.3)allowsustofindthemotionofthebodyaboutitscenterofmass.Thisisgeneralaroundanaxisthroughthecenterofmassbutofvaryingdirectionandwithvaryingangularvelocity.Thesolutionis,ingeneral,quitecomplicated.Weshallconsiderthesimplestcaseshere.

Weimmediatelynoticeanimportantpropertyoftherigidmotions:theworkoftheinternalforcesisalwayszero.Indeed,theinternalforcescomeincouplesactingonpairsofpointsinthedirectionofthelinejoiningthepoints.Theworkdonebyoneofthetwoforagivendisplacementofthebodyisequaltotheforcetimestheprojectionofthedisplacementofthepointonwhichitactsonthedirectionoftheforce.Thelatteristhelinejoiningthetwopoints.Theworkdonebythecoupleofforcesisthenequaltothemagnitudeoftheforcetimesthedifferencebetweentheprojectionsofthetwodisplacementsonthejoiningline.Butthisdifferenceisthechangeinthedistancebetweenthetwopoints,andthisiszero,ifthebodyisrigid.

8.2 AppliedForcesSupposethatseveralexternalforcesareactingonarigidbody.Aswehavejust

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seen,themotionofthebodyisdeterminedbytheirresultantandtotaltorque.Clearly,thereisaninfinitenumberofsystemsofforceshavingthesameresultantandthesametorque.Alltheseforcesystemsappliedtothesamerigidbodyproducethesamemotion,whenstartingfromthesameinitialconditions.Consequently,fromtheobservationofthemotion,wecanknowtheresultantforce(fromthecenterofmassacceleration)andthetotaltorque(fromtheangularacceleration),butnotthesingleactingexternalforces.Wedefineasequivalentanysystemofappliedforceswiththesameresultantandtorque.Noticesuchforcesystemsareequivalentforthemotionofarigidbody,buttheydonothavethesameeffectsifappliedtoanon-rigidbody.Considertheverysimpleexampleofacoupleonthesameline.Theresultantforceandtorquearezero.Actingonarigidbody,theytendtoapproachorseparatethetwopoints,namelytochangetheirdistance.Thisdistance,thebodybeingrigid,cannotvary.Butifthebodyisarubberband,thedistancevariesandbothforcesdowork.

Wenowshowafewsimplepropertiesoftheforcesystemsthatwillbeusefulinthefollowing.

AforcesystemhasresultantFandtotaltorqueaboutthefixedpointΩ,MΩ.WeshowthatthetorqueaboutanyotherfixedpoleΩ′is

(8.5)WithreferencetoFig.8.2,wecaneasilyseethattherelationbetweenthe

torquesaboutthetwopolesofthegenericforceFiis

Fig.8.2 Aforceandtwodifferentpoles

which,summedonalltheforces,givesEq.(8.5)

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Corollary1Thetorqueofaforcesystemofzeroresultantisindependentofthepole,namelyMΩ=MΩ′.

Corollary2Iftwoforcesystemshavethesameresultantandthesametorqueaboutthesamepole,theyhavethesametorqueaboutanypole.

ConsideragenericforcesystemofresultantFandtorqueaboutΩ,MΩ.ThesystemisequivalenttoaforcesystemofaforceFappliedtothepoleΩplusatorquecoupleMΩ.Thedemonstrationisimmediate.Thetwosystemshavethesameresultantandthesametorque,asthetorqueofFaboutthepoleisnull.

AsystemofmutuallyparallelforcesFiappliedtodifferentpointsPiofpositionvectorsriisequivalenttotheirresultantFappliedtothepointC,havingthepositionvector

(8.6)

ThepointCiscalledthecenteroftheforcesystem.Thedemonstrationofthetheoremiseasy.Firstofall,thetwosystemsobviouslyhavethesameresultant.Asforthetorque,letustaketheoriginOasthepole,asinFig.8.3.Theforcesbeingparallel,wecancallutheircommonunitvectorandwriteFi=Fiu.ThetorqueaboutOis

Fig.8.3 Asystemofparallelforces

which,bydefinitionofthecenteroftheforces,becomes

whichprovesthetheorem.Theweightforcesarearelevantexampleofparallelforces.Considera

systemofnmaterialpoints(theargumentisalsovalidforacontinuoussystem)Piofpositionvectorsriandmassesmi.TheweightsmigareparallelforcesappliedtothepointsPi.Thepositionvectorofthecenteroftheforcesis

(8.7)

Weseethatthecenteroftheweightforces,calledthebarycenter,issimplythecenterofmassofthesystem.Themotionofarigidbodyundertheactionoftheweightsofallitspartscanbedescribedasifasingleforcewasacting,itstotalweightappliedtothecenterofmass.Thisproperty,whichwehavealreadyused,substantiallysimplifiesseveralproblems.

Notice,tobeprecise,thatthecoincidencebetweencenterofmassandcenteroftheweightforcesexistsforbodiesthatarenottoolarge,suchthattheweightsofalltheirpartscanbeconsideredtobeparallel.Thisisalmostalwaystrueinpractice.

8.3 EquilibriumoftheRigidBodiesAconfigurationofarigidbodyissaidtobeofequilibriumif,leavingthebodyatrestinthatconfiguration,itkeepsitindefinitely.Thenecessaryandsufficientconditionfortheequilibrium,inaninertialframe,isthattheexternalresultantforceandtheexternalmomentarezero.Indeed,ifthebodyisinequilibrium,theaccelerationofitscenterofmassiszero;hence,theresultantforceiszero.Inaddition,theangularmomentumthatisinitiallyzeromustremainassuch.Hence,thetotalmomentiszero.Ontheotherhand,iftheresultantiszero,thecenterofmassdoesnotchangeitsvelocity,whichisinitiallyzero,and,ifthemomentiszero,theangularmomentumisconstantandremainszero,ifitissoinitially.

Noticethatthetwoconditionsareindependentfromoneanother.For

example,acoupleofforceshavezeroresultantandnon-zeromoment,whileaforceappliedtothepolehaszeromomentandnonzeroresultant.

ExampleE8.1Considerarigidbodyonahorizontalplaneundertheactionofitsweight.Thepositionisofanequilibriumpositioniftheverticalthroughthecenterofmassofthebodyintersectsitssupportbase.Indeed,theexternalforcesaretheweightsofitselementsandtheconstraintforces.Theformerareequivalenttothetotalweightappliedtothecenterofmass,thelatterarenormaltothebaseandconsequentlyareasystemofparallelforcestoo.Consequently,theyareequivalent,withtheirresultantNappliedtotheircenterofforcesD,asshowninFig.8.4a.TheconstraintautomaticallyadjustsitsreactioninsuchawaythatthemagnitudeofNandthecenterDguaranteetheequilibrium,inotherwords,thatmgandNareacouplewiththesamelineofapplication.ThisimpliesthatN=−mgandthatDshouldbeontheverticalfromC.ThisispossibleifthefootofthisverticalisbetweenAandB,namelyinsidethebase.Theinsertinthefigureshowsapossibleconfigurationoftheconstraintforces.TheyareappliedbetweenAandB.Consequently,theircentermustbeapointofAB.

Fig.8.4 aAnequilibriumposition,banon-equilibriumposition

IntheconfigurationofFig.8.4b,theequilibriumisnotpossible.EveniftheconstraintnormalreactionNisconcentratedintheextremepointBofthebasis,thisisnotenoughtoproduceacoupleofzeromoments.Thebodyoverturns.Duringthefall,thenormalreactionislessthantheweight,becausethecenterofmassisacceleratingdownwards.Thedifferencemg−Nisequaltotheaccelerationofthecenterofmasstimesthemassofthebody.

Thecenteroftheconstraintforcescan,however,bebroughtoutsidethesegmentAB,andtheequilibriumisalsoguaranteedintheconditionsofFig.8.5b,ifpartoftheconstraintforcesisdirectedupwards.Wecan,for

example,driveanailinA,asinFig.8.5,orattachahook.IfRisthereactionofthenail,orofthehook,andNthereactionoftheplane,theequilibriumiswhentheresultantforceandmomentarezero,namely .

Fig.8.5 TheconstraintRguaranteesequilibrium

ExampleE8.2TheladdershowninFig.8.6oflengthlissupportedbyaverticalwall,atanangleofα.Supposethefrictiononthewalltobenegligible,whilethecoefficientofstaticfrictiononthehorizontalplaneisµS.Letusdiscusstheequilibriumconditions.

Fig.8.6 Theforcesactingonaladder

InFig.8.5,Cisthecenterofmass,andAandBarethefootholds.Wetakethereferenceframewiththex-axishorizontalintheplaneofthefigure,thez-axishorizontaldirectedoutofthefigureandthey-axisverticalupwards.Theexternalforcesare:theweightmg,appliedtothecenterofmass,theconstraint

reactionappliedinB,whichweconsiderdecomposedinaverticalcomponent,N,andahorizontalcomponent,Ft,andfinally,theconstraintreactionappliedinA,NAthatishorizontal(nofrictionhere).Atequilibrium,theirresultantiszero:

Thisequationgivestwoindependentrelations,itsxandycomponents,thezcomponentbeingidenticallyzero.Thetworelationsare whichgivestheunknownN,and ,whichlinkstheothertwounknowns.Wenowstatethattheexternalmomentshouldbezerotoo,namely

WehavewrittenthesignsinthisequationtakingintoaccountthatNAmustbeinthepositivexdirection,becausethewallcanonlypush.Consequently,NAtendstorotatetheladderclockwiseandthezcomponentofitsmomentisnegative.Ontheotherhand,fortheabovewrittenequation,fortheequilibriumofthehorizontalforces,Ftmustbeintheoppositexdirection.Thezcomponentofitsmomentisconsequentlypositive.SolvingthetwoequationsforFtandNA,weimmediatelyhave .

Thefrictionforcecannotbetoolarge,namely Ontheotherhand,Consequently,tobeinequilibrium,theleaningangleshould

notbetoolarge,namely Forlargerangles,theladderslidesdown.Wehaveassumedtheverticalwalltobesmoothanditsreactiontobe

normal.Ifthereisfriction,astherealwaysisinpractice,thereisaverticalcomponenttothewallreactiontoo.Wewouldhaveonemoreunknown,withthesamenumberofequations.Undertheseconditions,theproblemisundetermined.Indeed,thereisaninfinitenumberofpairsofthetwotangentialreactionsthatleadtoequilibrium.Anotherexampleofanundeterminedproblemistheproblemoffindingtheconstraintreactionsonthefourwheelsofacar,orthefourlegsofatable,onaplane.Theseproblemshaveasolutionifmoreinformationisavailable,suchasthenatureoftheelasticforcesofthetiresonthecarorthelengthsofthelegsofthetable.

8.4 RotationAboutaFixedAxisAnimportantandrelativelysimpleclassofrigidmovementsistheclassofmovementsaboutafixedaxis.Considerarigidbodyofarbitraryshape,asrepresentedinFig.8.7,whichcanmovearoundtheaxisa,whichisfixedinan

inertialframe.Letuabetheunitvector,arbitrarilychoseninoneofthetwodirectionsofa.Theconfigurationofthebodyisdefinedbytherotationangle,whichwecallϕ,aroundtheaxisa,relativetoafixedplane,whichwechooseastheoriginoftheangles.

Fig.8.7 Arigidbodywithafixedaxis

WenowchooseapointΩontheaxisasthepoleofthemomentsandcallMΩthetotalexternalmomentandLΩthetotalangularmomentumaboutΩ.Thedynamicequationis

(8.8)Wenowtakethedotproductofthetwomemberswiththeunitaryvectorof

therotationaxisua.Wehave

(8.9)Inthisequation,wehavetheprojectionsonthea-axisoftheexternal

momentandoftheangularmomentum,namely

(8.10)Thesequantitiesarecalledtheexternalmomentorthetorqueabouttheaxis

andtheangularmomentumabouttheaxis.Bothquantitiesarethecomponents

ofapseudo-vector.Theycanhavebothsigns.ItcanbeeasilyshownthattheyareindependentofthechoiceofthepoleΩ,provideditisontherotationaxis.

WecanwriteEq.(8.9)as

(8.11)whichexpressesthetheoremoftheangularmomentumaboutanaxis.Inotherwords,therateofchangeoftheangularmomentumaboutafixedaxis,inaninertialframe,isequaltotheexternalmomentaboutthesameaxis.

Letusfindtheexpressionoftheangularmomentum.Theangularvelocity,whichwecallω,isparalleltotheaxis.Itsmagnitudeanditssignrelativetotheaxiscanvaryintime,butnotitsdirection.Westartbyconsidering,forsimplicity,thebodyconsistingofparticlesofmassmi,inthepositionsrirelativetoΩ,distancefromtheaxisr′iandvelocityvi,asshowninFig.8.7.Thetrajectoryofthegenericparticleisacirclenormaltotheaxisofradiusr′i.Itsvelocityistangenttothiscircleandhasthemagnitude .

Weprofitbythefactthattheangularmomentumabouttheaxisisindependentofthepoleontheaxisandtakeit,foreachparticle,inthecenterOiofitsorbit.Theangularmomentumoftheparticleaboutthispoleis

which,asinfigure,hasthedirectionoftheaxis.Whatweneedisitscomponentontheaxis.Itssignisthesameasthesignoftheprojectionontheaxisoftheangularvelocity,ωa.Wehave .Wenowsumoveralltheparticlesandobtainthetotalangularmomentumabouttheaxis

(8.12)

wherewehaveintroducedthequantity

(8.13)

whichisthemomentofinertiaofthebodyabouttheaxisa.Wenowconsiderthebodyasacontinuousdistributionofmasses.Insteadof

pointparticlesofmassmi,weconsiderinfinitesimalvolumeedV,inthepositionrandhavingmassdm=ρ(r)dV,whereρisthedensity(thatcanbedifferentfrompointtopoint).Followingthesameargumentsasforthediscretebody,onefindsthesameresult

(8.14)butnowwithanintegralinplaceofthesum,namely

(8.15)InSect.8.7,weshallcalculatethemomentsofinertiaofseveralbodiesof

simplegeometry.Weobserveherethatthemomentofinertiadependsontheaxis,notonlyonthebody.Whatmattersishowthemassesaredistributedabouttheaxis.TheequationofmotionEq.(8.11)canbewritteninequivalentforms.

(8.16)andalso

(8.17)where

(8.18)istheangularacceleration.

Thelastexpressionlooksverysimilartothedynamicalequationforapointmovingalongastraightline.Ifxisitscoordinate,mthemassandFxthecomponentoftheactingforce,theequationofmotionis,asweknow,

Equation(8.17)isthesamedifferentialequationwithϕinplaceofx(andconsequently,angularvelocityandaccelerationinplaceofthelinearones),theexternalmomentabouttheaxisinplaceoftheforceandthemomentofinertiainplaceofthemass.Consequently,thesolutionstoEq.(8.17)arethesameasthoseforthelinearmotionofapoint.

ThesimplestcaseiswhentheexternalmomentabouttheaxisMaisconstant.Then,theangularaccelerationα=Ma/Iaisconstanttooand,analogoustotheuniformlyacceleratedrectilinearmotion,wehave

(8.19)whereϕ0andω0aretheangleandtheangularvelocity,respectively,att=0.

ExampleE8.3Figure8.8showsarigiddisk,sayapulley,thatcanrotatearoundahorizontal

axisapassingthroughitcenterofmass.Awire,towhichamassmisattached,iswrappedaroundthepivot.Theradiusofthepivotisr.Theexternalmomentabouttheaxisisclearlyconstant,Ma=mgr.Supposethedisktobeinitiallyatrestandchoosetheoriginoftheanglessuchthatϕ0=0.Themotionisthen

.Namely,theanglethroughwhichthesystemhasturnedisproportionaltothesquareofthetime.

Fig.8.8 Apulleyandaweight

8.5 ConservationAngularMomentumAboutanAxisWestillconsiderarigidbodywithafixed(inaninertialframe)rotationaxisa.Iftheexternalmomentabouttheaxisiszero,theangularmomentumabouttheaxisisconstant,namely

(8.20)Theexternalmomentabouttheaxisiszero,apartofthetrivialcaseof

absenceofforces,intwoimportantcases:(1)thedirectionsofalltheexternalforcesareparalleltotheaxis,and(2)theapplicationlinesofalltheexternalforcesmeettheaxis.Inthesecases,forEq.(8.14),asthemomentofinertiaisconstant,theangularvelocityisconstanttoo

(8.21)Noticethat,forzeroexternalmomentabouttheaxis,Eq.(8.20)isalsovalid

fornon-rigidbodies,whileEq.(8.21)isnot.Asimpleexperimentfollows.Apersonsitsonaturntablestoolholdinginhishandstwoheavyobjectswitharms

horizontallyoutstretched.Asecondpersonpushesthefirstinrotation.Thefirstbringshandsandheavyobjectsnearhischest.Hisangularvelocityincreasessubstantially.Theinitialmomentofinertiaofthebodywas,say,I1andwasquitelargebecauseheavymasseswerefarfromtheaxis,whilethefinalone,I2,ismuchsmallerbecausethemassesareclosetotheaxis.Wecansaythattheexternalmomentabouttheaxisiszero,ifweneglectfrictions,becausetheexternalforces,theweights,areparalleltotheaxis.Theangularmomentumisconserved,and,ifω1andω2aretheinitialandfinalvelocities,wehave

andconsequently,asI2 I1,ω2 ω1.Thistrickisusedbyskatersintheirfigures.

ExampleE8.4Asanexample,considerthesysteminFig.8.9,whichshowsanelectricalmotorfixedonasupportthatcanrotateaboutaverticalaxis,coincidingwiththeaxisofthemotor.Themotorhastwoparts:theexternalone(stator)isfixedtotheplatform,whiletheinternalone(rotor)isfreetorotateandhasaflywheel(Vinthefigure).Thetwopartsarecoaxialrigidbodieswithmomentsofinertia,I1beingtheinternalandI2theexternal.

Fig.8.9 Anelectricalmotor

Supposethat,startingfromrest,weswitchonthemotorforsometimeandthenswitchitoff.Weneglectfrictions.Weobservethatthetwopartsrotateatangularvelocitiesω1andω2,respectively.

Theinitialangularmomentumiszero.Thefinaloneiszeroaswell,becauseduringtheactionofthemotor,theforcesareonlyinternal.Hence,again,

or .Wecanmeasuretheinitialandfinalangularvelocities,repeattheexperiencewithdifferentflywheels,andverifyifthepredictioniscorrect.

8.6 WorkandKineticEnergyWecontinueourstudyoftherigidbodyrotatingaboutthefixedaxisa,representedinFig.8.7.Itsgenericparticleofmassmi,asshowninFig.8.10,movesinacircle.WecallOiitscenterandr′ithepositionvectoroftheparticlefromit.

Fig.8.10 Themotionofaparticleofarigidbodyrotatingaboutanaxis

WenowcalculatethetotalmomentabouttheaxisoftheexternalforcesFiactingontheparticle.Westartfromthemomentτiaboutanypoleontheaxis.Oncemore,wetakethecenterOiofthetrajectoryofmiasthepole.TheforceFicanbethoughtofasthesumofthreecomponents,oneparalleltotheaxis,onetor′i,andonetangenttothetrajectory.Thecontributionofthefirstisnormaltotheaxisandhasnoaxialcomponent.Thecontributionofthesecondiszero,becauseitisparalleltothearm.Theonlycontributionisthethird.

Wecalluttheunitvectortangenttothetrajectorywithpositivedirectioninaccordancewiththedirectionofincreasingangles(whichisnotnecessarilythedirectionofmotion).LetFtibethecomponentoftheexternalforceonut.Thecomponentofτiontheaxisisthen,inmagnitudeandsign, .

Considernowtheinfinitesimalrotationofthebodyalongtheangledϕ,and

calculatethecorrespondingtotalworkoftheforces.Asweknow,thebodybeingrigid,thetotalworkoftheinternalforcesiszero.Asfortheworkoftheexternalforces,westartwiththeworkononeparticle.Thedisplacementoftheparticleisdsi=r′idϕandtheelementarywork .Tofindthetotalworkoftheexternalforces,wehavenowonlytoaddupalltheparticles.Takingintoaccountthatdϕisthesameforallandcalling ,wehave

(8.22)Thisimportantrelationtellsusthattheelementaryworkoftheexternal

forcesforaninfinitesimalrotationisequaltotheexternalmomentabouttheaxistimestherotationangle.Again,wehavefoundananalogywiththeelementaryworkoftheforceonapointFxdx.

Theworkforafiniterotation,sayfromϕ1toϕ2,isobtainedbyintegration

(8.23)

Fortherotationsaboutafixedaxis,thekineticenergytheoremhasasimpleexpression.RecallingEq.(8.16),wewrite

Forafiniterotation,theworkisequaltothedifferenceofthekineticenergies

(8.24)

Weseethatthekineticenergyofarigidbodyrotatingaboutafixedaxisis(onceagainsimilartothematerialpoint)

(8.25)

8.7 CalculatingInertiaMomentsInthissection,weshallcalculatethemomentsofinertiaofafewgeometricallysimplebodies.Weshallconsiderallofthemashomogeneous,namelyhavingdensityindependentonpoint.Consequently,theirgeometriccenterscoincidewiththeircentersofmass.

Cylindricalbar.Figure8.11arepresentsabarofmassmandlengthL.We

assumeittobethin,namelyoftransversedimensionsmuchsmallerthanthelength.Weassumethefacestobeperpendiculartothegeometricalaxis.Theshapeofthefacesisarbitrary.Theycanbecircles,squares,anything.WecalculateitsmomentofinertiaabouttheaxiscnormaltothebarthroughitscenterC.

Fig.8.11 Calculatingthemomentofinertiaofathinbaraboutacentraltransverseaxis

Wetakeacoordinatexalongthebaroriginatinginitscenter.Wecutthebarintoinfinitesimalslicesbetweenxex+dxofmassdm.Asthediameterofthesliceisverysmall,wecanconsiderallthepointsofthesliceatthesamedistancefromtheaxisc.Themassofthesliceisclearlydm=(m/L)dx.Wenoticethattherearetwoslicesatthesamedistancefromc,onitstwosides.Theircontributiontothemomentofinertiais Weintegrateitonhalfofthebar,namelyfrom0toL/2,andobtain

(8.26)

Ring.Figure8.12representsathinringofmassmandradiusR.WeassumethediameterofthesectiontobesmallcomparedtoR.AllthepointsofasectioncanbeconsideredatthesamedistanceRfromthecenter.

Fig.8.12 Calculatingthemomentofinertiaofathinringaboutthecentralaxis

WecalculatethemomentofinertiaabouttheaxiscnormaltotheplaneoftheringthroughitscenterC.Asallthemasssitsatthesamedistance,weimmediatelyhave

(8.27)Cylindricalsurface.Themomentofinertiaofacylindricalsurface(namely

ofnegligiblethickness)aboutthegeometricalaxisisgivenbyEq.(8.27)aswell,becauseallthemassesinthiscasearealsoatthesamedistanceRfromtheaxis.

Homogenousdisk.Figure8.13representsadiskofradiusRandmassm.Wecalculatethemomentofinertiaaboutthegeometricaxiscshowninthefigure.Wedividethediskintoinfinitesimalringsofraysbetweenrandr+dr.Theareaofaringis2πrdr,tobecomparedwiththeareaπR2oftheentiredisk.Themassoftheringisthen Itscontributiontothe

momentofinertiais Integrating,weobtain

Fig.8.13 Calculatingthemomentofinertiaofadisk

(8.28)

Homogenouscylinder.Figure8.14showsahomogenouscylinder.Itcanbethoughtofasapileofdisks.Hence,themomentofinertiaaboutthesymmetryaxisisgivenbyEq.(8.28).

Fig.8.14 Homogenouscylinderanditsaxis

Homogeneousrectangularparallelepiped.Figure8.15representaparallelepipedofuniformdensityρ,massmandsidelengthsa,bandc.WecalculatethemomentofinertiaabouttheaxesthroughthecenterCparalleltothesides.Thesewecallx,yandzandtakeasreferenceaxes.

Fig.8.15 Calculatingthemomentofinertiaofaparallelepipedaboutthreecentralaxes

Asamatteroffact,itwillbeenoughtocalculatethemomentofinertiaaboutoneaxis,sayz,andthiswillbeanalogousforalltheaxes.Wehave

Analogousexpressionsholdingfortheotheraxes,wecanconcludethat

(8.29)Homogeneouscube.Themomentofinertiaaboutone,ofthethree,

symmetryaxesisaparticularcaseofwhatwehavejustfound.IfListhelengthoftheside,wehave

(8.30)Homogeneoussphere.Wegiveonlytheresultwithoutdevelopingthe

calculation.Themomentofinertiaaboutanaxisthroughthecenteris

(8.31)

8.8 TheoremsontheMomentsofInertia

Inthissection,weshallshowtwotheoremsthatwillhelpinseveralcasesofcomputingmomentsofinertia.Thefirstone,theSteinertheorem,afterJacobSteiner(1796–1863),concernsrigidbodiesofarbitraryshape,whilethesecondoneisforthinbodies,namelyofnegligiblethickness.

Theoremoftheparallelaxes,orSteinertheorem.Thetheoremoftheparallelaxesstatesthatthemomentofinertiaaboutanarbitraryaxisisequaltosumofthemomentofinertiaabouttheparallelaxisthroughthecenterofmassandtheproductofthemassofthesystemandthesquareofthedistancebetweenthetwoaxes.

Figure8.16representsarigidbodyofarbitraryshape.ThecaxisisthroughitscenterofmassC.ThemomentofinertiaaboutcisIc,theonerelativetotheparallelaxisa,atdistanceh,isIa.Lethbethevectorfromatocinaplanenormaltotheaxes.Consideranarbitraryelementofthebody,ofmassdmandtheplanenormaltotheaxesthroughtheelement.Inthisplane,letrc′andra′bethevectorstodmfromcanda,respectively.Clearly, .

Fig.8.16 Arigidbody,anaxisthroughthecenterofmassandaparallelaxis

ThecontributiondIaofthemomentofinertiaaboutais

TakingintoaccountthatthelasttermisdIcandintegratingonthebody,wehave

Theintegralinthefirsttermisthemassofthebody,whilethesecondtermisthecomponentontheconsideredplaneofthepositionvectorofthecenterofmassfromthecenterofmass,andiszero.Wehave

(8.32)whichistheparallelaxestheorem.Noticethatmh2isapositivedefinitequantity.Foralltheaxesofagivendirection,themomentinertiaisminimumfortheaxisthroughthecenterofmass.

ExampleE8.5ConsidertherightcylinderinFig.8.17,ofmassmandradiusR,itscentralaxiscanditsgeneratora.

Fig.8.17 Momentofinertiaofacylinderaboutagenerator

ThemomentofinertiarelativetocisgivenbyEq.(8.28).Hence,fortheparallelaxestheorem, .

Theoremoftheperpendicularaxes.Themomentofinertiaofathinbodyaboutanaxisperpendiculartoitsplane

throughthepointOofthisplaneisequaltothesumofitsmomentsofinertiaabouttwomutuallyperpendicularaxespassingthroughO.

ConsiderthebodyrepresentedinFig.8.18.Oisanarbitrarypointofthebodythatwetakeastheoriginofthecoordinatesaxes,znormaltotheplane,

andxandyintheplane.Themomentofinertiaaboutzis whereris

thedistanceoftheelementdmfromz.As ,wehave,

namely

Fig.8.18 Athinbodyandtwoperpendicularaxesinitsplane

(8.33)whichisthetheoremoftheperpendicularaxes.

ExampleE8.6Calculatethemomentofinertiaofarectangularplateofsidesaandbabouttheperpendicularaxisthroughitscenter,asinFig.8.19.

Fig.8.19 Arectangularplateandtheconsideredaxes

Equation(8.29),withc=0,gives and and,forthetheoremoftheperpendicularaxes

(8.34)whichisthethirdofEq.(8.29)

ExampleE8.7

CalculatethemomentofinertiaofacircularplateofradiusRaboutadiameter,asinFig.8.20.

Fig.8.20 Circularplateandaxes

ThemomentofinertiaaboutthecentralperpendicularaxisIzisgivenbyEq.(8.24).Ontheotherhand,obviously,Ix=Iyand,forthetheoremoftheperpendicularaxes, thusgivingus

(8.35)

ExampleE8.8Findthemomentofinertiaofacirculardiskaboutanaxistangenttoitsrim,asinFig.8.21.

Fig.8.21 Acirculardiskandanaxistangenttoitsrim

Wejusthavetoapplythetheoremoftheparallelaxestotheresultwejustfoundtohave

Momentofinertiaofacylinderaboutthenormalaxisthroughthecenter.

Considerthe(homogeneous)cylinderofradiusRandlengthLrepresentedinFig.8.22.

Fig.8.22 Thecylinderanditslongitudinalandperpendicularcentralaxes

Wewantthemomentofinertiaabouttheaxisyinthefigure.Thisisthesamesituationaswediscussedintheprevioussection,buthere,wedonotassumethesectionofthecylindertobenegligible.Wecallλthelineardensity,namelythemassperunitlengthofthecylinder.Consideraninfinitesimalslicebetweenxandx+dx.Itsmassisdm=λdx.WecanuseEq.(8.35)tofindthemomentofinertiaofthesliceabouttheaxisthroughitparalleltoy(dottedinthefigure).Forthetheoremofparallelaxes,wehavedIybyaddingtoitx2dm,namely

Integratingalongtheentirelength,namelyinxfrom−L/2toL/2,wehave

(8.36)

8.9 TorsionBalanceThetorsionbalanceisaverysensitiveinstrumentusedtomeasuresmallmomentsand,consequently,smallforces.WehavealreadyseenhowitwasusedbyCavendishinSect.4.7andbyEötvösinSect.5.8.Weshalldiscussitinmoredetailnow.Figure8.23showstheschemeofthedevice.ArigidbarABissuspendedfromaverticalwirethroughitscenter.Theequilibriumpositionofthebarisdeterminedbytheconfigurationofthewireatrest.

Fig.8.23 Schematicsofthetorsionbalance

Whenweapplyamomentτ,thebarrotatesaboutitscenter.Therotationgivesorigintoanelasticmomentτeinthewireintheoppositedirection,proportionaltotherotationangleϕ

(8.37)wheretheminussignindicatesthattheelasticmomenttendstobringthebarbackintoitsoriginalposition.Theelasticconstantkdependsonthelengthandthesectionofthewireandonitsmaterial.Wecanchoosethisconstantwhenwedesignthebalance,dependingonthetorqueswehavetomeasure.Forexample,thinquartzwirescanbeusedforsensitivitiesdowntoseveralfemtonewton.

Thenewequilibriumisreachedwhentherotationangleissuchthattheelasticmomentisequaltotheappliedone,τ=τe.Hence,wecanmeasureτbymeasuringϕ,andknowingk.

Themostaccuratemeasurementofkisdoneusingadynamicalmethod.Werotatethebaratanangleϕ0andletitgo.Itisthemotionofarigidbodyaboutafixedaxisundertheactionoftheexternaltorqueτe.IfIisthemomentofinertia,theequationofmotion,Eq.(8.16),is

(8.38)or

(8.39)with

(8.40)Werecognizethedifferentialequationoftheoscillator.Itssolutionisan

harmonicmotionintheangularcoordinateϕwithperiod

(8.41)Theperiodcanbemeasuredwithhighaccuracy,becausewecanmeasureit

overmanyoscillationsandcountthem.Onceweknowtheperiodandthemomentofinertiabyconstruction,weknowtheelasticconstant.

8.10 CompositePendulumThecompositependulumisarigidbody,ofmassm,whichcanrotatearoundafixedhorizontalaxis,anaxiswhichshouldnotbethroughthecenterofmass.InFig.8.24,Oisthetraceoftheaxis,Cisthecenterofmassandϕistheangletothevertical,takentobepositivecounter-clockwise.Thedistanceofthecenterofmassfromtheaxisish.

Fig.8.24 Thecompositependulum

WetakethepoleforthemomentstobethefixedpointO.Twoforcesactonthependulum,theweight,whichwecanthinkofasbeingappliedtothecenterofmass,andtheconstraintreaction,appliedtotheaxisofrotation.Thisisacylinderofradiusr,asshownintheinsertofthefigure.TheconstraintreactionisappliedtothepointPofitslateralsurface.Inthepresenceoffriction,theforcehasadirectiondifferentfromthedirectionofthesegmentOPanditsmomentaboutOisdifferentfromzero.If,however,thefrictionisnegligible,asweshallassume,thedirectionoftheforceisOPanditsmomentiszero.Theexternalmomentonthesystemis,undertheseconditions,themomentoftheweight,which,attheangleϕ,is−mghsinϕ.Theequationofmotionis

(8.42)whereIisthemomentofinertiaabouttheaxis.Forsmallangles,wecanapproximatethesinewiththeangle,obtaining

(8.43)with

(8.44)Equation(8.43)isequaltothatofthesimplependulum.Hence,themotion

ofthecompositependulumisaharmonicmotioninϕ.Itsperiodis

(8.45)

Thedeviceisused,inparticular,tomeasureg,knowingfromconstructiontheotherquantitiesinEq.(8.45).

Theperiodofthecompositependulumisequaltotheperiodofthesimpleoneoflength

(8.46)whichisthencalledthereducedlengthofthecompositependulum

8.11 DumbbellWehavediscussedseveralexamplesofrotationsofrigidbodiesaroundafixedaxis.However,theaxiswillmoveifwedonotprovidethepropersupportstokeepitfixed.Ingeneral,theaxisissupportedbyamassivebodyatrest,onwhichtheaxisrotatesthroughanumberofballbearingstoreducethefrictionsasmuchaspossible.Therelevantkinematicquantitiesaretheangularvelocityandtheangularmomentum.Botharevectorquantities.Theformerisbydefinitionparalleltotheaxis,thelatternotnecessarilyso.Uptonow,wehaveusedonlythecomponentontheaxisoftheangularmomentum.Ingeneral,therearealsocomponentsperpendiculartotheaxis,which,inaddition,varyintime.Consequently,anexternalmomentmustbepresent.Thisistheactionofthesupports.Weshallnowturnourattentiontothisaction.

WeshallstartfromtheparticularlysimplecaseofthedumbbellinFig.8.25.Itismadeoftwoequalspheresofmassmattheextremeendsofarigidbaroflength2dofnegligiblemass.

Fig.8.25 Adumbbellrotatingaboutasymmetrycentralaxis

Theaxis,ainthefigure,isverticalthroughthecenterofthesystemOperpendiculartothebar,namelyasymmetryaxisofthebody.Thefrictionsintherotationarenegligible.Thesystemisverysimilartothosewehavealreadydiscussed.

Weindicatewith and thepositionvectorsofthetwomassesfromOandwithuatheunitvectoroftheaxis.Theangularvelocityhasthedirectionoftheaxisω=ωua.

TheangularmomentumaboutthefixedpointO, .

Consideringthat wecanwrite

(8.47)whereIaisthemomentofinertiaabouta.Inthiscase,theangularmomentumisparalleltotherotationaxis.Theexternalmomentiszero.Indeed,themomentsoftheweightsofthetwomassesareequalandoppositeandweareneglectingthefrictions.Undertheseconditions,angularmomentumandangularvelocityareconstantintime.Ifinitiallythesystemrotatesatacertainangularvelocity,itwillcontinuetodosoforever.Theballbearingsthatkeeptheaxismustsupportthetotalweight,butdonotexertanymoment.

Wenowsupposethefixedrotationaxistobestillthroughthecenter,butnotperpendiculartothebar,attheangle,sayπ/2−θ,withit,asinFig.8.26.Theangularvelocitystillhasthedirectionoftheaxis,ω=ωua.Ifr1andr2arethepositionvectorsofthetwomasses,theangularmomentumaboutOis

Fig.8.26 Adumbbellrotatingaboutacentral,non-symmetry,axis

(8.48)Lookingatthefigure,weseethatthetwotermsareequalbothinmagnitude

andindirection.Thelatteristhedirectionperpendiculartothebarintheplaneofthebarandtheaxis.Wecallnitsunitvector.Theplanerotateswithangularvelocityω.Themagnitudeoftheangularmomentumis

(8.49)Themomentofinertiarelativetotheaxisisnow .

MultiplyingEq.(8.49)bycosθ,andindicatingwithLatheangularmomentumabouttheaxis,wehave

(8.50)Wealreadyknewthisresult.Theaxialangularmomentumisequaltothe

momentofinertiatimestheangularvelocity.However,theaxialangularmomentumisonlyoneofthecomponentsoftheangularmomentumvector.Equation(8.49)givesitsmagnitude,whileitsdirectionisn.Wehave

(8.51)Eveniftheangularvelocityisconstant,theangularmomentumvectorisnot.

Itrotatesatconstantvelocityontheconeofsemi-vertexangleθaroundthefixedaxis.Consequently, andtheexternalmomentisnotzero.Itisduetothesupportingballbearings.

Letuslookmorecloselyatthesituation.WedecomposethemomentumofEq.(7.59)initscomponentsparallelandperpendicular,transverse,totheaxis

asshowninFig.8.27afortheangularmomentum.ItscomponentparalleltotheaxisLPisconstant,andconsequently,MP=0.LTisconstantinmagnitudeandrotatesaroundtheaxisataconstantangularvelocity.Itsderivativeis

Fig.8.27 aTheangularmomentumanditscomponents,btheexternaltorque

AsseeninFig.8.27b,thisderivativeisalsoavectorrotatingatconstantangularvelocityωinaplaneperpendiculartotheaxis.Itisat90°withLT.Thisderivativeisjusttheexternalmoment,whichisexertedbytheballbearings.Theseactwithtwoforces,F1andF2inthefigure,ofconstantmagnitudeandrotatingdirection.

Thesituationissimilar,forexample,whentherotationaxisofthereelofacarisnotexactlythesymmetryaxis.Theperiodicstressontheballbearingswouldinducevibrationsinthevehicle.

8.12 AngularMomentumAboutaFixedPoleLetussummarizewhatwehaveestablisheduptothispointonthemotionsofrigidbodies.Thesimplestistherotationaboutafixedaxis.Inthiscase,theconfigurationofthebodyisdefinedbyasingle(angular)coordinate.Itsrateofchangeistheangularvelocity.Theaxialangularmomentumisthecomponentontheaxisoftheangularmomentumaboutanypointoftheaxis(andisindependentofitschoice).Theaxialangularmomentumisequaltotheangularvelocitytimesthemomentofinertiaabouttheaxis.Therateofchangeoftheaxialangularmomentumisequaltothecomponentofthemomentofthe

externalforcesontheaxis.Thisisthedifferentialequationrulingthedynamicsofthesystem.Inthelastsection,wesawtheconsequencesoftheangularmomentumcomponentsperpendiculartotheaxisontheconstraintsthatguaranteethestabilityoftheaxis.Weshallnowfurtherstudytherelationbetweenangularvelocityandangularmomentumandthemotionofarigidbodyaboutafixedpoint,whichisatrestinaninertialreferenceframe.Wecallitthepole,O.

Therigidmotionaboutafixedpoleisstillarotationwithanangularvelocityω,whichnowmayvarybothinmagnitudeandindirection.Inotherwords,ineveryinstant,thebodyrotatesaboutan“instantaneousrotationaxis”thatpassesthroughOandhasthedirectionofω,whichcontinuouslychanges.WechoosetheinertialreferenceframeasshowninFig.8.28,withoriginO.WeshallalsotakeOasthepole.

Fig.8.28 MotionofarigidbodyaboutafixedpointO

Weconsiderasystemofmaterialpoints.Acontinuoussystemcanbetreatedthroughthesameargumentswithintegrationsinplaceofsums.Letmibethemassofthegenericpointandriitspositionvector.Itsvelocityis

(8.52)whichisobviouslythesameforallthepoints.TheangularmomentumofthepointaboutOis

(8.53)WenowuseEq.(1.29)toexpressthedoublecrossproductinthelast

member,obtaining

(8.54)Thesumofthesequantitiesisthetotalangularmomentumwewanttofind.

Indoingthat,wewouldfindasetofquantitiesanalogoustothemomentofinertiaaboutanaxis.Thesearethenineelementsofa3×3matrix.WeshallworkontheCartesiancomponents.Westartwiththexcomponentofthejustfoundequation.Aftersimplification,wehave

Wenowaddupallthepoints,obtaining

where,inthelastmember,wehaveintroducedthequantities

(8.55)

Thefirstquantityisimmediatelyrecognizedasthemomentofinertiaaboutthex-axis,whilethesecondandthirdonesaretheproductsofinertia.Thesameargumentfortheothertwocomponentsoftheangularmomentumleadtoanalogousexpressions.Thefinalresultcanbeexpressedinacompactformwiththematrixformalismas

(8.56)

The3×3matrixismathematicallyatensorandiscalledthetensorofinertia.ItselementsinthefirstlinearegivenbyEq.(8.55),andthoseoftheothertwobyanalogousexpressions.Weshallnotneedtoknowitsmathematicalproperties.Weonlynoticethatthematrixissymmetric,namelytheelementsinsymmetricpositionsaboutthediagonalareequal,Ixy=Iyx,etc.

Thesituationlooksquitecomplicated,butwecanmakeitsimplerwithanappropriatechoiceofthedirectionsofthecoordinateaxes.Thisisbecausethematrixofinertiaissquareandsymmetric.Indeed,mathematicsshowsthatthistypeofmatrixcanalwaysbeputindiagonalformbyarotationoftheaxes.Westillrefertox,y,zassuchaxes,pose,forsimplicity,Ix=Ixx,Iy=Iyy,Iz=Izz,andwrite

(8.57)

AnotherformofEq.(8.57)thatweshalluseis

(8.58)Wehavefoundthattheangularmomentumoftherigidbodyaboutthepole

Oisthesumofthreevectors.Eachofthemisdirectedasoneoftheaxes,withthemagnitudeequaltotheproductofthecomponentoftheangularvelocityonthataxisandthemomentofinertiaaboutthataxis.Thisistrueonlyfortheparticularchoiceofaxesthatmakesthematrixofinertiadiagonal.ThesearecalledtheprincipalaxesofinertiarelativetoO.Theirpositionisfixedrelativetothebodyandtheymovewithit.Consequently,thereferenceOxyzisNOTgenerallyaninertialone.IfthepoleOisthecenterofmass,theprincipalaxesarecalledcentralaxesofinertia.

Weshallnowstatewithoutdemonstrationafewimportantpropertiesoftheprincipalaxesofinertia.

Firstly,asintuitionsuggests,ifthebodyhassymmetryaxesrelativetoO,thesearealsotheprincipalaxes.

Forexample,theprincipalaxesofahomogeneousrectangularparallelepipedrelativetoitscenterOaretheaxesparalleltoitssidesthroughO.Iftwosidesoftheparallelepipedareequal,twoofitsmomentsofinertiaareequal,sayIx=Iy.ConsidernowanaxisthroughOinanarbitrarydirectionintheplanexy(namelydefinedbythetwoequalmomentsofinertia).ItcanbeshownthatthemomentofinertiaaboutitisI=Ix=Iy,evenifthataxisisnotasymmetryaxis.Wenoticethatthesymmetryofthemomentsofinertiaislargerthanthesymmetryofthedistributionofthemasses.

Considerasasecondexamplearighthomogenouscylinder.Itsgeometricaxisisbothasymmetryaxisandacentralaxisofinertia.Anyaxisintheplaneperpendiculartoitthroughthecenterisacentralaxistoo.Hence,again,thereareinfinitecentralaxes.

Therearealsocasesinwhichallthreemomentsaboutthecentralaxesareequal.Consider,forexample,thesymmetryaxesofahomogenouscubeparalleltoitssides.Theseareclearlycentralaxesofinertia,withequalmoments.However,anyotheraxisthroughthecenterisalsoacentralaxisofinertiawiththesamemoment.Again,weseethatthesymmetryofthemomentsofinertiaislargerthanthesymmetryofthedistributionofthemasses.Theformerisforacube,asphericalsymmetry.Obviously,alltheaxesthroughthecenterofahomogeneousspherearecentralaxesofinertia.

Considernowagainahomogeneouscylinder,withheighthandbaseradiusR.Weputtheoriginofthereferenceinitscenterandthezonitsaxis.Theothertwo(central)axesareonthenormalsection.Wealreadyknowtheexpressionofthemomentsofinertia,Eqs.(8.28)and(8.36),whichgive

Wenoticethatallofthemareequalif Intheseparticularcases,all

theaxesthroughthecenterarecentralaxesofinertia.Allthemomentsofinertiaaboutthemareequal.Again,thesymmetryofthemomentsofinertiaislargerthanthesymmetryofthemasses.Inotherwords,iftherearesymmetryaxes,theseareprincipalaxesofinertia,butaprincipalaxisofinertiamaynotbeasymmetryaxis.Indeed,anyrigidbodyofwhatevershape,withnosymmetryatall,likeanirregularstone,hasthreeprincipalaxesofinertiaaboutanypointatrestwithit,evenoutsidethebody.

WestatewithoutproofthattheprincipalaxesofinertiaaboutapointOandthoseaboutanotherpointO′arenotparallel,ingeneral.

WeshallnowdiscussafewimportantaspectsofEq.(8.58).First,ittellsusthatangularvelocityandangularmomentumarenot,ingeneral,parallelvectors.However,theyaresoiftherotationisaroundaprincipalaxis,namelyωisparalleltoaprincipalaxis.Consequently,theprincipalaxesarealsocalledpermanentrotationaxesorspontaneousrotationaxes.Considerarotationaboutafixedpointinaninertialframe.Itsgenericmotionisarotationaboutaninstantaneousaxisthroughthefixedpoint,whosedirectionvariescontinuouslyintime.Asaconsequence,theangularmomentumaboutthepointvariestoo.Thisimpliestheexistenceofanon-zeroexternalmoment.

Considernowarigidbodywithafixedpointwhichisotherwisefree.Theexternalmomentiszero.Consequently,itsangularmomentumaboutthefixedpointisconstant.If,atacertaininstant,thebodyrotatesaboutaprincipalaxiswithangularvelocityω,itissimply Lbeingconstant,ωisconstanttoo,inmagnitudeanddirection.If,onthecontrary,thebodyrotatesaroundanon-principalaxis,Lisconstant,butωisnotnecessarilyso.

Thesameargumentsarevalidforthemotionofarigidbodywithoutanyconstraint,providedthecenterofmassischosenasthepole,forEq.(7.60).

8.13 KineticEnergyInthissection,weshalldiscussthekineticenergyofarigidbodymovingaboutafixedpointO,whichisnotnecessarilyinaninertialframe.Figure8.29shows

thesituationatacertaininstant.Thevectorωistheinstantaneousangularvelocity,which,ingeneral,variesbothinmagnitudeanddirection.

Fig.8.29 Arigidbodymovingaboutafixedpoint

Asusual,wethinkofthebodyasbeingmadeofmaterialpointsofmassmi.Thekineticenergyofthegenericpointis ,whereυ

iisthemagnitudeofthevelocityofthepointandr′Iisitsdistancefromtheinstantaneousrotationaxis.Weobtainthekineticenergyofthebodyaddingupallthepoints.IfIωisthemomentofinertiaabouttheinstantaneousrotationaxis,wehave

(8.59)Wehadalreadyfoundthisexpression,Eq.(8.25),inthecaseofrotation

aboutafixedaxis.Ifthereferenceisaninertialoneandifthebodyisnotsubjecttoexternal

forces,thekineticenergyisconstantintime,butthedirectionoftheangularvelocityrelativetothebodydoes,ingeneral,vary.Alsoingeneral,bothωandIωvary,whiletheproductofthesquareoftheformerandthelatterareconstant.Inpractice,Eq.(8.59)isnotveryuseful.LetusfindamoreusefulexpressionproceedinginawaysimilartowhatwedidinSect.8.12fortheangularmomentum.WeworkinthereferenceframeofFig.8.29,withorigininthefixedpointO.ThevelocityofthegenericpointPiatthepositionvectorriis

(8.60)Thekineticenergyofthepointis

Weshouldnowaddupallthepoints.Intheaboveexpression,wehave,forexample,theterm Addingupthepoints,thisgives andis

analogousfortheotheraxes.Thesumsofthetermswiththeproductsoftwocoordinatesgivetermspropositionaltotheproductsofinertia.ItisthenconvenienttochoosethecoordinatesontheprincipalaxesrelativetoO,becausetheproductsofinertiaarezero.Withthischoice,wehave

(8.61)whichwecanwrite,recallingEq.(8.58),as

(8.62)Inthisexpression,thecomponentsontheaxesnolongerappear.

Consequently,itisvalidindependentofthereferenceframe.Wealsonoticethat,inabsenceofexternalforces,bothkineticenergyandangularmomentumareconserved.Consequently,thecomponentoftheangularvelocityonLOisconstanttoo.

8.14 RotationAboutaFixedAxis.ForcesontheSupportsWeoftendeal,inpractice,withsymmetricrigidbodiesthatrotateaboutafixedaxisathighangularvelocities.Thisisthecasewiththerotatingpartsofelectricandinternalcombustionengines,withthereelsofcarsandbikes,turbines,helices,etc.Therearetwovectorquantitiesinthegame:theangularvelocity,whichis,byconstruction,paralleltotheaxis,andtheangularmomentum,whichcanhaveadirection.WehaveseenanexampleofthissituationinSect.8.11.InSect.8.12,wehaveseenthatangularvelocityandangularmomentumareparallelonlyiftherotationaxisisaprincipalaxisofinertia.Ifthisisnotthecase,anexternalmomentmustbeappliedtomaintaintherotationaxisasfixed.Thisisdonethroughthemechanicalstructurethatsupportstheaxis,ingeneral,throughaballbearingtoreducefrictions.

Tobeconcrete,considertheexampleinFig.8.30.TherotationaxisisthroughthecenterofmassCofthebody,butisnotthesymmetryaxis.Theaxis

iskeptinpositionbytwoballbearings,representedinthefigure.Thecentralaxesofinertiaarethesymmetryaxisofthedisk,thatweshalltakeascoordinatez,andanytwomutuallyperpendiculardirectionsintheplanethroughCnormaltoz,whichwetakeascoordinatesxandy.Thefigureisashotofthemovementwhenthexaxisgoesthroughtheplaneofthefigure.

Fig.8.30 Rotationofarigidbodyaboutacentralnon-principalaxis

Thetotalforceexertedbythesupportsisjustequaltotheweightofthebody,bothifitrotatesandifitisatrest.Itwillnotenterintoourarguments.

WeshalltakeasthepoleofthemomentsoftheforcesandoftheangularmomentumthecenterofmassC,whichisalsoafixedpointinthiscase.Thesymmetryaxisofthebodyformsanangleαwiththerotationaxis.Consequently,angularmomentumandangularvelocityarenotparallel.Weshallsoonfindthedirectionoftheformer.

Weobservethattheangularmomentumcanbeusefullydecomposedinonecomponentparallelandoneperpendiculartotheaxis.Thedirectionofthelatterrotatesaroundtheaxiswithangularvelocityω.

Thecomponentoftheangularmomentumontheaxisis,withobviousmeaningofthesymbols,

(8.63)Tovarythemagnitudeoftheangularvelocity,wemustapplyamoment

paralleltotheaxis.Thisiswhatenginesdo,whentheyaccelerateordecelerate.Asamatteroffact,theballbearingsareusedtodecreasethefriction,which,however,cannotbecompletelyeliminated.Thefrictionmomentopposesthemotion.Ifweabandonthebodyinrotation,weobserveitsangularvelocitygraduallydecreasingduetothemomentofthefrictions.

Wenowstudytherotationofthecomponentsnormaltotheaxisoftheangularmomentumandofthemomentexertedbythesupport.Weassumethefrictionstobenegligibleandthemomentoftheforcestobeperpendiculartothe

axis.Consequently,boththemagnitudeoftheangularvelocityandtheaxialcomponentoftheangularmomentumareconstant.

Equation(8.58)becomes,inthecaseunderconsideration,

(8.64)Ifθistheanglebetweentheangularmomentumandtherotationaxis,as

seeninFig.8.30,wehave

(8.65)BoththeratioIx/Izandtherelationbetweenαandθdependontheshapeof

thebody.Ifthebodyisadisk,aswesawinSect.8.8,Ix/Iz=1/2,andEq.(8.65)gives If,asisoftenthecase,theanglesaresmallandwecanapproximatethetangentwithitsargument,itis Hence,theanglebetweenangularmomentumandrotationaxisisconstantintime.Inaddition,aswehavealreadyobserved,thecomponentoftheangularmomentumontheaxisisalsoconstantand,asaconsequence,themagnitudeoftheangularmomentumisconstant.Inconclusion,thenormalcomponentoftheangularmomentumisconstantinmagnitudeandrotatesaroundtheaxiswithangularvelocityω.Thedynamicalequationis

(8.66)whereMCistheexternalmomentexertedbytheballbearings.Thecoupleofforcesisshowninthefigure.Intheconsideredinstant,theplaneofthecoupleistheplaneofthefigure.Themagnitudeofthemomentis .Andalso,writingEq.(8.63)as ,

(8.67)Inconclusion,thestressonthesupportisperiodic,withperiod2π/ω,and

proportionaltothesquareoftheangularvelocity.Ifthelatterincreases,forexample,byafactoroften,themomentincreasesbyonehundred.

Wenowconsiderarotationatconstantangularvelocityaroundafixedaxis,whichisprincipalofinertia,butnotthroughthecenterofmass,asinFig.8.31.Inthiscase,theangularmomentumisparalleltotheaxisand,consequently,isconstantintime.Themomentexertedbytheballbearingsiszero.Theforcetheyexert,however,mustbeequaltothecentripetalforcethatisnecessarytomaintainthecenterofmassinitscircularmotion,namely

Fig.8.31 Rotationaboutaprincipalnon-centralaxis

(8.68)whererCisthepositionvectorofthecenterofmassrelativetothepointOontheaxis(seefigure)anduCisitsunitvector.Theforceisexertedbytheballbearings.Itsdirectionrotatesatangularvelocityω,itsmagnitudeisconstant,proportionaltothesquareoftheangularvelocity.

Inconclusion,theballbearingsduringtherotationmustdevelopforcesthatperiodicallyvaryindirection,havingresultantFCandtotalmomentMC.Theformeriszeroifthecenterofmassisontheaxis;thelatteriszeroiftherotationaxisisaprincipalaxisofinertia.Botharezeroiftheaxisiscentralofinertia.Clearly,thisistheconfigurationengineerstrytorealize,especiallyifthevelocitiesarehigh.Undersuchconditionsthesystemissaidtobedynamicallybalanced.Dynamicbalanceisobtained,forexample,forcarwheels,byinsertingsmallleadcounterweightswherenecessaryalongthetirerim.

8.15 RollingMotionThewheelsofabikeorofacarmovingdownthestreetnormallyrollwithoutslipping.Ifthefrictionbetweenwheelandstreetislowerduetorainorsnow,slippingcansetin,asituationthatshouldobviouslybeavoided.Thewheelcanbeconsideredadisk.Thehubisacentralaxisofinertia.

Consider,forexample,abikewheel.Ifweliftthebikeandbeginrotatingthewheel,whichdoesnottouchtheground,itrotatesarounditsaxis.Ifweputitdownandrideit,themotionofthewheelisthesumofatranslation,withthevelocityofitscenter,andofarotation.

Weshallconsiderrollingwithoutslippinghere.Ifthisistrue,ineveryinstant,thecontactpointofthewheelwiththegroundisstill.Figure8.32representsthewheelatacertaininstantinfullcolorandinfournearinstants,

1.

2.

twobeforeandtwoafterthatinpalecolor.Asonecansee,theextremeofthespokenearthegroundisalmostatrest,whilealltheotherpointsmove,tothedegreethattheyarefartherfromthecontactpoint.

Fig.8.32 Abikewheelmovingdowntheroad

Asamatteroffact,therearetwoequivalentwaystodescribetherollingmotion,showninFig.8.33.

Fig.8.33 Twopossiblerepresentationsofrollingwithoutslipping

atranslationwiththevelocityofthecenterofmasswithasuperposedrotationaroundthesymmetryaxiswithangularvelocityω

arotation,againwithangularvelocityω,aroundtheinstantaneousrotationaxis,whichistheaxisparalleltothesymmetryoneincontactwiththegroundintheconsideredinstant.

Thetypeofmotionwearediscussing,rollingwithoutslipping,cantakeplaceforcylindricalandsphericalshapes.Tobeconcrete,weshallcontinueconsideringacylinder,ofradiusR,rollingonaplane,withreferencetoFig.8.34.

Fig.8.34 Cylinderrollingonaplane

Wetakethexaxisonthegroundinthedirectionofthemotion.Ifthereisnoslipping,themagnitudeυCofthevelocityvCofthecenterofmassandtheangularvelocityωarelinkedbytherelation

(8.69)Thedirectionoftheangularvelocityvectorωisnormaltotheplanedrawn

towardstheinside.IfRisthepositionvectorofthecenterCrelativetothecontactpointA,wecanwrite

(8.70)Wenowfindtheexpressionofthekineticenergyofthebodyinbothofthe

above-mentionedpointsofviewandverifythattheresultisthesame.Inthefirstpointofview,thekineticenergyisthesumofthekineticenergy

“ofthecenterofmass”, wheremisthemassofthecylinder,andthatof

themotionrelativetothecenterofmass, whereICisthemomentofinertiarelativetothecentralaxis

(8.71)Inthesecondpointofview,themotionisapurerotation,withthesame

angularvelocity.Themomentofinertiais,forthetheoremofparallelaxes,.Hence,thekineticenergyisgivenbythelastmemberofEq.(8.71).

8.16 RollingonanInclinedPlaneAnimportantexampleofrollingmotionisthedescentofarigidsphereonaninclinedplane.Figure8.35representsthesystem.Theplaneformstheangleθwiththehorizontal,andtheradiusofthesphereisR.Theforcesactingonthesphereareitsweightmg,appliedtothecenterofmassC,andthereactionoftheconstraintappliedtothecontactpointA.Thelattercanbedecomposedintwocomponents,onenormal,N,andonetangent,Ft.Noticethat,inthecaseofthelatter,thefrictionforcemustbepresentinordertopreventslipping.Asamatteroffact,themagnitudeofFtcannotbelargerthanµSN,whereµSisthecoefficientofstaticfriction.Afterthat,slippingoccurs.Weshallassumethattheconditionofpurerollingissatisfied.

Fig.8.35 Asphererollingonaninclinedplanewithoutslipping

Weshalldealwiththeproblemthroughthreedifferentmethods.Method1.Weconsiderthemomentoftheforces,MA,aboutthe

instantaneousrotationaxisthroughthecontactpointA.IfIAisthemomentofinertiarelativetothisaxis,wecanwritethedynamicalequation

(8.72)Themomentoftheconstraintreaction,whichisappliedinA,iszero.The

momentoftheweightis,inmagnitude, andwehave .Thevelocityofthecenterofmassis becausethemotiondoesnotincludeslipping,anditsaccelerationis Substitutingintheaboveequation,wefind

but,forthetheoremofparallelaxes, ,whereICisthemoment

ofinertiaaboutthecentralsymmetryaxis.Consequently,

(8.73)Method2.Weconsiderthemomentsaboutthehorizontalcentralaxis

(throughC),MC,andusetheequation

(8.74)ThemomentoftheweightiszerobecauseitisappliedtoC.Themomentof

thenormalreactionNisalsozerobecausetheforceisparalleltothearm.ThemagnitudeofthetangentreactionoftheconstraintisFtR.Wecanwrite

(8.75)Thisequationcontainstwounknowns,theangularaccelerationandFt.A

secondequationisgivenbythetheoremofthecenterofmassmotion

(8.76)Recallingthat wefindbackforaCEq.(8.73)andforFt

(8.77)Method3.Intheprocess,weareconsideringthatthemechanicalenergyis

conserved.Indeed,evenifanon-conservativeforceispresent,suchasthefriction,itsworkiszero,becausethecontactpointA,whereitisapplied,doesnotmove.SupposethatthebodystartsfromrestatthepointOoftheplaneattheheighth(seeFig.8.35).WecallxacoordinatealongtheinclinedplanedirecteddownwardswiththeorigininO.ThevelocityofthecenterofmassisWetakethezeroofthepotentialenergyath=0.Initially,theenergyofthebodyisonlypotential,anditsvalueismgh.Whenthebodyisatthegenericcoordinatex,itspotentialenergyismg(h−xsinθ).Itskineticenergyisthesumofthekineticenergiesofthecenterofmass, andoftherotationaboutthe

centerofmass, Theenergyconservationequationisthen

or

(8.78)fromwhichweobtainthecenterofmassvelocityatthegenericx

(8.79)

Attheendoftheinclinedplane,thecenterofmassvelocityisthen

(8.80)

Theratio thatappearsinthisexpressionhasthephysicaldimensionsofalengthsquared.Thislength,k,iscalledtheradiusofgyrationofthebodyaboutthecentralaxis,namely

(8.81)Usingthisquantity,thefinalcenterofmassvelocityis

(8.82)

Usingenergyconservation,wehavedirectlyfoundthecenterofmassvelocity.Takingitstimederivative,wegetbackEq.(8.73)writtenintermsofthegyrationradius.

(8.83)Inthedenominatorsoftheexpressions,wehavefoundwehavetheratioof

twolengths,thegyrationradiusandthegeometricradiusofthebody.Thisratiodependsonthedistributionofthemasses,asweshallnowseeinsomeexamples.Noticethattheaccelerationandthefinalvelocityfromagivenheightaresmallerforlargervaluesofk/R.Indeed,aswehaveseen,partoftheinitialpotentialenergybecomeskineticenergyofthetranslation,whilepartbecomeskineticenergyoftherotation.Theratiobetweenthesetwoenergiesis

(8.84)

Forexample,usingtheexpressionsforthemomentsofinertiawefoundinSect.8.7,wefindforanemptycylinderk2=R2and forafullhomogeneouscylinderk2=R2/2and andforafullhomogenousspherek2=2R2/5and .Ingeneral,theemptybodiesdescendslowly,followedbythefullones.Thisisbecause,forthesametotalmass,theformerhavelargermomentsofinertia,andconsequently,thefractionofkineticenergyassociatedwiththerotationislarger.Toenhancetheeffect,wecanbuild

thedeviceshowninFig.8.36a,whichisadiskwithacylindricalaxis.TheradiusRofthelatterismuchsmallerthanthatofthedisk.Theaxislaysontwoparallelinclinedrails.Theratiok/Rcanbemadeverysmall,obtainingaquiteslowdownwardacceleration.Contrastingly,intheconfigurationofFig.8.36b,theinstantaneousaxisofrotationisclosetothecentralaxisandthelargerfractionofthekineticenergytheenergyofthecenterofmass.

Fig.8.36 Thefractionofkineticenergyinrotationisalarge,bsmall

Weshallnowanalyzewhentheconditionsofpurerollingaresatisfied.WehavealreadyfoundtheexpressionEq.(8.77)forthetangentialforcethattheconstraintmustprovide.Wenowwriteitintheform

(8.85)

ThemaximumtangentialforcetheconstraintcanprovideisThenormalreactionshouldequilibratethenormalcomponentoftheweight,becausethereisnoaccelerationinthatdirection,namely .Hence,

.Theno-slippingconditionisthen

Thisisaconditionontheslopeangleθ,namely,simplifying,

(8.86)Supposewestudythemotionofasphererollingonaninclinedplaneandwe

graduallyincreaseitsslope.WhenwereachslopeslargerthanthevalueofEq.(8.86),weobservethecontactpointslippingontheinclinedplane.

LetusbrieflygobacktowhatwesawinSect.2.12,astohowGalileiexperimentallyestablishedthatthevelocityofasphereattheendofaninclinedplaneisindependentonitsslope,dependingonlyonthedroph.Hedidnotknow,thatpartofthekineticenergyisintherotationmotion.However,wecannowshowthatthisconclusionwasindependentofthat.IntheconfigurationofFig.8.35,thevelocityofthesphereafteradrophis

(8.87)

tobecomparedtothatofamaterialpoint

(8.88)Consequently,themotionofthecenterofmassofthesphereisthesamefor

amaterialpointwith5/7ginplaceofg.Wenotice,inaddition,thatheverylikelywasusingacross-sectionofthebeamsimilartoFig.8.36bforwhichthefactorinfrontofgiscloserto1.However,thisfactorisirrelevant,becausethescopeofhisexperimentswasthestudyoftheacceleratedmotion,notthemeasurementofthegravityacceleration.

8.17 GyroscopesAgyroscopeisarigiddiscwithafixedpoint.Often,butnotalways,thefixedpointisthecenterormassor,atleast,apointonthesymmetryaxis.Theconstructionissuchthattherotationaxisisfreetoassumeanyorientation.Ifthefixedpointisthecenterofmass,theexternalmomentiszero,andconsequently,theangularmomentumisconservedwhenthediskrotates.Thedirectionoftheaxisisunaffectedbytiltingorrotationofthemounting.Forthisproperty,gyroscopesofthistypeareusefulformeasuringormaintainingorientation.Anotherexampleofagyroscopeisthespinningtop.

ThegyroscopeinFig.8.37isthediskinthecenter.Themounting,calledaCardanmounting,afterGirolamoCardano(1501–1576),guaranteesacompletefreedomtorotateinanydirectionwiththecenterofmassfixed.Thesupportismadeofthree“gimbals”orrings.Theoutergimbalisahalfcircular,orfullycircular,ringfixedonthesupportbasis.Thesecondgimbalismountedontheouterone.Itisfreetopivotaboutanaxisinitsownplane(ainthefigure)thatisalwaysperpendiculartothepivotaxisoftheoutergimbal.Thethirdgimbalismountedonthesecondoneandisfreetopivotaboutanaxisinitsownplane

perpendiculartothefirstaxis(binthefigure).Finally,theaxisofthediskismountedonthethirdgimbal,freetopivotaroundanaxisinitsplaneperpendiculartothesecondaxis(cinthefigure).Thisisacentralaxisofthediskand,assuch,apermanentrotationaxis.

Fig.8.37 AgyroscopewithCardanmounting

Allthepivotsarejoinedthroughballbearingstominimizethefrictions.Noticethatinthefigure,thethreeaxesarenotonlymutuallyperpendicular,butalsothatbisvertical,andaandcarehorizontal.Thelatterconditionisnotnecessary,however.Indeed,ifonetakesthebasisinone’shandandrotatestheexternalsupport,bwillnotbeverticalandaandcwillnotbehorizontal,buttheyremainmutuallyperpendicular.

Ifwetakethediskinourhand,wefeelhowitcanberotatedinanydirectionwithoutanyeffort.Indeed,thediskisinanindifferentequilibriumconfigurationand,aswejustsaid,thefrictionsarenegligible.Wecangivearapidspinningmotiontothediskbywrappingmanyturnsofwirearounditsaxisandthendrawingitquickly.Therotationcanlastalongtime,becausethefrictionsareverysmall.

TheCardanmountingisnotnecessaryforgyroscopeshavingthefixedpointonthesymmetryaxis,butnotinthecenterofmass.Themostwell-knownexampleisthespinningtop.Thetop,themotionofwhichweshallstudysoon,isabodyofapproximatelyconicshapesupportedonahorizontalplanespinningaboutitsaxis.Ifthefrictionbetweenthetipofthetopandtheplaneisenough,

thesupportpointremains(approximately)atrestandthetopisagyroscope.Weanticipatethatthemotionsofthegyroscopes,whenweapplyanexternal

actionontothem,lookquitestrange.Gyroscopesdonotbehaveasourintuitionwouldsuggesttous.Tounderstandthem,weshouldfixourattentiononthefactthatthecharacteristickinematicquantitiesofarigidbodyinrotationaretheangularvelocityandtheangularmomentum.Botharevectors.Payattentiontothefactthat,tomodifytheangularmomentum,weneedtoapplyatorque,oracoupleofforces,ratherthanoneforce.Theinducedchangeofangularmomentum(anothervector)hasthedirectionoftheappliedtorque,whichisperpendiculartotheforce.Ifweapplyatorqueparalleltotheangularmomentum,wemodifyitsmagnitudeandnotitsdirection,whereasifweapplythetorqueperpendiculartotheangularmomentum,wemodifyitsdirectionandnotitsmagnitude.

Letusnowdiscussafewsimpleexamples.Inthefirstcase,representedinFig.8.37,thefixedpointisthecenterofmass

andtheaxisisthesymmetryaxis,whichisanaxisofpermanentrotation.TheangularmomentumLCandtheangularvelocityωareparalleland

(8.89)Iftheexternalmomentiszero,theangularmomentumisconstantandthe

angularvelocityaswell:

(8.90)Weobservethat,ifwetakethesupportinonehandandchangeits

orientation,thespinningdirectionrelativetotheground,whichisaninertialframe,doesnotchange.Thesupportgimbalschangedirectionabouttheinvariabledirectionoftherotationaxis(c,inthiscase).

Torpedoes,forexample,makeuseofthisproperty.Onemountsagyroscopeinsidethetorpedoandguaranteesacontinuousspinningwithamotor.Ifthetorpedodeviatesfromthestraighttrajectory,duetoasubmarinecurrentorsomeotherfactor,thedirectionofthespinningaxischangesrelativetothetorpedo.Aservomechanismthenentersintoactiontomodifytherouteacingonthehelm.

Thesecondcaseisthesamegyroscopeinthepresenceofanappliedtorque.Wecan,forexample,suspendamassmtoapointAofthecaxisatacertaindistancefromthecenter,asinFig.8.38.TheangularvelocityandtheangularmomentumarestillparallelandEq.(8.89)isstillvalid.Butnow,theangularmomentumvaries,accordingtotheequation

Fig.8.38 Agyroscopewithanexternaltorque

(8.91)OurintuitionsuggeststhatwewouldseethepointAlowerundertheaction

oftheweight.Butthisisnotwhatweobserve.PointAdoesnotlower,but,onthecontrary,slowlymovesinahorizontalcircle.Thismotionoftherotationaxisciscalledprecession.

Tounderstandthis,aswehavealreadystated,wemustthinkaboutthedirectionoftheappliedmoment,notoftheforce.Letusstartconsideringtheinstantinwhichtheaxisofthegyroscopeisstillatrestandweapplytheweight.Theverticalweightforceexertsonthegyroscopeamoment,ortorque,thedirectionofwhichishorizontalandperpendiculartothecaxisand,consequently,perpendiculartotheangularmomentum.Inthetimeintervaldt,thevariationofangularmomentumis,forEq.(8.91),

whichhasadirectionperpendiculartoLCinthehorizontalplane,asshowninFig.8.39.Consequently,LCvariesindirectionandnotinmagnitude.Thecaxis,whichhasthedirectionoftheangularvelocityand,consequently,oftheangularmomentum,alsorotatesinaccord.Thetorque,whichisperpendiculartotheaxis,rotatesaswell,becausetheweightisappliedtothepointAoftheaxes.Thetorquealwaysremainsperpendiculartotheangularmomentum.Consequently,thejustdescribedsituationissuchineveryinstant,notonlyinthe

initialone.

Fig.8.39 Theprecessionofthegyroscope

Theangularvelocityoftheprecessionmotion,whichwecallΩ,isdirectedvertically,inourcase,upwards.Tohaveitdownwards,namelytohavethegyroscopeprecedingintheoppositedirection,wejusthavetoattachtheweightattheotherextremeofthecaxis,orhavethediskspinningintheoppositedirectionandtheweightstillinA.

Themotionofthegyroscopeisarotationwithangularvelocityequaltothe(vector)sumofωandΩ.Rigorouslyspeakingthen,therotationaxisisnotexactlythesymmetryaxis.However,inpractice,thespinningvelocityisalwaysmuchlargerthatoftheprecession.TherotationisinagoodapproximationaboutthesymmetryaxisandwecanconsiderEq.(8.89)valid.WithreferencetoFig.8.39b,wecanwrite .IflisthedistanceofpointAfromthecenterC,themagnitudeofthemomentisMC=mglandtheprecessionvelocityis

Thethirdcasewewillconsideristhefollowing.Thesuspensionpointisnotthecenterofmass,butisonthesymmetryaxisanyway.Theexternalmomentisnotzero;itsdirectionisalwaysperpendiculartotherotationaxis.Figure8.40ashowshowsuchconditionscanberealized.Theaxisofthediskendswithasmallsphere.Thespherelaysonaconcavesupportontopofacolumn,allowingtheaxistospinandtochangeitsdirectionfreely.Wenowgivetothegyroscopearapidspinaboutitsaxis,keepingithorizontalwithourhand.Whenweabandontheaxis,itdoesnotfalldownwards,butrotatesinaprecessionmotioninthehorizontalplane.Theanalysisofthemotionisidenticaltotheprecedingcase,withtheonlydifferencebeingthattheweightisnowtheweightofthe

gyroscopeitself.

Fig.8.40 aAgyroscopewithsuspensionpointonthesymmetryaxis,butnotinthecenter.bprecessionandnutation

Noticehowthebehaviorofthesystemiscompletelydifferentwhenthediskisspinningfromwhenitisnot.Inthelattercase,ifwetaketheextremeoftheaxisinourhandandthenabandonit,theaxisfalls,rotatingintheverticalplane.Ifwedothesamewiththediskspinning,theaxisrotatesinthehorizontalplane.Theactingtorque,themomentoftheweight,isequalinbothcases,andsoisthechangeoftheangularmomentuminanytimeintervaldt.This,however,inthecaseofthespinningdisk,addstoapre-existentangularmomentum,modifyingitsdirection,while,contrastingly,inthecaseofnospinning,thechangeissolelytotheangularmomentum,whichconsequentlyhasthedirectionofthetorque.

Tobesure,theangularvelocityisthesumofωandΩ,and,consequently,isnotexactlyparalleltoaprincipal,permanentrotationaxis.Thejust-madedescriptionisvalidonlyinafirstapproximation.Letuslookmorecarefullyintotheissue.

Asamatteroffact,thegyroscope’sstrangeimmunitytoitsownweightisnotcompletelytrue.IfwesetthegyroscopespinningwiththepointAinourhand,whenweabandonit,itinitiallyfallsdownverticallyabit.However,assoonastheprecessionstarts,theextremeArisesagain,reachingthehorizontalplane,asshowninFig.8.40b.Thisisnotall,however.Theaxisdoesnotremainhorizontal.Theprecessionhassloweddownsomewhatduetotheriseoftheaxis

andisnolongerfastenoughtoneutralizetheweight.Theextremefallsagaintotheheightofthefirstdescent,theprecessionvelocityincreasesandtheextremerisesagain,andsoon;themotioncontinueswithaseriesofupanddownoscillations,whichareideallyallequal.Themotionissimilartothemotionoftheheadofsomebodythatnods,andiscallednutation,whichmeans‘nodding’inLatin.

Weshallnotfurtheranalyzethismotion,whichisquitecomplex.Rather,weshallmakeafewobservations.WhenagyroscopespinsaboutitssymmetryaxiswithangularvelocityωandprecedesatthesametimewithangularvelocityΩ,itstotalangularvelocityisnotparalleltothesymmetryaxis.Consequently,angularvelocityandangularmomentumarenotexactlyparallel.Theeffectsaregenerallysmallbecauseω Ω,butitisthebasisofthenutationphenomena.

Consideragyroscoperotatingaboutanaxisabitdifferentfromasymmetryaxisinabsenceofexternaltorque.Inthatcase,theangularmomentumisconstantandtheangularvelocityrotatesaroundit,describingacone.Inourcase,however,anexternaltorqueexists.Itisthemomentoftheweightthatisdirectedhorizontally,perpendiculartotheaxis.Theverticalcomponentoftheangularmomentumisconstant,becausetheexternaltorqueishorizontal.Themagnitudeoftheangularmomentumisconstanttoo,becausethetorqueisperpendiculartoitsdirection.Consequently,theangularmomentumvectorrotatesuniformlyinthehorizontalplane.Thisistheprecession.Theangularvelocitycontemporarilydescribesaconearoundtheangularmomentum.Theextremeoftheaxisdescribeacycloidcurve,asshowninFig.8.40b.Thisisthenutation.

Wecanlookatthephenomenonfromanotherslightlydifferentpointofview.Whenweabandontheaxishorizontalofthespinninggyroscope,theprecessionstarts.ThisaddstotheangularmomentumthevectorquantityIΩΩwhereIΩisthemomentofinertiaabouttheverticalaxisthroughO.Theexternaltorquebeinghorizontal,theverticalcomponentoftheangularmomentumisconserved.Consequently,thespinningaxismustfallabit,orevenbetter,rotatedownwards,insuchawaythatICωhasaverticalcomponentequalandoppositetoIΩΩ(seeFig.8.41).AnoscillationstartsinwhichIΩΩincreasesanddecreasesalternatively,andsodoestheanglewiththehorizontalofICω.

Fig.8.41 Thevectorsplayingrolesinthenutation

Asalastexampleofprecession,weconsiderthetop.Thetopisarigidbodyofapproximatelyconicalshape,endingwithatip.Initially,wegivethetoparapidspinaboutitssymmetryaxiswithangularvelocityω.ThetipOlaysonahorizontalfloor,asinFig.8.42.WeassumethatthefrictionisenoughtokeeppointOatrest.Weobservethat,beyondspinning,thetopalsohasaprecessionmotion,withangularvelocitythatweshallcallΩ.

Fig.8.42 Atopanditsprecession

LetrbethepositionvectorofthecenterofmassCrelativetoOandmgtheweightofthetop,applied,asusual,tothecenterofmass.TheconstraintforcesareappliedinO,whichwechooseasthepoleofthemoments.Consequently,theconstraintforcesdonotcontributetotheexternalmoment.Wehave

(8.92)ThemomentMOishorizontal,perpendiculartothespinaxis.Consequently,

thedirection,butnotthemagnitudeoftheangularmomentum,varies.Moreprecisely,theangularmomentumrotateswithangularvelocityΩ.Hence,forthePoissonformula

(8.93)

ConsideringthatΩ ω,wecanassumethat and,fortheaboveequations,that Now,bothΩandgarevertical,somethingwecanexpressas .Substitutinginthelastexpression,wehave

or

andfinally,forthemagnitudes

(8.94)Thiscorrespondstotheperiodoftheprecession

(8.95)Letuslookattheordersofmagnitude.Weapproximatethetopwitha

homogeneouscylinderofradiusR=2cm.Letr=3cmbethedistancefromthecenterofmasstothetip.Supposethatthespinningangularvelocityisω=120s−1(thatis,about20turnspersecond).Letuscalculatetheprecessionperiod.ThemomentofinertiaisI=mR2/2.Wehave

Thecorrespondingprecessionangularvelocityis whichisqiutesmallincomparisontothespinningvelocity.

8.18 CollisionsBetweenMaterialSystemsInChap.7,westudiedthecollisionphenomena.Inthatdiscussion,weconsidered,foreachcollidingbody,onlythemotionofitscenterofmass.Wedidnotconsidertherethemotionofeachbodyrelativetoitscenterofmass.Forexample,whenafootballplayerkickstheball,themomentumoftheballvaries.However,theplayermaywishtogiveanangularmomentumtotheballaswell,tomakeitfollowacurvedtrajectory.Ingeneral,inacollision,boththelinearandtheangularmomentumofeachbodyvary.Weshallnowdiscussthisaspectofcollisions.Weshalllimitthediscussiontothecasesinwhichoneofthebodiescanbeconsideredaspoint-like,whiletheotherisextendedandrigid.

Therearetwodifferentpossiblesituations:thetargetbodymaybefreeor

maybesubjecttoconstraints.Inthefirstcase,butnotinthesecond,wecanneglecttheexternalforcesduringthecollision,asinthecaseofpoint-likeobjects.Consequently,thetotallinearmomentumandtheangularmomentumareconserved.Thelatterdoesnothaveanyroleincollisionsbetweenpoint-likeobjects.Itaddsnothingtothelinearmomentumconservation.Theangularmomentumconservationhas,contrastingly,observableconsequencesincollisionsbetweenextendedobjects.Evenbetter,weobservethattheangularmomentumconservationinanisolatedsystemisaconsequenceofaparticularaspectoftheaction-reactionlaw,namelythatactionandreactionhavethesameapplicationline.Thisaspectcannotbeexperimentallycontrolledwithcollisionsbetweenpoint-likebodies.Weneedtolookattheangularmomentumconservationincollisionsbetweenextendedbodies.

Ifconstraintsarepresent,thereareexternalforcesactingduringthecollision.Consider,forexample,aballhittingabarpivotedonanaxisinitiallyatrest.Themotionafterthecollisionisboundtobearotationabouttheaxis.Theforcesexertedbytheconstraintsmustbesuchastoequilibratesomeofthecomponentsoftheimpulsiveforcesthatdevelopduringthecollision.Consequently,theirintensityisgreat,andcannotbeneglected.Thecollisionwithaconstrainedbodyisnot,consequently,aprocessinanisolatedsystem.Linearandangularmomentum,ingeneral,arenotconserved.

Weshalllimitthediscussiontotwoexamples.

ExampleE8.9AhomogeneousdiskofmassMandradiusRlaysonahorizontalplane.Itisinitiallyatrest.Abulletofmassmandvelocityvi1hitsthediskonitsrimtangentially,asinFig.8.43,andsticks.Findthemotionofthesystemafterthecollision.

Fig.8.43 Abullethittingadisk

Therearenoconstraints.Thesystemcanbeconsideredasisolated.Linear

andangularmomentumareconserved.AsinFig.8.43,wetakeareferenceframeatrestonthesupportplane,withthexaxisinthedirectionoftheinitialvelocityofthebulletandthroughthecenterAofthedisk.Theyisperpendiculartoxinthesupportplaneandthezaxisisperpendiculartoboth.Theangularmomentumofthesystem,whichwetaketobeaboutthecenterofmassC,isinthezdirection.Afterthecollision,wehaveonelonebodymovingatthevelocityofthecenterofmassvC.Theangularmomentumconservationgivesthetwoequations

Thesecondequationtellsusthattheycomponentofthevelocityofthecenterofmassiszero,whichisobvious.Thefirstequationgivesthevelocityofthecenterofmass

(8.96)Wechoosethecenterofmassasbeingthepoleoftheangularmomentums.

Itsycoordinatedoesnotvaryduringthemotionandisgiven,bydefinition,by

(8.97)Theinitialangularmomentumisthatofthebullet,becausethediskisnot

moving.Itsdirectionisoppositetothezaxisanditsmagnitudeis

(8.98)Inthefinalstate,thesystemdiskplusthebulletrotateswithangularvelocity

ω,whichwemustdetermine.Itsangularmomentumaboutthecenterofmassis.Theangularmomentumconservationthengives

(8.99)whichgivesusωonceweknowthemomentofinertiaIC.Thisisthesumofthemomentsoftheinertiaofthebullet,Ibandofthedisk,Id.Theformeris

,andthelattercanbefoundwiththetheoremof

parallelaxes,giving Inconclusion,

andfinally,fromEq.(8.99),

ExampleE8.10SupposenowthediskinthepreviousexampleisconstrainedbyaverticalaxisthroughitscenterA,aboutwhichitcanrotatefreely.Inthiscase,thesystemisnotisolated.Neitherthelinearnortheangularmomentumarenecessarilyconserved.Inthiscase,theexternalforcesaretheconstraintones,whichareappliedtothepointA.TheirmomentaboutAiszero.Consequently,theangularmomentumaboutAisconserved,sayLA,f=LA,i.

Inthefinalstate,thesystemisarigidbody,diskplusbullet,rotatingwiththeangularvelocityω,stilltobefound.Wecanwrite .Here,IAisthemomentofinertiaofthesystemaboutA,namelyHence,thefinalangularvelocityis

Noticethedifferencefromthepreviousexample.

8.19 TheVirtualWorksPrincipleInthissection,weshalldiscussamethodthatoftenturnsouttobeusefulforestablishingtheequilibriumconditionsformechanicalsystems,rigidornot.Themethodisbasedontheso-calledvirtualworksprinciple.

Avirtualdisplacementofamechanicalsystemisdefinedasanyinfinitesimaldisplacementcompatiblewiththeconstraintstowhichthesystemissubject.Forexample,forarigidbodypivotedonafixedaxis,arotationofaninfinitesimalangleabouttheaxis,orforacarriageonarail,aninfinitesimaltranslationinthedirectionoftherail,etc.

TheworkdWithatwouldbedoneforthatdisplacementbytheithforceactingonthesystemiscalledthevirtualworkofthatforce.Thevirtualworksprinciplestatesthatamechanicalsystemisinequilibriuminagivenconfigurationifthesumofthevirtualworksdonebytheforcesactingonthesystemforanyvirtualdisplacementfromthatconfigurationiszero.

(8.100)

Weshalldiscussafewexamples.

ExampleE8.11Figure8.44showsarigidbar,pivotedinO.Twoforces,F1andF2,areappliedtoitsextremesA1andA2perpendiculartothebar.ThedistancesoftheextremesfromOareb1andb2,respectively.

Fig.8.44 aFindingtheequilibriumconditionforalever,bthesamewithtwoweights

Thevirtualdisplacementsds1andds2oftheextremesareinfinitesimalarcsofthecirclesofthecenterinOofradiusesA1andA2.Indeed,theonlydegreeoffreedomistherotationangleϕabouttheaxis.

LetusstartwithF1.Itsvirtualworkforthedisplacementds1is,whereτ1,zisthemomentoftheforceaboutthez

rotationaxis,withpositivedirectionpointingoutsidethepageofthedrawing.Onceds1ischosen,thedisplacementds2ofA2isfixed.ThevirtualworkoftheforceF2is .Accordingtothevirtualworkprinciple,theconfigurationisofequilibriumif Werecognizetheknownresultthattohaveequilibrium,thetotalmomentabouttherotationaxismustbezero.

Asamatteroffact,thevirtualworksprincipleisaconsequenceoftheenergyconservationlaw.Toseethat,letuslookatthepresentexamplefromaslightlydifferentpointofview,asinFig.8.44b.TheforceF1istheweightofablockplacedinA1.WewanttoestablishtheequilibriumbyplacinganotherblockofweightF2inA2.WhichisthevalueofF2requestedforthat?Weimaginethatwhenweplacethesecondblock,A2movesds2downandA1movesds1up.Thecorrespondingvariationofpotentialenergyis

Beingthetotalenergyconserved,thevariationofpotentialenergymightbecompensatedbyanoppositevariationofkineticenergy.However,inthevirtualchangeweareconsidering,thesystemisatrestbothbeforeandafterthedisplacementandthekineticenergyisalwayszero.Weconcludethatthepotentialenergycannotvary, Thisiswhatthevirtualworks

principlestates.

ExampleE8.12Figure8.45ashowstwoblocksofmassesm1andm2restingontwoinclinedplanestiltedtothehorizontalattheanglesθ1andθ2andconnectedbyarope.Frictionsarenegligible.Wewanttoknowwhichistheratioofthetwomassestohaveequilibrium.

Fig.8.45 aTwoblocksinequilibriumondifferentslopes,bthebasisoftheStevinargument

Wethinktomoveblock1ofdsupwardsontheplane.TheworkdonebytheweightisdW1=−m1g(sinθ1)ds.Atthesametime,block2movesonitsplaneofthesamedsdownwards,becausewewanttheropetoremaininvariant.TheworkofitsweightisdW2=+m2g(sinθ2)ds.Theconstraintforcesarenormaltothedisplacementsanddonowork.ThevirtualworksprinciplesthenrequiresforequilibriumthatdW1+dW2=0.Theratioofthemassesmustbe

Historically,aswehavealreadymentioned,Galileiestablishedthelawofthefreefallwithexperimentsoninclinedplanesofdifferentslopes.Hethenextendedthevalidityofthelawtotheverticalmotionwiththeexactaboveargument,withoneoftheanglesbeingequalto90°.

HiscontemporarySimonStevin(1548–1620)demonstratedtheruleasdrawninFig.8.45b.Thechainisinequilibrium.Inthecaseofthisparticularrighttriangle,fiveringsbalancethreerings.Inthosetimes,thetrigonometricfunctionswerestillnotknown.

ExampleE8.13

8.1.

Figure8.46showsarigidbarofmassmpivotedatitslowestpointandheldbyarope.Thebarholds,inturn,ablockofmassM.FindthetensionToftherope.Thisproblemalsohasonedegreeoffreedom,therotationangleθ.Torespecttheconstraint,wecanonlydiminishit,sayby−dθ.Theactingforcesarethree:theweightofthebockMg,theweightofthebarmgandthetensionoftheropeT.

Fig.8.46 Findtheequilibriumconfiguration

Thedisplacementcorrespondingto−dθoftheblockisd(bcosθ)=bsinθdθupwards.TheworkofitsweightisdW1=−Mgbsinθdθ.Theweightofthebarisappliedtoitscenterofmass.ItsworkisdW2=−mg(b/2)sinθdθ.Theapplicationpointoftheropehasthedisplacementd(asinθ)=acosθdθandtheworkofthetensionisdW3=acosθdθ.

ImposingdW1+dW2+dW3=0,wehave .

8.20 ProblemsFig.8.47representsarigidbarb,andv1andv2arethevelocitiesofitsextremes.Isitpossible?

Fig.8.47 Problem8.1

8.2.

8.3.

8.4.

8.5.

8.6.

8.7.

ArigidbaroflengthL=8mandmassm=100kglaysontwosupportsatdistancesL1=2mandL2=1mfromthetwoextremes.FindtheforcesF1andF2onthetwosupports.

Onwhichofthefollowingelementsdoesthemomentofinertiaofabodydepend?Themassofthebody,theshapeofthebody,theangularvelocityofthebody,thepositionoftheaxisrelativetothebody,ortheexternalresultantforce?

Arigidbodyrotatesaboutafixedaxis.Howmuchdoesitskineticenergyvaryiftheangularvelocitydoubles?

Twomaterialpointsofmassesm1andm2arelinkedbyarigidbaroflengthLandnegligiblemass.Findthemomentofinertiaaboutaperpendicularaxisthroughthecenter.

Thedensityρ(r)ofacylinderoflengthLandradiusRvarieslinearlywiththedistancerfromtheaxisfromthevalueρ1ontheaxistothevalueρ2=3ρ1onthelateralsurface.Findthemomentofinertiaabouttheaxis.

Figure8.48representsathinannularsheetofradiiR1andR2.Findthemomentofinertiaabouttheaaxis.

8.8.

8.9.

8.10.

8.11.

8.12.

Fig.8.48 Problem8.7

Arigidcylinderrollsonaninclinedplanewithoutslipping.Itsdensityisnotnecessarilyuniform.Canthekineticenergyrelativetothecenterofmassbelargerthanthatofthecenterofmass?

Twomaterialpointsofmassesm1andm2arefixedtotheextremesofarigidbaroflengthLandnegligiblemass.Wewanttobringthebarintorotationwithangularvelocityωaboutanaxisperpendiculartothebarthroughoneofitspoints.Howshouldwechoosethispointsoastohavetheminimumkineticenergyforthegivenangularvelocity?

UnderwhichconditionsdotheangularvelocityωandtheangularmomentumLofarigidbodyhavethesamedirection?

Inwhichcasesisequation validforarigidbody?

Inwhichcasesisthekineticrotationenergyofarigidbodygivenby

?

8.13.

8.14.

8.15.

8.16.

8.17.

AhomogeneoussphereofradiusRandmassmrotatesaboutanaxisthroughitscenterCwithangularvelocityω.FindtheangularmomentumaboutC.Doestheangularmomentumdependonthepole?

AhomogeneoussphereofradiusRandmassmrotateswithoutslippingonahorizontalplane.Itsaxisadvanceswithvelocityv.ThepointsX,Y,andZinFig.8.49areintheverticalplanecontainingthecenterofthecylindershowninthefigure.TheirheightsareR/2lowerthanC,equaltoCandR/2higherthanC,respectively.Findtheangularmomentumofthecylinderabouteachofthesepoints.

Fig.8.49 Problem8.14

Abaroflengthl=3mofmassm=50kgisinitiallyverticalatrestwithoneextremeOlayingontheground.WithOmaintainedatrest,thebarfallstotheground.FindtheangularmomentumaboutOandthevelocityoftheotherextremeattheinstantinwhichthebarhitstheground.

Arigidhomogeneoussphereissetfreeonaplaneinclinedat40°withthehorizontal.Atwhichvaluesofthefrictioncoefficientwillthesphererollwithoutslipping?

Ayo-yo(Fig.8.50),whichweconsidertobeahomogenouscylinder,ofmassm=100g,hangsfromawirewrappedarounditsaxis.Theaxisishorizontal.Assumetheradiusofthewrappingtobeequaltotheradiusof

8.18.

8.19.

thecylinder.Theyo-yoisreleasedatrest.(a)Whatistimetthatittakestodroptoh=50cm?(b)WhichisthetensionTofthewireduringthedescent?

Fig.8.50 Theyo-yoofProblem8.17

AhomogeneousdiskofradiusRinaverticalplanecanrotateaboutitsgeometricalaxis(Fig.8.51).ThefrictionontheaxisisnotnegligiblebutexertsatorqueMaabouttheaxis,independentoftheangularvelocity.Aparticleofmassmstickstotherimofthecylinderattheleveloftheaxis.Thesystemisreleasedatrest.(a)Whichistheminimumvalueofmforthecylindertostartrotation?(b)Whichisthevalueofmatwhichitrotatesaquarterofaturnandstops?

Fig.8.51 Problem8.18

AhomogeneousdiskofradiusRandmassmrotatesaboutitsgeometricaxiswithangularvelocityω.Thefrictionsoftheaxisslowitdownuntilit

8.20.

8.21.

comestorest.Howmuchworkhavetheydone?

AblockofmassMonaninclinedplaneattheangleθisheldinequilibriumbyasetofpulleys,asshowninFig.8.52.Usingthevirtualworksprinciple,findthevalueofthemassmofthecounterweightneededtoinsuretheequilibrium.

Fig.8.52 ThesystemofExercise8.20

ThesysteminFig.8.53ismadeoftwoidenticaldumbbells.Eachofthemconsistsoftwosmallspheres,eachofmassm=0.3kg,separatedbyabarofnegligiblemassoflengthl=1m.Thedumbbellsmoveonahorizontalplanewithnegligiblefrictionwithequalandoppositevelocitiesυ=1m/s.Twospheres,asshowninthefigure,collideelastically.(a)Describethemotionafterthecollision.Findtheangularvelocities(magnitudeanddirection).(b)Howlongdoestherotationlast?(c)Thenwhathappens?

Fig.8.53 ThesystemofProblem8.21

1.1.

1.2.

1.3.

1.4.

1.5.

1.6.

1.7.

1.8.

Solutions(a)No.(b)Yes,ifΔVhasthesamedirectionandverseasV.

ΔV=−2V,ΔV=0and|ΔV|=2V.

(a)Δv=(4,0,3),(b)|Δv|=5,(c)Δυ=3.9.

(a)<υ>=υ,(b)<v>=0.

(a) ,a(t)=6jm/s2;(b)υ(t=2s)=12.2m/s.

R=υ2/a.

(a)v(t)=−iAωsinωt+jAωcosωt;a(t)=−iAω2sinωt−jAω2cosωt;υ(t)=Aω,a(t)=Aω2;(b) ,velocityisperpendiculartothepositionvector(c) ,accelerationisparallelandoppositetothepositionvector;(d)x(t)2+y(t)2=A2=constant.ThetrajectoryisacirclewithcenterintheoriginandradiusA.Themotionisuniforminanticlockwisedirection;(e)directionchangestoclockwise.

(a)Thefirststepinsolvingthistypeofproblemsisdrawingthevectorstheycontain,asinFig.1.v1isthecyclistvelocity,v2isthewindvelocityrelativetoground,v2−v1isthewindvelocityasfeltbythecyclist.

1.9.

1.10.

1.11.

1.12.

Vectorsandangledrawnwithcontinuouslinesareknown.Withthesinelawweget andβ=139.5°.Consequently,thewindblowsfrom40.5°fromNorthtoEast.(b)Thenewapparentdirectionofthewind(thevelocityofthecyclistis−v1)is anditsapparentdirectionis35.6°fromSouthtoEast.

Fig.1 Velocitiesandrelativevelocityofproblem1.8

Shewillcrossat3milestowardssternin18′.

(a)Therotationaxisintheplanexzisat27°tothex-axis(b)20rad.(c)themagnitudeofωgrowsproportionallytothesquareoftime,itsdirectionisconstant.

(a)α=78.5°,(b)t=11.5s(thesmallersolutionmustbechosen);(c)s=1.15km.

Themotionisthesumofatranslationatthevelocityvandarotationaboutthewheelaxis.Hence,υA=(υ,υ,0);υB=(2υ,0,0);υC=(υ,−υ,0).

2.3.

2.4.

2.5.

2.6.

2.7.

2.8.

2.9.

2.10.

2.11.

3.94N

TheequationofmotionofthebodyofmassMis−T+Mg=Maandforthebodyofmassmis−T+mg=ma.Hence .

Thekineticenergyofthehammeris(1/2)mυ2whenithitsand0attheend.Thechangeofkineticenergyisequaltotheworkdoneonthenail,whichinturnisequaltothemeanforcetimesthedisplacements.Themeanforceisthen .

F,fortheactionreactionlaw.

−Foneachhand,independentlyoftheaccelerationsofthespheresbeingdifferent,asaconsequenceoftheaction-reactionlaw

Statement1is,ingeneral,false.Statement2istruefortheringsontheguidesbandc,forenergyconservation.Forthesamereasonthestatementisfalsefortheguidea,becausethatringcannotreachB.

Theinitialkineticenergytransformsintoelasticenergyofthepoleand

2.12.

2.13.

2.14.

2.15.

3.1.

3.3.

theninpotentialgravitationalenergyoftheathlete. .(NBInpractice,theathleteraisesevenmoredoingworkwithhisarms.)

Thetworopeshaveequaltensions.Theybreakatthesametime.

Thelightersphererisesfourtimesmore(energyconservation).

(a)F(t=0)=0.09N;(b)Fmax=10N.

Iftherotationplaneishorizontal,thewireisonaconeatanangle,sayθ,withthehorizon,inordertobalancetheweightmgwiththeverticalcomponentofthetension,Tsinθ.Hencetheradiusofthecircleislcosθ.Wehavetwoequations .

Eliminatingθ,wehave

Ifthecircleisvertical,itsradiusisl.Thetensionvariesalongthecircle,reachingitsmaximuminthelowestpoint.Drawthesituation.Inthispoint ,hence .

IntheSIunits,ω0=10s−1,ν=1.6s−1,T=0.63s.

ExpandEq.(3.46)inseriesofthesmallquantityγ/ω0=0.02as

.ω1issmallerthanω0offive

partsinonehundredthousand.

3.4.

3.5.

3.6.

3.8

3.9.

3.10.

(b)Eachspringexpandsx/2,therestoringforcereducesto½,theproperangularfrequencyis√2smaller.(c)Theproperangularfrequencyis√2largerthanin(a).

(a)k=1kN/m;(b)withxincentimetersandtinsecondsx(t)=5cos10t;(c)inthesameunitsx(t)=5cos10t+10cos10t.

TaketheaverageonaperiodofEq.(3.70)andcomparethemembers.

InavectordiagramlikeinFigs.3.7and3.8thetwoforcesarerepresentedbyrotatingvectorsatthesameangularvelocity.Theanglebetweenthem,whichisthedifferencebetweentheirphases,ϕ,isconstant.Thephasedifferencebetweenforcesisthesameasbetweendisplacements.Fromthegeometrywehave andϕ=133°.

Initialvelocityisυ=28m/sandthekineticenergyUk=390kJ.Thisistheworkoftheforcein90m.Themagnitudeoftheforceis4.3kN(43%oftheweightofthecar).With15%slope,in100mthecardescendsh=15mandthepotentialenergydecreasesbymgh=150kJ.Tostopthecartheworkofthebrakingforceshouldbe430kJ.After100mthekineticenergyisreducedto110kJandthevelocityis15m/s(53km/h).

Theverticalforcesareequalandopposite.ThehorizontalforcesarethetensionofthewireT,whichisthecentripetalforceofmagnitudemυ2/ldirectedtowardsOandthefrictionofmagnitudeµdmgdirectedoppositetovelocity.Themagnitudesofbothareequalto4N.Theanglebetweenthemis90”.Hencethemagnitudeoftheresultantis5.7Nanditsdirectionisat135”withvelocity.

3.11.

3.12.

3.13.

4.1.

4.2.

4.3.

4.4.

4.5.

Atthelimitvelocityυlimthedragforceisequaltotheweightmg.Ifthetermproportionaltothevelocitydominates,R=C1aυ,υlim=1.3×108a2m/s.Fora=1mmυlim=130m/s,fora=0.1mmυlim=1.3m/s.Thetermproportionaltovelocityisdominantonlyinthesecondcase.Ifthetermproportionaltothesquarevelocityofthedragdominates,υlim=217√am/s.Hence,fora=1mm,υlim=6.9m/s.Neglectingthetermproportionaltovelocityisjustified.

(a)T=mυ2/R−mg.Thecentripetalforceisthesumoftheweightandthetension,whichintheconsideredpointhavethesamedirection,verticaldownwards.Ifthevelocityissmallerthanthecriticalonethemotionisnotcircular.(b)T=mυ2/R+mg.

(a)h=2R/3.(b)sameonthemoon,itdoesnotdependong.

0.5s

g(h2−h1)=∆ϕ.Hence1000/9.8orabout100m.

Neithervelocitynoraccelerationareconstant.

Massdoesnotvary,weightdiminishes.

Answersarefoundputtingthecentripetalforceequaltothegravitational

4.6.

4.7.

4.8.

4.9.

4.10.

4.11.

4.12.

attraction.

(almost3h).

Ifristheradiusoftheorbitofthesatellite,itsvelocityis .

Rememberthat .Theperiodofthesatelliteisthen

andr−RE=1500km.

Theradiusofthespheresisr=0.60m,thedistancebetweentheircentersisd=1.23m.ThegravitationalforceisF=4.4×10−3Nandtheshrinkingofthespringis90µm.

, (noticethatthisvalueis

muchhigherthanthedensityofthecrust,whichisabout2000kg/m3showingthatthecentralpartoftheearthmustbemuchdenserthantheaverage;itismadeofiron).

, ,alittledenserthan

water.

.

TheorbitalvelocityofIoisυI=18km/sandconsequently

4.14.

5.1.

5.2.

5.3.

5.6.

.Weseethatthe

contributionstothepotentialofthedistantmassesareimportant,differentlythanfortheforce.Indeed,thepotentialdecreasesas1/r,theforceas1/r2.

(a).Vertically.(b)Attheanglearctang(a/g)tothevertical,forward.

(a)Duringthebraking,theaccelerationofthetrainisat=−3ms−2.Inthereferenceframeofthetrain,theforcesactingonthecasearetheinertialforce−matandthefrictionforce−µdmg.Itsaccelerationrelativetothetrainisar=1ms−2andtheabsoluteoneaa=−2ms−2.(b)Duringthetimetbofthebraking,thecasemovesrelativetothetrainwithaccelerationarstartingfromrest.Itsspeedis10m/s,bothrelativetothetrainandtheground(trainhasstopped).(c)Thecasetravelsafirstpaths1=50mduringbraking(acceleratedrelativemotion)andasecondones2whenthetrainhasstopped.Inthesecondpath,theaccelerationisa′=−2ms−2,taking5tostop.Thetimetostopiss2=25m.

Theaccelerationoftheliftis2.8ms−2upwards.Nothingcanbesaidonvelocity.

Theangularvelocityisω=3.45rad/s,thecentrifugalforceattherimism1.8N,wheremisthemassoftheinsect.Theforceofstaticfrictionism0.98N.Itdoesnotmakeit.

5.7.

6.1.

6.2.

6.3.

6.4.

6.5.

6.6.

Notreally,becausethelateralshiftofthepointwherethegroundishitisabout5mm.

SupposethatthedirectionfromthelamptothemirroristhesameasthevelocityvO′.(theanalysisoftheappositecaseisquitesimilar).ThetimeinStakenbythepulseforitsround-tripisalwaysΔt0=2l/c.TheobserverinSseestheclockofS′movinginthedirectionofthelength;this

is .Inadditionheseesthat,whilethepulseistravelling,the

mirrorrecedeswithspeedυO′.Call thetimetoreachthemirror,we

have ,hence .Whenthepulse

comesback,theobserverinS′seesthedetectorapproaching.Ifwecall

thereturntime,wehave ,hence

.Theperiodoftheclockisthesumofthetwo.

(a)13.5µs.(b)1.9µs.(c)560m.

Thelimitfort→∞isc. constantintime.

υf=0.62c,M=2.1m.

γ=1010,about5min.

p=1.42meV/c.

6.7.

6.8.

6.9.

6.10.

7.1.

7.2.

7.3.

7.4.

7.5.

7.6.

p=0.295meV/c.

E=0.852meV.

γ=2.75,β=0.93.

γ=105, .

Itiszero.

Itmoveswithaccelerationg.

1m/s.Yes,thecollisionwascompletelyinelastic.

i=5×104Ns,F=105N.

m2/m1=3,υCM/υi=1/4.

(a) ,(b) .

7.7.

7.8.

7.9.

7.10.

7.11.

7.12.

7.13.

7.14.

7.15.

7.16.

0.74RE.

(a) .(b)10.

.

ThevelocityafterthecollisionisV=(0,2,2)equaltothecenterofmassvelocity.(b)50J,30J,20J.

(a)θ=−29°, .(b)Not.

(a)(0,0,14)Nm,(b)IfαistheanglebetweenvectorsrandF,

,henceb=2.8m,(c) ,henceFn=1.4N.

5m,3.2m,2.05mand1.31m.TheinitialenergyisU=mg5,thefollowingonesare0.8U,(0.8)2U,(0.8)4U,(0.8)6U.

Themaximumenergytransferis15J,correspondingtoΔx=32cm.

(14/6,11/6,11/6).

∆LO=(0,−800,0)kgm2s−1.

7.17.

8.1.

8.2.

8.4.

8.5.

8.6.

8.7.

8.8.

8.9.

, .

No.

Theexternalresultantforceiszero.Thetotalexternalmomentaboutoneofthesupportpointsiszero.F1=590N,F2=390N.

Quadruple.

I=µL2,whereµisthereducedmass.

.

Usetheparallelaxestheorem. .

ApositiveanswerwouldrequireI/R2>M,whereMisthemassandRistheradiusofthecylinder.Clearly,thisisimpossibleforanydistributionofthemasses.

Atthedistancefromm1of ,whichisthecenterofmass.

8.11.

8.13.

8.14.

8.15.

8.17.

8.18.

8.19.

WhenbothMOandωhavethedirectionofaprincipalaxisofinertiaandwhenOisfixedoristhecenterofmassandthethreeprincipalaxesaboutitareequal.

,whichdoesnotdependonthepole,ifitisstill.

, ,L=0.

, .

Therearetwounknown,thetensionofthewireandtheaccelerationofthecenterofmass.Usetheequations(7.49)and(7.59)andsolvethem.Therearetwoalternativesforthesecondequation,namelytakingthepoleinthecenterofmassorinthepointΩwherethewiredetachesfromtheyo-yo.Inthelattercase,takeintoaccountthatthevelocityofthepoleisparalleltothetotallinearmomentum.

, .

(a) ,(b)

8.20.

8.21.

(a)Bothdumbbellsrotatewithcounter-clockwiseangularvelocity,andtheircentersareatrest(angularandlinearmomentumconservation).Themagnitudeoftheangularvelocityω=2υ/l=2rad/s.(b)Theyrotatehalfaturn,thencollideagain.Ittakest=π/ω=1.57s.(c)Thesecondcollision,whichissymmetrictothefirst,blockstherotationsandthetwodumbbellsseparatewithtranslationsofspeedoppositetotheinitialones.

Index

AAbsolutereferenceframeAcceleratedmotionAccelerationAccelerationoftransportationActionandreactionActionlineAction-reactionlawAdams,JohanAdditionofvelocitiesAlmagestAmpère,AndréMarieAngularfrequencyAngularmagnificationAngularmomentumAngularmomentumaboutanaxisAngularmomentumaboutanaxistheoremAngularvelocityAnomalyAphelionApplicationpoint

ArchimedesArealvelocityAristarchusAristarchusofSamosArmAstronomianovaAstronomicalunitAtomicmassunitAtomicnumberAveragevalueAxialvectorAxisofpermanentrotationAzimuth

BBallisticpendulumBarycenterBaseunitsBilateralBoundorbitBoundvectorBraheBrahe,Tycho

CCardanmountingCartesianframeCassiniCassini,GiovanniDomenicoCavendishCavendishconstantCavendish,HenryCelestialequatorCelestialsphereCenterofforcesystemCenterofmassCenterofmassframeCenterofmassmomentum

CenterofmassmotionCenterofmomentaframeCenteroftheforcesCentralaxesofinertiaCentralcollisionCentralfieldCentrifugalforceCentripetalaccelerationCentripetalforceCircularuniformmotionCMframeCoefficientofkineticfrictionCoefficientofrestitutionCoefficientofstaticfrictionCollisionCommnetariolusCompletelyinelasticcollisionCompositependulumCompositionofforcesConfigurationConjugatediameterConservationofangularmomentumConservationoflinearmomentumConservativeConservativeforceContactforceContractionofthelengthsCo-ordinateCo-ordinateaxisCopernicusCopernicus,NicolausCoriolisaccelerationCoriolisforceCoriolis,GustaveCoriolistheoremCoupleCouplearmCovarianceCriticaldamping

CriticalvelocityCrossproductCurvatureCurvatureradius

DD’Alembert,JeanBaptisteDampedoscillationDampedoscillatorDarkmatterDeRevolutionibusDecaytimeDeferentDegreesoffreedomDellaPorta,GiovanniBattistaDensityDerivativeofavectorDerivedunitsDescartes,RenéDeterminantDialogueDiameterDicke,RobertDimensionalequationDirectionalderivativeDissipativeDissipativeforceDotproductDoublestarDynamicalequationsDynamicallybalanceDynamometer

EEccentricityEclipticEinsteinEinstein,Albert

ElasticcollisionElasticconstantElasticdeformationElasticenergyElasticforceElastichysteresisElasticlimitElectromagneticwavesElectromagnetismElectronvoltEllipseEllipticorbitsEnergyEnergyconservationEnergydiagramsEnergy-momentumvectorEnergyofmotionEötvösEötvösexperimentEötvös,LorándEpicycleEpicycloidEquantEquilibriumEquipollentsegmentEquipotentialsurfacesEquivalenceprincipleEquivalentforcesystemEtherEuclideanspaceEventExponentialExternalmomentaboutanaxis

FFaraday,MichaelFictitiousforceField

FieldofforceFitzGerald,GeorgeFixedaxisForceForcecentrifugalForcedoscillatorForcefieldForcemomentFoucault,LéonFoucaultpendulumFour-momentumFour-vectorsFracturestrengthFreefallFreefallaccelerationFreefalltoEastFrequencyFrictionFrictionangleFullwidth

GGalileiGalilei,GalileoGalileitransformationsGalleGalle,JohanneGeiger,HansGeneralrelativityGimbalGlobularclusterGoniometerGradientGravitationalattractionGravitationalconstantGravitationalfieldGravitationalforceGravitationalmass

GravitationalpotentialGravityaccelerationGroupGyroscope

HHalley,EdmundHandnessHarmonicHarmonicmotionHarmonicoscillationHarmonicoscillatorHarmonicemundiHerschel,JohnHerschel,WilliamHertzHertz,HeinrichHertz,HeinrichRudolfHightideHilbert,DavidHomogeneityprincipleHomogeneousHooklawHooke,RobertHookelawHuygens,ChristiaanHyperboles

IImpactparameterImpulseImpulse-momentumtheoremImpulsiveforcesInclinedplaneIndependenceofmotionsInelasticcollisionInertiaInertialaw

InertialforceInertialframeInertialmassInertialreferenceframeInitialphaseInitialpositionInstantaneousrotationaxisInteractionInteractionpotentialenergyInterferencefringesIntervalInvariantIsochronismIsolatedsystem

JJoule,JamesJuleJupiterJupitersatellites

KKeplerKepler,JohannesKeplerlawKeplerproblemKilogramKineticenergyKönig,Samuel

LLaboratoryframeLASERrangingLatusrectumLawofinertiaLeft-handedLengthofthependulum

LeverruleLeVerrierLeVerier,UrbainLifelineLightconeLight-likeLinearmomentumLinearregimeLineintegralLinesofforceLorentzLorentzfactorLorentzgroupLorentz,HendrikLorentztransformationsLowtide

MMarsMarsden,ErnestMassMassenergyMasslessparticlesMaterialpointMatrixMatrixminorMatrixorderMatrixproductMaxwellMaxwellequationsMaxwell,JamesClerkMechanicalenergyMechanicaloscillatorMercuryMetreMetrologyMichelsonMichelson,Albert

MichelsoninterferometerMichelsonMorleyExperimentMicrometerMoleculeMomentMomentofacoupleMomentofinertiaMomentumMoonMorley,EdwardMorsepotentialMotionuniformlyaccelerated

MotionaboutafixedpoleMotionperiodicMultiplesofunitsMuon

NNaturallengthNeutralequilibriumNewtonNewtonconstantNewtonlawNewton,IsaacNon-conservativeNormNormalreactionNutation

OObjectivelensOperationaldefinitionOppositevectorOpticalleverOrientedsegmentØrsted,HansChristianOrthogonalmatrix

OscillationamplitudeOscillationsOsculatingcircleOver-damping

PParabolaParallelaxestheoremParallelforcessystemParallelogramruleParsecParticlePascalPascal,BlaisePendulumPerihelionPerihelionofMercuryPeriodPeriodicmotionPermanentdeformationPermanentrotationaxesPerpendicularaxestheoremPhasePhaseoppositionPlanemotionPlanetsPlasticdeformationPlasticregimePoincaréPoincaré,HenryPoissonformulaPoisson,Siméon-DenisPolarco-ordinatesPolePositionPositionvectorPotentialenergyPower

PrecessionPrecessionofperihelionPrincipalaxesofinertiaPrincipiaProductProductsofinertiaProperangularfrequencyProperlengthPropertimePseudo-EuclideanspacePseudoscalarPseudovectorPseudscalarPtolemyPtolemy,ClaudiusPurerolling

QQuadraturetidesQuantityofmotion

RRadianRadiusRadiusofgyrationRectilinearuniformReducedlengthReducedmassReferenceframeRelativevelocityRelativisticmechanicsRelativityprincipleRelativitytheoryResolvingpowerResonanceResonancecurveResonancefrequencyRestenergy

RestlengthRestoringforceResultantReynoldsnumberReynolds,OsborneRight-handedRigidbodyRigidmotionsRollingRollingfrictionRollingresistanceRollingresistancecoefficientRosetteRotationRotationcurveRoto-translationRutherford,Ernest

SSaturnScalarScalarproductScalartripleproductScalenon-invarianceScatteringangleSecondSecondNewtonlawSemi-latusrectumSemi-majoraxisSiderealyearSidereusnunciusSimultaneitySistèmeInternationalSourcesofthefieldSpaceSpaceinversionSpace-likeSpacerotation

Space-timeSpecialrelativitySphericalsymmetrySpinningtopSpiralgalaxySpontaneousrotationaxesSpringconstantSquarematrixStableequilibriumStaticfrictionStatictranslationStationaryfieldStationaryoscillationStationarysolutionSteinerStevinStrainStressStronginteractionSubmultiplesofunitsSymmetrypropertiesSynchronizeclocksSynodicperiodSyzygy

TTargetparticleTelescopeTensionTensorofinertiaTide-generatingforceTidesTimeTimedilationTimeintervalTime-likeTimetranslationsTop

TorqueTorqueaboutanaxisTorsionbalanceTotalangularmomentumTotalenergyTotalmechanicalenergyTotalmomentTotalmomentumTotaltorqueTrajectoryTranslationTriplevectorproductTuningforkTunnellingTwo-bodysystemTwonewsciences

UUnder-dampingUnificationUniformcircularUniformfieldUniformmotionUniformtranslationmotionUnilateralUnitvectorUniversalgravitationUnstableequilibriumUraniburgobservatory

VvanderWaalsforcevanderWaals,JohannesVariablespeedmotionVarignonexperimentVarignon,PierreVectorVectorcomponents

VectordiagramVectordirectionVectormagnitudeVectormomentVectornormVectorproductVectorsumVelocityVelocityoflightVelocityoftransportationVenusVirtualdisplacementVirtualworkVirtualworksprincipleViscosityViscousdragViscousforceViscousresistanceViviani,VincenzovonMayer,Juilus

WWallis,JohnWaterchronometerWattWatt,JamesWeakinteractionWeightWindcirculationWorkWren,Christofer

YYoungmodulusYoung,Thomas

Z

Zenith