CLASSICAL MECHANIC

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<p>Classical MechanicsAn introductory courseRichard FitzpatrickAssociate Professor of PhysicsThe University of Texas at AustinContents1 Introduction 71.1 Major sources: . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71.2 What is classical mechanics?. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71.3 mks units . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91.4 Standard prexes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101.5 Other units . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111.6 Precision and signicant gures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121.7 Dimensional analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122 Motion in 1 dimension 182.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 182.2 Displacement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 182.3 Velocity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 192.4 Acceleration. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 212.5 Motion with constant velocity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 232.6 Motion with constant acceleration . . . . . . . . . . . . . . . . . . . . . . . . . . . 242.7 Free-fall under gravity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 263 Motion in 3 dimensions 323.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 323.2 Cartesian coordinates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 323.3 Vector displacement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 333.4 Vector addition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 343.5 Vector magnitude. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3523.6 Scalar multiplication . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 353.7 Diagonals of a parallelogram . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 363.8 Vector velocity and vector acceleration . . . . . . . . . . . . . . . . . . . . . . . . . 373.9 Motion with constant velocity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 393.10 Motion with constant acceleration . . . . . . . . . . . . . . . . . . . . . . . . . . . 393.11 Projectile motion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 413.12 Relative velocity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 444 Newtons laws of motion 534.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 534.2 Newtons rst law of motion. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 534.3 Newtons second law of motion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 544.4 Hookes law. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 554.5 Newtons third law of motion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 564.6 Mass and weight . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 574.7 Strings, pulleys, and inclines . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 604.8 Friction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 674.9 Frames of reference . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 705 Conservation of energy 785.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 785.2 Energy conservation during free-fall . . . . . . . . . . . . . . . . . . . . . . . . . . 785.3 Work. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 815.4 Conservative and non-conservative force-elds . . . . . . . . . . . . . . . . . . . . . 885.5 Potential energy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9235.6 Hookes law. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 935.7 Motion in a general 1-dimensional potential . . . . . . . . . . . . . . . . . . . . . . 965.8 Power . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 996 Conservation of momentum 1076.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1076.2 Two-component systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1076.3 Multi-component systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1126.4 Rocket science . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1156.5 Impulses . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1186.6 Collisions in 1-dimension . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1216.7 Collisions in 2-dimensions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1277 Circular motion 1367.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1367.2 Uniform circular motion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1367.3 Centripetal acceleration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1387.4 The conical pendulum . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1417.5 Non-uniform circular motion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1427.6 The vertical pendulum. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1487.7 Motion on curved surfaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1508 Rotational motion 1608.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1608.2 Rigid body rotation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1608.3 Is rotation a vector? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16248.4 The vector product . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1668.5 Centre of mass . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1688.6 Moment of inertia . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1728.7 Torque. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1798.8 Power and work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1848.9 Translational motion versus rotational motion . . . . . . . . . . . . . . . . . . . . . 1868.10 The physics of baseball . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1868.11 Combined translational and rotational motion. . . . . . . . . . . . . . . . . . . . . 1909 Angular momentum 2049.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2049.2 Angular momentum of a point particle . . . . . . . . . . . . . . . . . . . . . . . . . 2049.3 Angular momentum of an extended object . . . . . . . . . . . . . . . . . . . . . . . 2069.4 Angular momentum of a multi-component system. . . . . . . . . . . . . . . . . . . 20910 Statics 21710.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21710.2 The principles of statics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21710.3 Equilibrium of a laminar object in a gravitational eld . . . . . . . . . . . . . . . . 22010.4 Rods and cables. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22310.5 Ladders and walls . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22610.6 Jointed rods. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22811 Oscillatory motion 23711.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23711.2 Simple harmonic motion. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 237511.3 The torsion pendulum . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24111.4 The simple pendulum . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24211.5 The compound pendulum . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24511.6 Uniform circular motion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24612 Orbital motion 25312.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25312.2 Historical background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25312.3 Gravity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26212.4 Gravitational potential energy. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26512.5 Satellite orbits . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26812.6 Planetary orbits. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26913 Wave motion 27913.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27913.2 Waves on a stretched string . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27913.3 General waves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28413.4 Wave-pulses. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28513.5 Standing waves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28913.6 The Doppler effect . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29161 INTRODUCTION1 Introduction1.1 Major sources:The sources which I consulted most frequently whilst developing this course are:AnalyticalMechanics: G.R. Fowles, Third edition (Holt, Rinehart, &amp; Winston, NewYork NY, 1977).Physics: R. Resnick, D. Halliday, and K.S. Krane, Fourth edition, Vol. 1 (John Wiley&amp; Sons, New York NY, 1992).EncyclopdiaBrittanica: Fifteenthedition(EncyclopdiaBrittanica, ChicagoIL,1994).Physicsforscientistsandengineers: R.A.Serway, andR.J.Beichner, Fifthedition,Vol. 1 (Saunders College Publishing, Orlando FL, 2000).1.2 What is classical mechanics?Classical mechanicsisthestudyofthemotionofbodies(includingthespecialcaseinwhichbodiesremainatrest)inaccordancewiththegeneral principlesrst enunciated by Sir Isaac Newton in his Philosophiae Naturalis Principia Math-ematica (1687), commonly known as the Principia. Classical mechanics was therstbranchofPhysicstobediscovered, andisthefoundationuponwhichallother branches of Physics are built. Moreover, classical mechanics has many im-portant applications in other areas of science, such as Astronomy (e.g., celestialmechanics), Chemistry (e.g., the dynamics of molecular collisions), Geology (e.g.,the propagation of seismic waves, generated by earthquakes, through the Earthscrust), and Engineering (e.g., the equilibrium and stability of structures).Classi-cal mechanics is also of great signicance outside the realm of science.After all,the sequence of events leading to the discovery of classical mechanicsstartingwith the ground-breaking work of Copernicus, continuing with the researches ofGalileo, Kepler, and Descartes, and culminating in the monumental achievements71 INTRODUCTION 1.2 What is classical mechanics?of Newtoninvolved the complete overthrow of the Aristotelian picture of theUniverse, whichhadpreviouslyprevailedformorethanamillennium, anditsreplacementbyarecognizablymodernpictureinwhichhumankindnolongerplayed a privileged role.In our investigation of classical mechanics we shall study many different typesof motion, including:Translational motionmotion by which a body shifts from one point in space toanother (e.g., the motion of a bullet red from a gun).Rotational motionmotion by which an extended body changes orientation, withrespect to other bodies in space, without changing position (e.g., the motionof a spinning top).Oscillatory motionmotion which continually repeats in time with a xed period(e.g., the motion of a pendulum in a grandfather clock).Circular motionmotion by which a body executes a circular orbit about anotherxed body [e.g., the (approximate) motion of the Earth about the Sun].Ofcourse, thesedifferenttypesofmotioncanbecombined: forinstance, themotionof aproperlybowledbowlingball consistsof acombinationof trans-lationalandrotationalmotion, whereaswavepropagationisacombinationoftranslational and oscillatory motion. Furthermore, the above mentioned types ofmotion are not entirely distinct: e.g., circular motion contains elements of bothrotational and oscillatory motion. We shall also study statics: i.e., the subdivisionof mechanics which is concerned with the forces that act on bodies at rest andin equilibrium. Statics is obviously of great importance in civil engineering: forinstance, the principles of statics were used to design the building in which thislecture is taking place, so as to ensure that it does not collapse.81 INTRODUCTION 1.3 mks units1.3 mks unitsThe rst principle of any exact science is measurement. In mechanics there arethree fundamental quantities which are subject to measurement:1. Intervals in space: i.e., lengths.2. Quantities of inertia, or mass, possessed by various bodies.3. Intervals in time.Any other type of measurement in mechanics can be reduced to some combina-tion of measurements of these three quantities.Eachofthethreefundamental quantitieslength, mass, andtimeismea-sured with respect to some convenient standard. The system of units currentlyused by all scientists, and most engineers, is called the mks systemafter the rstinitials of the names of the units of length, mass, and time, respectively, in thissystem: i.e., the meter, the kilogram, and the second.The mks unit of length is the meter (symbol m), which was formerly the dis-tance between two scratches on a platinum-iridium alloy bar kept at the Inter-national Bureau of Metric Standard in S`evres, France, but is now dened as thedistance occupied by 1, 650, 763.73 wavelengths of light of the orange-red spectralline of the isotope Krypton 86 in vacuum.The mks unit of mass is the kilogram (symbol kg), which is dened as the massof a platinum-iridium alloy cylinder kept at the InternationalBureau of MetricStandard in S`evres, France.The mks unit of time is the second (symbol s), which was formerly dened interms of the Earths rotation,but is now dened as the time for9, 192, 631, 770oscillations associated with the transition between the two hyperne levels of theground state of the isotope Cesium 133.In addition to the three fundamental quantities, classical mechanics also dealswith derived quantities,such as velocity,acceleration,momentum,angular mo-91 INTRODUCTION 1.4 Standard prexesmentum, etc. Each of these derived quantities can be reduced to some particularcombinatio...</p>