chapter 3 work and energy. §3-1 work §3-2 kinetic energy and the law of kinetic energy §3-3...
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Chapter 3 Work and Energy
§3-1 Work
§3-2 Kinetic Energy and the Law of Kinetic Energy
§3-3 Conservative Force, Potential Energy
§3-4 The Work-Energy theorem Conservation of Mechanical Energy
§3-5 The Conservation of Energy
§3-1 WorkWork
a
b
F
rdFdWcos
rdF
-- -- element workelement work
rd
2r
1r
0
oror dsFdW cos
b
adAAaabb ：：
b
ardF
1.Work1.Work ----variable forcevariable forceF
Equal to the displacement times Equal to the displacement times the component of force along the the component of force along the displacement.displacement.
In In Cartesian coordinate system
b
ardFW
)( dzFdyFdxF zy
b
a x 2.Work done by resultant force2.Work done by resultant force
...21 FFF
If
Then b
ardFW
b
ardFF
...)( 21
...21 b
a
b
ardFrdF
...21 WW The work done by the resultant force = the algebraic sum of the works done by every force.
3. Power3. Power
The work done per unit time The work done per unit time
t
AN
t
0
limdt
rdF
dt
dA vF
4.Work done by action-reaction pair of forces4.Work done by action-reaction pair of forces
1m2m
1r
2r
12f
21f
1rd 2rd
1rd
2rd
'rd
O
12' rdrdrd
1121 rdfdW
2212 rdfdW
21 dWdWdW
'21 rdf
)( 1221 rdrdf
与参考点的选择无关与参考点的选择无关 relative displacement relative displacement
§3-2 Work-kinetic energy theorem Work-kinetic energy theorem
rdF
rdamrdmacos
a
ta
na
rdmat
rddt
dvm
mvdv
1. WKE Theo. of a particle1. WKE Theo. of a particle
a
b
22
2
1
2
1ab mvmv
b
ardFW
b
a
v
vmvdv
F
rd
The total work done on a particle The total work done on a particle = = the increment of its kinetic energythe increment of its kinetic energy
2
2
1mvEk
--Work-kinetic energy theorem
DefinitionDefinition -- -- Kinetic energyKinetic energy
kakb EEW
( the Law of kinetic energy)
1a1b1m
2a2b2m
12f
21f
1F
2F
According to aboveAccording to above
1
111211 )(
b
ardfFW
11 kakb EE
2
221222 )(
b
ardfFW
22 kakb EE
For For mm11
…
+
2. WKE Theo. of particle system2. WKE Theo. of particle system
For For mm22 …
2
2
1
1
2
2
1
12211122211
b
a
b
a
b
a
b
ardfrdfrdFrdF
)()( 2121 kakakbkb EEEE
kakbinex EEWW
exW -- Work done by external force
inW -- Work done by internal force
Final KE Initial KE
The sum of the works done by all external forces The sum of the works done by all external forces and internal forces and internal forces = = the increment of the the increment of the system’s KE.system’s KE.
-- System’s work-kinetic energy theorem-- System’s work-kinetic energy theorem
Extend this conclusion to the system including Extend this conclusion to the system including n n particlesparticles
kakbinex EEWW
[[ExampleExample] A particle with mass of ] A particle with mass of mm is fixed on t is fixed on the end of a cord and moves around a circle in hohe end of a cord and moves around a circle in horizontal coarse plane. Suppose the radius of the crizontal coarse plane. Suppose the radius of the circle is ircle is RR. And . And vvoo vvoo/2 /2 when the particle moves owhen the particle moves o
ne revolution. Calculate ne revolution. Calculate The work done by fricThe work done by friction force. tion force. frictional coefficient. frictional coefficient. How ma How many revolutions does the particle move before it reny revolutions does the particle move before it rests?sts?
· ·v
R
mgf Opposite to the moving directionOpposite to the moving direction
rdfW
Rmg 2 208
3mv
We getWe getRg
v
16
3 20
SolutionSolution
2
02
2
1
2
1mvmvW 2
020
2
1)
2(
2
1mv
vm
208
3mv
According to WKE theo.,According to WKE theo.,
Suppose the P moves Suppose the P moves n n rev. before it rests.rev. before it rests.
RnmgWn 2According to work-kinetic energy theorem,According to work-kinetic energy theorem,
202
102 mvRmgn
3
4n (rev)(rev)
We haveWe have
§3-3 Conservative force Potential energy §3-3 Conservative force Potential energy
1. Conservative force1. Conservative forceThe work done by The work done by Cons. forceCons. force depend only on the depend only on the initial and final positionsinitial and final positions and not on the path. and not on the path.
The integration of The integration of Cons. forceCons. force along a close along a close
path path ll is equal to zero. is equal to zero. 0lrdF
The potential energy can be introduced The potential energy can be introduced when the work is done by the Cons. Force.when the work is done by the Cons. Force.
Otherwise, non-conservative forceOtherwise, non-conservative force 0lrdF
((11) PE of weight) PE of weight
gmP
2. Potential energy2. Potential energy
jmgP
Gravitational forceGravitational force
x
y
O
a
b
rd
P
rdPdW
cosrdmg mgdy
oror
)( ba yymg b
a
y
yab mgdyW
DefinitionDefinition
mgyE p --PE of weight--PE of weight
then pbpaab EEW
the work done by GF =the reduction of PE of weight
The The pointpoint of of zero PE of weightzero PE of weight is arbitrary is arbitrary
PE of weight at point PE of weight at point aa == the work done by GF the work done by GF
moving moving mm from from aa to to zero PE pointzero PE point..
thenthen 0
0aya mgdyW amgy
apa mgyE 0aW
IfIf 0by
((22) Elastic PE) Elastic PE
0 xmF
kxF Elastic forceElastic force
rdFdW
kxdx
ax
a
bx
b
b
a
x
xab kxdxW 22
2
1
2
1ba kxkx
DefinitionDefinition --Elastic PE--Elastic PE2
2
1kxE p
thenpbpaab EEW
The The pointpoint of of zero elastic PEzero elastic PE ：： relaxed position of spring relaxed position of spring ((x=0x=0))
the work done by EF =the reduction of elastic PE
((33) Universal gravitational PE) Universal gravitational PE
rr
mmGF ˆ
221
Universal gravitational forceUniversal gravitational force
M
a
b
ar
brr
rdr m
dr
F
rd
rdFdW
)cos(2
rdr
GmM
drr
GmM2
b
a
r
rab r
drGmMW
2 )11
(ba rr
GmM
DefinitionDefinitionr
mMGE p
whenwhen br aWar
GmM
The The pointpoint of of zero UGPEzero UGPE ：： the distance of both particles is infinitythe distance of both particles is infinity( ( r r
--------UGPEUGPE
thenthen pbpaab EEW
paE
The PE of a particleThe PE of a particle at at a pointa point is is relativerelative and and
the change of a particlethe change of a particle from from one pointone point to to
another pointanother point is is absoluteabsolute..
RemarksRemarks
Only conservative force can we introduce Only conservative force can we introduce
potential energy.potential energy.
The done by conservative force The done by conservative force = =
the reduction of PE
)12 ppco EEW （pE
PE belongs to the system.PE belongs to the system.
Gravitational forceGravitational force
Elastic forceElastic force
Universal gravitational forceUniversal gravitational force
Conservative internal force
The frictional force between bodies is non-conservative internal force
Internal force Internal force ==Conservative IFConservative IF++non-Cons.IFnon-Cons.IF
noincoinin AWW
§3-4 The work-energy theorem Conservation of Mechanical Energy
System’s work-kinetic energy theoremSystem’s work-kinetic energy theorem
kakbinex EEWW
pcoin EW )( papb EE
)()( pakapbkbnoinex EEEEWW
pk EEE LetLet
-- -- mechanical energy of the systemmechanical energy of the system
The sum of the work done by the external The sum of the work done by the external forces and non-conservative forces equals forces and non-conservative forces equals to the increment of the mechanical energy to the increment of the mechanical energy of the system from initial state to final state.of the system from initial state to final state.
---- the work-energy theorem of a system the work-energy theorem of a system
abnoinex EEWW
--Conservation of mechanical energy
whenwhen
We haveWe have .ConstEE ab
0 noinex WW
[[ExampleExample] Two boards with ] Two boards with mass of mass of mm11 ，， mm2 2 ((mm22>>mm11) )
connect with a weightless connect with a weightless spring. spring. If the spring can pull If the spring can pull mm22 out of the ground after the out of the ground after the FF
is removed, How much the is removed, How much the FF must be exerted on must be exerted on mm1 1 at lest? at lest?
How is about the result if How is about the result if mm11 ，，mm22 change their position? change their position?
1m
2m
FF
1x2x
SolutionSolutionSuppose the length of the Suppose the length of the spring is compressed as spring is compressed as the the FF is exerted. And is exerted. And mm22 is is
pulled out of the ground as the pulled out of the ground as the length is just stretched length is just stretched after the after the FF is removed is removed
1x
2x
k
gmFx 1
1
k
gmx 2
2
11 xkgmF
22 xkgm
then2m
Chose the point of Chose the point of zero PE zero PE :: The spring is free length ( no information)The spring is free length ( no information)
Its mechanical energy is conservationIts mechanical energy is conservation
112
1 )(2
1xgmxk 21
22 )(
2
1xgmxk
Two boardsTwo boards++springspring++earth earth = = systemsystem
We can getWe can get gmmF )( 21
The result do not change if The result do not change if mm11 ，， mm22 change change
their position.their position.
§3-5 The Conservation of Energy
Friction exists everywhere
The frictional force is called as a non-
conservative force or a dissipative force which
exists everywhere. Its work depends on the
path and it is always negative. So if the
dissipative forces exist such as the internally
frictional force, it is sure that the mechanical
energy of the system decreases.
According to the work-energy theoremAccording to the work-energy theorem
abnoinex EEWW
The decrease of mechanical energy The decrease of mechanical energy is transformed into other kinds of energy such as heat energy because of friction. Which leads to the increase of temperature of system so that the internal energy of the system has an increment.
intE
In order to simplify this problem, if we suppose In order to simplify this problem, if we suppose WWexex=0=0
abnoin EEW We haveWe have 0
intEEEW banoin
0)(int ab EEERe-write above formula
The change of internal energy + the change of mechanical energy = conservation
So we can get the generalized
conservation law of energy as follow
Energy may be transformed from one kind to another in an isolated system. But it cannot be created or destroyed. The total energy of the system always remains constant.