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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
Work

F
1.Work
--variable force

dr
Equal to the displacement
times the component of force
along the displacement.
 r
r1 2

dW  F cos dr
or
 
a
 F  dr -- element work
dW  F cosds
ab：
A

b
a
dA 
0
b
a
 
F  dr
b

F

In Cartesian coordinate system
W 
b
a
  b
F  dr   ( Fx dx  Fy dy  Fz dz )
a
2.Work done by resultant force
  
If F  F1  F2  ...

b 
b 


Then W  F  dr   ( F1  F2  ...)  dr
a
a
b 
 b 
  F1  dr   F2  dr  ...  W1  W2  ...
a
a
The work done by the resultant force =
the algebraic sum of the works done by
every force.
3. Power
The work done per unit time
 
A dA F  dr  
N  lim



F

v
t  0  t
dt
dt
4.Work done by action-reaction pair of forces


dW1  f12  d r1


dW2  f 21  dr2
dW  dW1  dW2



 f21  ( dr2  dr1 )


 f21  d r '
m1
 
r1 dr1 r2
O
relative displacement
与参考点的选择无关

f 12

f 21 m2

 dr2

dr2

dr1

dr '



dr '  dr2  dr1
§3-2
Work-kinetic energy theorem
1. WKE Theo. of a particle
 
 
 F  dr  ma  dr

 ma cos  dr
a

 dr  b
at  a

 F
an
dv 

 mat dr  m dr  mvdv
dt
  vb
1 2 1 2
W   F  dr   mvdv  mv b  mv a
va
a
2
2
b
1
Definition E k  mv 2
2
-- Kinetic energy
W  Ekb  Eka
The total work done on a particle =
the increment of its kinetic energy
--Work-kinetic energy theorem
( the Law of kinetic energy)
2. WKE Theo. of particle system
According to above
For m1
W1  
b1
a1
 

( F1  f12 )  dr1
 Ekb1  Eka1 …
For m2 W2 
+

b2
a2
a1
a2
m1

f 21

F1
b1

f12
m2

F2
b2
 

( F2  f12 )  dr2  Ekb2  Eka2 …

b1
a1
  b2   b1   b2  
F1  dr1   F2  dr2   f12  dr1   f21  dr2
a2
a1
-- Work done by
Wex external force
a2
-- Work done by
Win internal force
 ( Ekb1  Ekb2 )  ( Eka1  Eka2 )
Final KE
Initial KE
Wex  Win  Ekb  Eka
Extend this conclusion to the system including
n particles
Wex  Win  Ekb  Eka
The sum of the works done by all external forces
and internal forces = the increment of the
system’s KE.
-- System’s work-kinetic energy theorem
[Example] A particle with mass of m is fixed on
the end of a cord and moves around a circle in
horizontal coarse plane. Suppose the radius of
the circle is R. And vo vo/2 when the particle
moves one revolution. Calculate The work
done by friction force. frictional coefficient.
 How many revolutions does the particle move
before it rests?
v
· R
·
Solution According to WKE theo.,
1 2 1
v0 2 1
2 1
2
 W  mv  mv 0  m( )  mv 0
2
2
2
2
2
3
2
  mv 0
8
f  mg Opposite to the moving direction
 
3
2
W   f  dr   mg  2R   mv 0
8
2
3v 0
We get  
16Rg

 Suppose the P moves n rev. before it rests.
Wn   mg  n  2R
According to work-kinetic energy theorem,
1
2
We have  nmg  2R  0  mv 0
2
4
n
3
(rev)
§3-3 Conservative force
Potential energy
1. Conservative force
The work done by Cons. force depend only on the
initial and final positions and not on the path.
The integration of Cons. force along a close
path l is equal to zero.
 
F

d
r

0

l
 
Otherwise, non-conservative force  F  dr  0
l
The potential energy can be introduced when
the work is done by the Cons. Force.
2. Potential energy
(1) PE of weight
ya
Gravitational force


P  mg


or P   mgj

dr


P
 
O
dW  P  dr

 mg dr cos    mgdy
yb
Wab    mgdy  mg ( ya  yb )
ya
b
x
Definition E p  mgy --PE of weight
then
Wab  E pa  E pb
the work done by GF =
the reduction of PE of weight
If
yb  0
0
then
Wa 0    mgdy  mgy a
ya
 E pa  mgya  Wa 0
PE of weight at point a = the work done by GF
moving m from a to zero PE point.
The point of zero PE of weight is arbitrary
(2) Elastic PE

F
Elastic force
F  kx
 
dW  F  dr
 kxdx
Wab  
xb
xa
0
a
m
xa xb
1 2 1 2
 kxdx  kxa  kxb
2
2
1 2
Definition E p  kx
2
b
--Elastic PE
x
then
Wab  E pa  E pb
the work done by EF =
the reduction of elastic PE
The point of zero elastic PE：
relaxed position of spring (x=0)
(3) Universal gravitational PE
Universal gravitational force

m1 m2
F  G 2 rˆ
r
 
dW  F  dr
GmM 
 2 dr cos(   )
r
GmM
  2 dr
r
b

dr
dr
  m
r  dr
rb

F

r
M
ra
a
dr
1 1
Wab  GmM  2  GmM (  )
ra r
ra rb
rb
mM
Definition E p  G
----UGPE
r
then
when
Wab  E pa  E pb
rb 
GmM
Wa 
 E pa
ra
The point of zero UGPE：
the distance of both particles is infinity( r 
Remarks
The PE of a particle at a point is relative and
the change of a particle from one point to
another point is absolute.
 Only conservative force can we introduce
potential energy.
 The done by conservative force =
the reduction of PE
Wco  
（E p 2  E p1 )   E p
 PE belongs to the system.
Gravitational force
Elastic force
Conservative
internal force
Universal gravitational force
The frictional force between bodies
is non-conservative internal force
§3-4 The work-energy theorem
Conservation of Mechanical Energy
System’s work-kinetic energy theorem
Wex  Win  Ekb  Eka
Internal force =Conservative IF+non-Cons.IF
Win  Wcoin  Anoin
Wcoin   E p  ( E pb  E pa )
Wex  Wnoin  ( Ekb  E pb )  ( Eka  E pa )
Let
E  Ek  E p
-- mechanical energy of the system
Wex  Wnoin  Eb  Ea
The sum of the work done by the external
forces and non-conservative forces equals to
the increment of the mechanical energy of
the system from initial state to final state.
-- the work-energy theorem of a system
when
Wex  Wnoin  0
We have
Eb  Ea  Const .
--Conservation of mechanical energy
[Example] Two boards with
mass of m1 ， m2 (m2>m1)
connect with a weightless spring.
 If the spring can pull m2 out
of the ground after the F is
removed, How much the F must
be exerted on m1 at lest? 
How is about the result if m1，
m2 change their position?
F
m1
m2
Solution
Suppose the length of the spring
is compressed x1 as the F is
exerted. And m2 is pulled out of
the ground as the length is just
stretched x after the F is
2
removed
then F  m g  kx
1
1
x2
x1
m2
m2 g  kx2
F  m1 g
 x1 
k
m2 g
x 2 
k
 Two boards+spring+earth = system
Its mechanical energy is conservation
Chose the point of zero PE :
The spring is free length ( no information)
1
1
2
2
k ( x1 )  m1 gx1  k ( x2 )  m1 gx2
2
2
We can get
F  ( m1  m2 ) g
 The result do not change if m1，m2 change
their position.
§3-5 The Conservation of Energy
Friction exists everywhere
The frictional force is called as a nonconservative 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 theorem
Wex  Wnoin  Eb  Ea
In order to simplify this problem, if we suppose
Wex=0
We have
Wnoin  Eb  Ea  0
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 Eint of the system has an
increment.
 Wnoin  Ea  Eb  Eint
Re-write above formula
Eint  ( Eb  Ea )  0
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.