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Chapter 12 Potential Energy
12-1 Conservative forces
The spring force, gravity and universal gravitation are
three examples of conservative force.
Definition :
c
c
b
a
d
 
 
 
 F  ds   F  ds   F  ds 
acbda
acb
 
  F  ds 
acb
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 
 F  ds
bda
a
b
d
 
 
 F  ds   F  ds  0
acb
adb
adb
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Chapter 12 Potential Energy
Frictional force is not a conservative force on
mechanical energy scope
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Chapter 12 Potential Energy
12-2 Potential energy
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Chapter 12 Potential Energy
Definition:
Define the change in potential energy associated with
the work by the conservative force as:
U  U f  U i  W
or W   U
Potential energy is a state quantity, and is represented by
the symbol U.
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Chapter 12 Potential Energy
For a particle in one dimensional motion acted on by a
conservative force F(x), it moves from the initial
coordinate x0 to the final coordinate x.
U  U ( x )  U ( x0 )  W
x
x0
x0
x
   F ( x )dx   F ( x )dx
x0
U ( x )  U ( x0 )   F ( x )dx
x
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Chapter 12 Potential Energy
Here U(x0) is potential energy of an arbitrary reference
point.
x0
U 1 ( x1 )   F ( x )dx  U ( x0 )
x1
x0
U 2 ( x 2 )   F ( x )dx  U ( x0 )
x2
x1
U 2 ( x 2 )  U 1 ( x1 )   F ( x )dx
x2
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Chapter 12 Potential Energy
1. The spring force :
F(x)=-kx
If we choose the reference
position x0 (x0 =0), and
assumed U(x0)=0.
1 2
U ( x )    kxdx  kx
x
2
0
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Chapter 12 Potential Energy
If we choose the reference position x0 of the block in the
spring being a distance x0 and declare U(x0)=0.
U ( x )  U ( x0 )  U ( x )  
x0
x
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1 2 1 2
 kxdx  kx  kx0
2
2
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Chapter 12 Potential Energy
2. The force of gravity:
F(y)=-mg
If we choose the
reference point y0 =0,
we define U(y0)=0.
0
U ( y )    mg dy  mgy
y
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Chapter 12 Potential Energy
If we choose the reference point y0 and define U(y0)=0,
y0
U ( y )  U ( y0 )  U ( y )    mg dy  mgy  mgy 0
y
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Chapter 12 Potential Energy
 The gravitational force
U (r )  

r
Mm
Mm
 G 2 dr  G
r
r
The relation between force and potential energy:
dU ( x )
F ( x)  
dx
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Chapter 12 Potential Energy
12-3 Conservation of mechanical energy
Consider an isolated system:
1. There are not external forces acted on it, or if there are
such forces, they do not work on the system.
2. There are internal forces, it exerts on one another in
system.
3. The internal forces are conservational forces.
Assume the system consists of two bodies, for example,
the block-spring, or the ball-earth.
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Chapter 12 Potential Energy
W=- ΔU, W= Δ K, (according to the work-energy theorem)
ΔK=- ΔU, Δ(K+U)=0, we obtain K+U=E=constant. E is
called mechanical energy of the conservative system.
This is the mathematical representation of law of
conservation of mechanical energy.
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Chapter 12 Potential Energy
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Chapter 12 Potential Energy
1
1
2
2
mv  U ( x )  mv0  U ( x0 )  E
2
2
Here we need not analyze the force or write down
Newton’s laws. Instead we look for something in
the motion that is constant: here the mechanical
energy is constant.
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Chapter 12 Potential Energy
Isolated conservative system consisting of many particles
Δ K total  Δ U total Δ K total  Δ U total  0
Δ ( K total  U total )  Δ Etotal  0
 Etotal  constant
or Ei  E f
This is the law of conservation of mechanical energy
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Chapter 12 Potential Energy
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Chapter 12 Potential Energy
12-4 Energy conservation in rotational motion
1. Rolling without slipping:
From König theorem,
N
1
K   mn vn2
n 1 2
1
1
1
1
2
2
2
 I cm  Mvcm  I cm  M 2 R 2
2
2
2
2
Here only one parameter is sufficient to determine the K
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Chapter 12 Potential Energy
Example: Find K of the rolling
cylinder when the center of mass
move down for height h from at
the rest.
Solution:
acm
2
vcm
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y
N
f
x
mg
θ
2
 g sin ;
3
4 gh
v cm 2 4 gh
h
2
 2acm

;  (
) 
2
sin
3
R
3R
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Chapter 12 Potential Energy
1
1
1
1
2
2
2
K  I cm  Mvcm  I cm  M 2 R 2
2
2
2
2
1 1
1 4 gh
2 4 gh
 ( mR )
 m
 mgh
2
2 2
3R
2
3
We can see that the mechanical energy is constant. It
means the frictional force does no work, although the
frictional force is needed.
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Chapter 12 Potential Energy
2. Rolling friction
An ideal wheel can roll without slipping on a horizontal
surface at constant translational and rotational velocity.
ω
vcm
f
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If the frictional force done work on
the wheel, then the vcm will decrease,
However the ω will increase, which
is not the case.
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Chapter 12 Potential Energy
Why will the rolling wheel decrease its velocity until the
rest? It is because of rolling friction.
N Ny
Nx
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The drag force N is not through the
center and has two components. Nx is
decrease vcm and Ny is decrease ω.
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Chapter 12 Potential Energy
12-5 One-dimensional conservation systems: the
complete solution
1. The potential energy curve and force curve:
1
U ( x )  mv 2  E
2
1
mv 2  E  U ( x )  0
2
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Chapter 12 Potential Energy
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Chapter 12 Potential Energy
Example: The potential energy function for the force
between two atoms in a diatomic molecule can be
expressed approximately as follows:
a
b
U ( x)  12  6 ; a and b are positive constants.
x
x
Find (1) the equilibrium separation between the atoms. (2)
the force between the atoms. (3) the minimum energy
necessary to break the molecule apart.
Solution: (1) The equilibrium occurs at the coordinate xm,
where U(x) is a minimum.
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Chapter 12 Potential Energy
dU
 12a 6b
2a 1 6
(
) x  xm  13  7  0; xm  ( )
dt
xm
xm
b
dU
d a
b
12a 6b
(2) Fx  
  ( 12  6 )  13  7 .
dt
dt x
x
x
x
(3) Dissociation energy is Ed, which is the difference of
potential energy between the equilibrium and x=∞.
U ( x m )  Ed  0,
12a 6b b 2
Ed  U ( x m )   12  6 
x m x m 4a
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Chapter 12 Potential Energy
2. General solution for x(t)
Now we consider motion of a particle of mass m moving in
one dimension and acted on by a spring of force constant k
1
2
E  mv  U ( x )
2
dx
2
2 1 2 1 2
v
[ E  U ( x )]  
( kx 0  kx )
dt
m
m 2
2
k
x(t )  x 0 cos
t
m
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Chapter 12 Potential Energy
12-6 Three-dimensional conservative systems
A particle moves along a path from i to f. A conservative
force F acts on the particle.
Δ U  U ( x f , y f , z f )  U ( x i , y i , zi )
f 

   F  dr    ( FX dx  Fy dy  Fzdz)   W
i

U ˆ U ˆ U ˆ
F ( x , y , z)  
i
j
k  U
x
y
z
F is the negative gradient of the potential energy U(x,y,z).
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Chapter 12 Potential Energy
Homework:
Exercises
11
23
Problems
13
14
16
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