Chapter 8 - Potential Energy and
Conservation of Energy
• Conservative vs. Non-conservative Forces
• Definition of Potential Energy
• Conservation Of Mechanical Energy
• Determining Potential Energy
– Gravitational – near the surface of the earth
– Gravitational – anywhere - escape velocity
– Elastic
• Determining the Force From Potential Energy Functions
• Work Done by Non-conservative Forces
• Power
Power
• Rate at which work is
done.
W
P
t
• Average Power
• Instantaneous Power
dW
P
F v
dt
Units
Physical
Quantity
Length
Dimension
Symbol
[L]
SI MKS
SI CGS
m
cm
US
Customary
ft
Mass
[M]
kg
g
slug
Time
[T]
sec
sec
sec
Acceleration
[L/T2]
m/s2
cm/s2
ft/s2
Dyne
g-cm/s2
pound (lb)
slug- ft/s2
Force
[M-L/T2] newton (N)
kg-m/s2
Energy [M-L2/T2]
Power [M-L2/T3]
Joule (J)
N-m
kg-m2/s2
Watt (W)
J/s
N-m/s
Erg
Ft-lb
Dyne-cm
2/s2
slug-ft
g-cm2/s2
HP =
550 ft-lb/s
Conservative vs. Non-conservative
• Conservative - A force is
said to be conservative if
the work done by the force
acting on a object moving
between two points “is
independent of the path”
the particle takes between
the points.
• Non-conservative “depends on the path”
dr
b
W   F  dr
a
Example: Gravity near the surface of the earth
F   mgjˆ
dr
dr  dxiˆ  dyjˆ  dzkˆ
b
W   F  dr
a
W  mg  y2  y1   mgh
Alternative definition
• A force is conservative
if the net work done
by the force on an
object moving around
any closed path is
zero.
Gravity is a conservative force!
A nonconservative force
Friction is a nonconservative force!
Potential Energy
• Energy associated with the position or
configuration of a system.
• The change in potential energy associated
with a particular conservative force is the
negative of the work done by that force.
U  U 2  U1   W
b
W   F  dr
a
Examples:
• Gravity
WG  mg  y2  y1 
U  mg  y2  y1 
U  mgy
• Springs
1 2
WS   kx
2
1 2
U  kx
2
1 2
U  kx
2
Differential form
One dimension:
2
U   F  dr
1
U    Fx dx
dU
Fx  
dx
Three dimensions:
U
Fx  
x
U
Fy  
y
U
Fz  
z
U ˆ U ˆ U ˆ
F
i
j
k  U
x
y
z
Potential Energy Summary
• Potential energy is only associated with
conservative forces. It is the negative of the work
done by the conservative force.
• The zero point of potential energy is arbitrary and
should be chosen where it is most convienient.
• Potential energy is not something a body has by
itself, but rather is associated with the interaction
of two or more objects.
Conservation of Mechanical Energy
W  K
W  U
Work-Energy Principle
Definition of Potential Energy
0  K  U
K1  U1  K 2  U2
E1  E 2
Problem solving strategy
Who is going faster at the bottom?
• Assume no friction
• Assume both have the same
speed pushing off at the top
Problem 1
• A Block of mass m is released from rest and
slides down a frictionless track of height h
above a table. At the bottom of the track,
where the surface is horizontal, the block
strikes and sticks to a light spring.
• Find the maximum distance the spring is
compressed.
• m = 2 kg, h = 1 m, k = 490 N/m
Problem 2
• A ball (mass m) on a
string (length L) is
released from rest with
the string horizontal.
What is the speed
when it reaches its
lowest point?
• What if the string was
not horizontal, instead
being released from
some angle q?
Energy conservation with dissipative forces
• Total energy is neither increased or
decreased in any process. Energy can be
transformed from one form to another, and
transferred from one body to another, but
the total amount remains constant.
W  WC  WNC  U  WNC  K
WNC  K  U
Example 3
• A roller coaster with mass of
1000 kg starts at a height of
40 m and is found to reach a
height of only 25 m before
coming to a stop. It traveled
a distance of 400 m.
Estimate the average friction
force.
• Is the friction force constant?
Problem 7
• A 2 kg block is attached to a light spring of force constant
500 N/m. The block is pulled 5 cm to the right and of
equilibrium. How much work is required to move the
block?
• If released from rest, find the speed of the block as it
passes back through the equilibrium position if
– the horizontal surface is frictionless.
– the coefficient of friction is 0.35.
Example
• A ball of mass 4.64 kg is taken to a position
3 moon radii above the surface of the moon
where it is dropped from rest. What is the
speed of the ball as it just starts to make
contact with the surface of the moon?
– Mm = 7.35 x 1022 kg
– Rm = 1.74 x 106 m
Gravitational potential energy again
Gm E m
F
rˆ
2
r
2
W   F  dl
1
W  U
GM E m
U r  
r
Escape velocity
GM E m
1
2
mvesc 
0
2
rE
vesc
2GM E

rE