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Chapter 8
Potential Energy and Conservation of Energy
In this chapter we will introduce the following concepts:
Potential Energy
Conservative and nonconservative forces
Mechanical Energy
Conservation of Mechanical Energy
The
conservation of energy theorem
will be used to solve a variety of
problems
As was done in Chapter 7 we use scalars such as work, kinetic
energy, and mechanical energy rather than vectors.
Therefore the
approach is mathematically simpler.
(81)
A
B
g
v
o
h
v
o
Work and Potential Energy:
A tomato of mass
m
is taken together with the earth as the
system we study.
The tomato is thrown upwards with initial speed
v
o
at
point A. Under the gravitational force, it slows down and
stops at point B.
Then it falls back to point A and reaches
the original speed
v
o
.
During the trip from A to B the gravitational force
F
g
does work
U
W
∆
= 
(82)
A
A
B
B
k
m
k
F
S
F
S
v
o
v
o
m
Consider our system with the mass
m
attached to a spring
of spring constant
k
. The mass has an initial speed
v
o
at
point A.
Under the spring force
F
S
m
slows down and stops at
point B (with a spring compression
x
).
Then the mass
reverses the direction of its motion and reaches point A.
Its speed reaches the original value
v
o
.
During the trip from A to B the spring force
F
s
does work
U
W
∆
= 
(83)
m
m
A
B
v
o
f
f
x
d
Conservative and nonconservative forces.
The gravitational force and the spring force are
called
“
conservative
”
because they can
transfer energy between kinetic energy and
potential energy with no loss.
Frictional and drag forces are called
“
nonconservative
”
for reasons that are
explained below.
Consider a block of mass
m
on the horizontal floor
.
The block starts to move
with initial speed
v
o
at point A. The coefficient of kinetic friction between the
floor and the block is
μ
k
.
The block will slow down and stop at point B after traveling a distance
d
.
During the trip from A to B the frictional force does work
(84)
Closedloop test
: A force is conservative if the net work done on a particle
during a round trip is always equal to zero (see fig.b).
In the examples of the tomatoearth and massspring system
W
net
= W
a1b
+
W
b2a
= 0 (because
)
So gravitational force and spring force are conservative forces!
Next, we prove the
path independence
of work done by a conservative force
Path Independence of Conservative Forces
How to decide whether a force is conservative or
nonconservative?
0
net
W
=
(85)
Thus, the work done by a conservative force does
not
depend on the path
taken but only on the initial and final points
choose a simple path
1
2
,
∆
= 
→
=

=

a b
a
b
b a
b
a
U
W
W
U
U W
U
U
1
2
=
=

a b
a b
a
b
W
W
U
U
Pathindependence of work done by
conservative forces
Work done by
conservative forces
(
W = 
∆
U
) is pathindependent, it results in
changes in the object's potential energy.
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