Potential Energy and Conservation of
Mechanical Energy
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Conservative and Nonconservative Forces
•
Potential Energy
•
Conservation of Mechanical Energy
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Homework
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First Definition of Conservative and
Nonconservative Forces
•
•
•
A force is conservative if the kinetic energy of a particle on which it acts returns to its initial value after any
round trip. A force is nonconservative if the kinetic
energy changes.
An example of a conservative force is the force exerted by a spring
An example of a nonconservative force is friction
2
Second Definition of Conservative and
Nonconservative Forces
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•
A force is conservative if the work done by the force
on a particle that moves through any round trip is
zero. A force is nonconservative if the work done by
the force on a particle that moves through any round
trip is not zero.
First and second definitions are equivalent from the
Work-Energy Theorem (W = ∆K)
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Third Definition of Conservative and
Nonconservative Forces
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A force is conservative if the work done by it on a
particle that moves between two points depends only
on these points and not on the path followed. A force
is nonconservative if the work done by it on a particle
that moves between two points depends on the path
taken between these two points.
The work done against friction depends on the path
taken since it is a nonconservative force
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Potential Energy
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When conservative forces are acting (e.g. spring force,
gravity)
∆K + ∆U = 0
where U = potential energy (has meaning only for
conservative forces)
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This means that the sum of the kinetic and potential
energies is a constant
U + K = a constant
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From the Work-Energy Theorem
W = ∆K = −∆U
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For one dimensional motion
∆U = −W = −
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Z
xf
xi
F (x)dx
Conservation of Mechanical Energy
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Since the sum of the kinetic and potential energies is
a constant when only conservative forces are acting,
we can write
Ki + U i = Kf + U f
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We can also calculate the force from the potential energy function
F (x) = −
•
dU (x)
dx
Check
dU (x) 
xf
xf 

∆U = − xi F (x)dx = − xi −
 dx
dx
Z
∆U =
Z
xf
xi
Z


dU (x)dx = U (xf ) − U (xi) = Uf − Ui
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Conservation of Mechanical Energy
(Cont’d)
Consider a particle moving from A to B along the x-axis
with a single conservative force acting on it
∆U = UB − UA
UB = ∆U + UA = −
Z
xB
xA
F (x) dx + UA
We cannot assign a value to UB until we assign one to
UA. Usually we choose UA = 0.
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Gravitational Potential Energy
Fg = −mg
Ug (y) = −
Z
y
0
Fg dy + Ug (0) = −
Z
y
0
(−mg) dy + Ug (0)
Ug (y) = mgy + Ug (0)
Let Ug (0) = 0
N ote :
⇒
Ug (y) = mgy
d (mgy)
dUg (y)
=−
= −mg
Fg = −
dy
dy
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Potential Energy of a Spring
Fs(x) = −kx
Us(x) = −
Z
x
0 Fs (x)dx
+ Us(0) = −
Z
x
0 (−kx) dx
+ Us(0)
1
Us(x) = kx2 + Us(0)
2
Let Us(0) = 0
N ote;
⇒
1
Us(x) = kx2
2
d 1
dUs(x)
= −  kx2 = −kx
Fs(x) = −
dx
dx 2

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
Example 1
What is the change in gravitational potential energy when
a 720-kg elevator moves from street level to the top of the
Empire State building, 380 m above street level?
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Example 1 Solution
What is the change in gravitational potential energy when
a 720-kg elevator moves from street level to the top of the
Empire State building, 380 m above street level?
∆U = Uf − Ui = mgyf − mgyi = mg(yf − yi)
2
∆U = (720 kg) 9.8 m/s (380 m) = 2.7 M J
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Example 2
The spring in a spring gun has a force constant k = 700
N/m. It is compressed 3.0 cm from its natural length, and
a 0.012-kg ball is put in the barrel against it. Assuming
no friction and a horizontal gun barrel, with what speed
will the ball leave the gun when released?
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Example 2 Solution
The spring in a spring gun has a force constant k = 700
N/m. It is compressed 3.0 cm from its natural length, and
a 0.012-kg ball is put in the barrel against it. Assuming
no friction and a horizontal gun barrel, with what speed
will the ball leave the gun when released?
Ki + U i = Kf + U f
1
1
0 + kx2 = mv 2 + 0
2
2
v
u
u
u
u
t
v
u
u
u
u
u
t
k
700 N/m
v=
x=
(0.03 m) = 7.25 m/s
m
0.012 kg
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Homework 12 - Due Fri. Oct. 8
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Read Sections 7.1-7.2
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Answer Questions 7.1 & 7.3
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Do Problems 7.2, 7.5, & 7.9
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