 # Gravitational and Elastic Potential Energy

```PHYC 160
Power, Gravitational
Potential Energy
Lecture #17
Gravitational Work
• Let’s examine the gravitational work done on
an object when it moves along some arbitrary
path:
y
final
WGravity
final
FGravity ds
initial
dxiˆ dyjˆ
initial
final
final
mgjˆ dxiˆ
initial
mgjˆ dyjˆ
initial
final
0 mg
x
mgjˆ
initial
mg y
dy ˆj ˆj
mg y final
yiniti
Work (by gravity) on a Pendulum
ds
ds
Mg
Mg
CPS Question 16-2
• How much work does the cable holding the bowling ball do
from the top of the arc to the bottom?
A) +mgh
FT
B) -mgh
C) 0
D) Not enough information to solve.
h
FT
ds
ds
Work-Energy Theorem
WTOTAL
WGravity WElastic WOther
KE
• Now, we have just broken the total work on an
object up into three terms, the first two being
from forces that we understand:
final
WGravity
FGravity ds
mg y f
FElastic ds
1 2
kx
2
yi
initial
final
WElastic
initial
Sign determined by the
dot product (+ when pushing
in same direction as motion,
- when opposite)
Power
• Power is the rate at which work is
done.
• Average power is Pav = ΔW/Δt and
instantaneous power is P = dW/dt.
• The SI unit of power is the watt (1
W = 1 J/s), but other familiar units
are the horsepower and the
kilowatt-hour.
Gravitational potential energy
• Energy associated with
position is called potential
energy.
• Gravitational potential
energy is Ugrav = mgy.
•
Figure 7.2 at the right shows
how the change in
gravitational potential
energy is related to the work
done by gravity.
Q7.1
A piece of fruit falls straight down. As it falls,
A. the gravitational force does positive work on it and the
gravitational potential energy increases.
B. the gravitational force does positive work on it and the
gravitational potential energy decreases.
C. the gravitational force does negative work on it and the
gravitational potential energy increases.
D. the gravitational force does negative work on it and the
gravitational potential energy decreases.
© 2012 Pearson Education, Inc.
The conservation of mechanical energy
•
•
•
The total mechanical energy of a system is the sum of its kinetic energy and
potential energy.
A quantity that always has the same value is called a conserved quantity.
When only the force of gravity does work on a system, the total mechanical
energy of that system is conserved. This is an example of the conservation of
mechanical energy. Figure 7.3 below illustrates this principle.
An example using energy conservation
•
Refer to Figure 7.4 below as you follow Example 7.1.
• Notice that the result does not depend on our choice for the
origin.
Q7.2
You toss a 0.150-kg baseball
straight upward so that it leaves
your hand moving at 20.0 m/s. The
ball reaches a maximum height y2.
What is the speed of the ball when
it is at a height of y2/2? Ignore air
resistance.
A. 10.0 m/s
v2 = 0
v1 = 20.0 m/s
m = 0.150 kg
B. less than 10.0 m/s but greater than zero
C. greater than 10.0 m/s
D. not enough information given to decide
© 2012 Pearson Education, Inc.
y2
y1 = 0
When forces other than gravity do work
•
Refer to ProblemSolving Strategy
7.1.
•
of Example 7.2.
Work and energy along a curved path
•
We can use the same
expression for
gravitational
potential energy
whether the body’s
path is curved or
straight.
Energy in projectile motion
•
Two identical balls leave from the same height with the
same speed but at different angles.
•
Follow Conceptual Example 7.3 using Figure 7.8.
Motion in a vertical circle with no friction
•
Follow Example 7.4 using Figure 7.9.
Q7.4
The two ramps shown are both frictionless. The heights y1 and y2 are
the same for each ramp. A block of mass m is released from rest at
the left-hand end of each ramp. Which block arrives at the right-hand
end with the greater speed?
A. the block on the curved track
B. the block on the straight track
C. Both blocks arrive at the right-hand end with the same speed.
D. The answer depends on the shape of the curved track.
© 2012 Pearson Education, Inc.
Moving a crate on an inclined plane with friction
•
•
using Figure 7.11 to the
right.
Notice that mechanical
energy was lost due to
friction.
Conservative and Non-conservative
• The nice thing about the conservation of mechanical
energy is that the change in the potentials only are
determined by the initial and final points of the path.
That’s because potentials always describe conservative
forces – forces where the work done by them in going
from one point to another is path independent.
Non- conservative forces
• An example of a non-conservative force is
friction. The work done by friction is definitely
dependent on the path.
• Let’s take the example of moving a book on a
table with kinetic friction:
Path 2
Path 1
• Since path 2 is longer, there will be more work
done by friction.
Work done by a spring
•
Figure 7.13 below shows how a spring does work on a block as
it is stretched and compressed.
Work from a Spring
• The same thing can be done with the work
from a spring:
final
final
WElastic
FElastic ds
initial
kxiˆ
dxiˆ
initial
final
kxiˆ dxiˆ
initial
final
k
initial
xdx iˆ iˆ
1
2
k x final
2
2
initial
x
Work from a Spring
• If we take the initial position to be the relaxed
position of the spring, then set xinitial = 0, we
have
1 2
WElastic
2
kx final
• And we can make the same definition of the
elastic potential energy:
U Elastic
WElastic
1 2
kx
2
• Where, again, this is the energy stored in the
spring-block system.
Elastic potential energy
• A body is elastic if it returns
to its original shape after
being deformed.
• Elastic potential energy is the
energy stored in an elastic
body, such as a spring.
• The elastic potential energy
stored in an ideal spring is Uel
= 1/2 kx2.
• Figure 7.14 at the right
shows a graph of the elastic
potential energy for an ideal
spring.
Q7.5
A block is released from rest on a
frictionless incline as shown. When the
moving block is in contact with the spring
and compressing it, what is happening to
the gravitational potential energy Ugrav and
the elastic potential energy Uel?
A. Ugrav and Uel are both increasing.
B. Ugrav and Uel are both decreasing.
C. Ugrav is increasing; Uel is decreasing.
D. Ugrav is decreasing; Uel is increasing.
E. The answer depends on how the block’s speed is changing.
© 2012 Pearson Education, Inc.
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