Conservation of Energy

PHYC 160
Elastic and Gravitational
Potential Energy
Lecture #18
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
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y2
y1 = 0
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.
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Motion in a vertical circle with no friction
•
Follow Example 7.4 using Figure 7.9.
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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.
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Moving a crate on an inclined plane with friction
•
•
Follow Example 7.6
using Figure 7.11 to the
right.
Notice that mechanical
energy was lost due to
friction.
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Work done by a spring
•
Figure 7.13 below shows how a spring does work on a block as
it is stretched and compressed.
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Stretching a spring
• The force required to stretch a
spring a distance x is proportional
to x: Fx = kx.
• k is the force constant (or spring
constant) of the spring.
• The area under the graph
represents the work done on the
spring to stretch it a distance X:
W = 1/2 kX2.
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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.
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Situations with both gravitational and elastic forces
• When a situation involves both gravitational and elastic forces,
the total potential energy is the sum of the gravitational potential
energy and the elastic potential energy: U = Ugrav + Uel.
•
•
Figure 7.15 below illustrates such a situation.
Follow Problem-Solving Strategy 7.2.
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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.
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Conservative and nonconservative forces
• A conservative force allows conversion between kinetic and
potential energy. Gravity and the spring force are conservative.
• The work done between two points by any conservative force
a) can be expressed in terms of a potential energy function.
b) is reversible.
c) is independent of the path between the two points.
d) is zero if the starting and ending points are the same.
• A force (such as friction) that is not conservative is called a
nonconservative force, or a dissipative force.
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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.
Force and potential energy in one dimension
• In one dimension, a conservative
force can be obtained from its
potential energy function using
Fx(x) = –dU(x)/dx
• Figure 7.22 at the right illustrates
this point for spring and
gravitational forces.
• Follow Example 7.13 for an
electric force.
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Energy diagrams
• An energy diagram is a
graph that shows both the
potential-energy function
U(x) and the total
mechanical energy E.
• Figure 7.23 illustrates the
energy diagram for a glider
attached to a spring on an
air track.
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Force and a graph of its potential-energy function
• Figure 7.24 below helps relate a force to a graph of its
corresponding potential-energy function.
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