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2015-07-23
PreClass Notes: Chapter 7, Sections 7.1-7.3
• From Essential University Physics 3rd Edition
• by Richard Wolfson, Middlebury College
• ©2016 by Pearson Education, Inc.
• Narration and extra little notes by Jason Harlow,
University of Toronto
• This video is meant for University of Toronto
students taking PHY131.
Outline
“A conservative force is a
force…that ‘gives back’ energy
that was transferred by doing
work.” – R.Wolfson
• 7.1 Conservative and
Nonconservative Forces
• 7.2 Potential Energy
• 7.3 Conservation of
Mechanical Energy
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Conservative and Nonconservative Forces
• A conservative force stores any work done against it, and
can “give back” the stored work as kinetic energy.
• For a conservative force, the work done in moving between
two points is independent of the path.
B
A
Conservative and Nonconservative Forces
• Because the work done by a
conservative force is path
independent, the work done in
going around any closed path is
zero:
𝐹 ∙ 𝑑𝑟 = 0
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Conservative and Nonconservative Forces
• A nonconservative force does not store work
done against it, the work done may depend on
path, and the work done going around a closed
path need not be zero.
Conservative and Nonconservative Forces
• Examples of conservative forces include
• Gravity
• The electric force
• The force of an ideal spring
• Nonconservative forces include
• Friction
• Pushing force of a human or animal
• Automobile engine
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Potential Energy
Stored energy held in readiness with
a potential for doing work
Examples:
• A stretched bow has stored energy
that can do work on an arrow.
• A stretched rubber band of a slingshot
has stored energy and is capable of
doing work.
Gravitational Potential Energy
Potential energy due to elevated position
Examples:
• coffee mug on the top
shelf
• water at the top of
Niagara Falls
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Potential Energy
Demo video
Potential Energy
 Consider two
particles A and B
that interact with
each other and
nothing else.
 There are two ways
to define a system.
 System 1 consists
only of the two
particles, the forces
are external, and the
work done by the
two forces change
the system’s kinetic
energy.
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Potential Energy
 System 2 includes
the interaction
within the system.
 Since Wext = 0, we
must define an
energy associated
with the interaction,
called the potential
energy, U.
 When internal
forces in the system
do work, this
changes the
potential energy.
Potential Energy
• The change in potential energy is defined as the
negative of the work done by a conservative force
acting over any path between two points:
B
U AB    F  dr
A
– Potential energy change is independent of path.
– Only changes in potential energy matter.
– We’re free to set the zero of potential energy at
any convenient point.
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Elastic Potential Energy
• Elastic potential energy stores the work done in
stretching or compressing springs or spring-like
systems:
2
U  12 kx
– Elastic potential energy increases quadratically
with stretch or compression x.
– Here the zero of potential energy is taken in the
spring’s equilibrium configuration.
Elastic Potential Energy
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Got it?
• A spring has a spring constant of 100 N/m. How
much potential energy does it store when
stretched by 10 cm?
A. 50 J
B. 10 J
C. 5 J
D. 0.5 J
E. 0.1 J
Potential Energy
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Gravitational Potential Energy
• Gravitational potential energy stores the work
done against gravity:
U  mg y
– Gravitational potential energy increases linearly
with height y.
– This reflects the constant gravitational force near
Earth’s surface.
Mechanical Energy
• Mechanical Energy is defined as the sum of the
kinetic plus potential energy:
Emech = K + U
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Conservation of
Mechanical Energy
K1 = 0
U1 = 10,000 J
𝐾1 + 𝑈1 = 𝐾2 + 𝑈2
K2 = 2,500 J
U2 = 7,500 J
K3 = 7,500 J
U3 = 2,500 J
K4 = 10,000 J
U4 = 0
Energy Bar Charts
Slide 10-34
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Conservation of Mechanical Energy
Got it?
• Can a system have negative potential energy?
A. No, because a negative potential energy is
unphysical.
B. No, because the kinetic energy of a system must be
equal to its potential energy.
C. Yes, as long as the total energy remains positive.
D. Yes, as long as the total energy remains negative.
E. Yes, because the choice of the zero of potential
energy is arbitrary.
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The Zero of Potential Energy
Problem Solving with Conservation of Energy
• Interpret the problem to make sure all forces are
conservative, so conservation of mechanical energy applies.
Identify the quantity you’re being asked to find, which may
be an energy or some related quantity.
• Draw the object in a situation where you can determine both
its kinetic and potential energy, then again in the situation
where one quantity is unknown. Set up the equation: E1 = E2
• Evaluate to solve for the unknown quantity, which might be
an energy, a spring stretch, a velocity, etc.
• Assess your solution to see that your answer makes sense,
has the right physical units, and is consistent with your
intuition.
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Example Problem
• Note that I have worked out Exercise 21 from Chapter 7.
• The 5-minute video is available at
https://youtu.be/lBN6yFf-2aU
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