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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

• 7.1 Conservative and

Nonconservative Forces

• 7.2 Potential Energy

• 7.3 Conservation of

Mechanical Energy

“A conservative force is a

force…that ‘gives back’ energy

that was transferred by doing

work.” – R.Wolfson

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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.

A

B

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 ∙ 𝑑 𝑟 = 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:

– 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.

B

ABA

U F dr

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Elastic Potential Energy

• Elastic potential energy stores the work done in

stretching or compressing springs or spring-like

systems:

– Elastic potential energy increases quadratically

with stretch or compression x.

– Here the zero of potential energy is taken in the

spring’s equilibrium configuration.

U 1

2kx2

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 stores the work

done against gravity:

– Gravitational potential energy increases linearly

with height y.

– This reflects the constant gravitational force near

Earth’s surface.

Gravitational Potential Energy

U mg y

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 + 𝑈1 = 𝐾2 + 𝑈2

K1 = 0

U1 = 10,000 J

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