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PreClass Notes: Chapter 13, Sections 13.3-

13.7

• 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

“Pushing a child on a swing, you

can build up a large amplitude by

giving a relatively small push once

each oscillation cycle. If your

pushing were not in step with the

swing’s natural oscillatory motion,

then the same force would have

little effect.”– R.Wolfson

• Simple Pendulum

• Circular motion and S.H.M.

• Energy in S.H.M.

• Damped Harmonic Motion

• Driven Oscillations and

Resonance.

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Simple Harmonic Motion

• Simple Harmonic Motion (S.H.M.) results

whenever the following equation applies:

• Double-time derivative of position = negative

constant × position

• If position is represented by x, then:

𝑑2𝑥

𝑑𝑡2= −𝜔2𝑥

• where 𝜔2 is a positive constant, and the angular

frequency of the oscillations is 𝜔.

• Almost every stable equilibrium will exhibit SHM

for small disturbances from equilibrium.

• Simple pendulum

• Point mass on massless

cord of length L.

• The tension force acts

directly toward the pivot, so

it provides no torque.

• The torque due to gravity

causes the angular

acceleration.

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Simple Pendulum

Simple Pendulum

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• What happens to the period of a pendulum if its

length is quadrupled?

A. The period is halved.

B. The period is doubled.

C. The period is quadrupled.

D. The period is quartered.

Got it?

• Simple harmonic motion can be viewed as one component of

uniform circular motion.

– Angular frequency in SHM is the same as angular

velocity in circular motion.

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Energy in Simple Harmonic Motion

Energy in Simple Harmonic Motion

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Energy in Simple Harmonic Motion

• In the absence of nonconservative forces, the energy of a

simple harmonic oscillator does not change.

– But energy is transfered back and forth between kinetic

and potential forms.

Energy in Simple Harmonic Motion

𝐸 = 𝐾max =12𝑚𝑣max

2

𝐸 = 𝑈max =12𝑘𝑥max

2=12𝑘𝐴2

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• If the total energy of a harmonic oscillator is reduced

by a factor of 3, the amplitude of the oscillations

A. increases by a factor of 3.

B. decreases by a factor of 3.

C. increases by a factor of 3.

D. decreases by a factor of 3.

E. remains unchanged.

Got it?

Simple Harmonic Motion is Everywhere!

• That’s because most systems near stable equilibrium have

potential-energy curves that are approximately parabolic.

– Ideal spring:

– Typical potential-energy curve of an arbitrary system: U 1

2kx2 1

2m 2x2

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Damped Harmonic Motion

• With nonconservative forces present, SHM gradually

damps out:

– Amplitude declines exponentially toward zero:

– For weak damping b, oscillations still occur at

approximately the undamped frequency

– With stronger damping, oscillations cease.

• Critical damping brings the system to

equilibrium most quickly.

2( ) cos( )

bt mx t Ae t

2

2

d x dxm kx b

dt dt

Damped Harmonic Motion2

( ) cos( )bt m

x t Ae t

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Damped Harmonic Motion

(a) underdamped

(b) critically damped, and

(c) overdamped oscillations.

Driven Oscillations

• When an external force acts on an oscillatory system, we

say that the system is undergoing driven oscillation.

• Suppose the driving force is F0cosωdt, where ωd is the

driving frequency, then Newton’s law is

• The solution is

where

and0

k

m is the natural frequency.

0

2 2 2 2 2 2

0

( )( ) /d d

FA

m b m

2

02cos d

d x dxm kx b F t

dt dt

( ) cos( )dx t A t

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Resonance

• When a system is driven by an external force at

near its natural frequency, it responds with large-

amplitude oscillations.

– This is the phenomenon of resonance.

– The size of the resonant response increases as

damping decreases.

– The width of the resonance curve (amplitude

versus driving frequency) also narrows with lower

damping.

Resonance

Resonance curves

for several damping

strengths; 0 is the

undamped natural

frequency k/m.

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Resonance

• Musical instruments are all based on the

phenomenon of resonance.

• A string of a particular length and tension will have

certain frequencies for which it resonates at large

amplitude and produces a certain frequency of

sound.

• A column of air of a certain length will have certain

resonance frequencies as well.