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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.