Essential University Physics Oscillatory Motion

Chapter 13 Lecture

Slide 13-1

Essential University Physics Richard Wolfson

2nd Edition

© 2012 Pearson Education, Inc.

Oscillatory Motion

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In this lecture you’ll learn

• To describe the conditions

under which oscillatory

motion occurs

• To describe oscillatory motion

quantitatively in terms of

frequency, period, and

amplitude

• To explain simple harmonic

motion and why it occurs

universally in both natural and

technological systems

• To explain damped harmonic

motion and resonance

Slide 13-3 © 2012 Pearson Education, Inc.

Oscillatory Motion

• A system in oscillatory motion undergoes repeating,

periodic motion about a point of stable equilibrium.

• Oscillatory motion is characterized by – Its frequency f or period T=1/f.

– Its amplitude A, or maximum excursion from equilibrium.

• Shown here are position-versus-time graphs for two different oscillatory motions with the same period and amplitude.

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– The paradigm example is a mass

m on a spring of spring constant k.

• The angular frequency

= 2p for this system is

– In SHM, displacement is a

sinusoidal function of time:

– Any amplitude A is possible.

• In SHM, frequency doesn’t depend on amplitude.

Simple Harmonic Motion

• Simple harmonic motion (SHM) results when the force

or torque that tends to restore equilibrium is directly

proportional to the displacement from equilibrium.

w = k m

( ) cosx t A t

2

2

d xm kx

dt

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Quantities in Simple Harmonic Motion

• Angular frequency, frequency, period:

• Phase

– Describes the starting time of the displacement-versus-

time curve in oscillatory motion:

x(t) = Acos(t + )

w =

k

mf =

w

2p=

1

2p

k

mT =

1

f= 2p

m

k

Slide 13-6 © 2012 Pearson Education, Inc.

Velocity and Acceleration in SHM

• Velocity is the rate of

change of position:

• Acceleration is the rate of

change of velocity:

( ) sindx

v t A tdt

2( ) cosdv

a t A tdt

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Other Simple Harmonic Oscillators

• The torsional oscillator

– A fiber with torsional constant

provides a restoring torque.

– Frequency depends on and

rotational inertia:

w = k I

• Simple pendulum

– Point mass on massless cord of

length L:

w = g L

2

2sin

dI mgL mgL

dt

2I mL for point mass

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SHM and Circular Motion

• 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.

As the position vector traces out a

circle, its x– and y– components are

sinusoidal functions of time.

r

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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.

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Simple Harmonic Motion is Ubiquitous!

• 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

2mw 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.

w = k m.

2( ) cos( )

bt mx t Ae t

(a) Underdamped, (b) critically damped,

and (c) overdamped oscillations.

2

2

d x dxm kx b

dt dt

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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 F0cosdt, where d is the

driving frequency, then Newton’s law is

• The solution is

where

0

k

m : 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 curves for several

damping strengths; 0 is the

undamped natural frequency k/m.

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• Oscillatory motion is periodic motion that results from a

force or torque that tends to restore a system to

equilibrium.

• In simple harmonic motion (SHM), the restoring force or

torque is directly proportional to displacement.

– The mass-spring system is the paradigm simple harmonic

oscillator.

– Damped harmonic motion occurs when nonconservative forces act on the oscillating system.

– Resonance is a high-amplitude oscillatory response of a system driven at near its natural oscillation frequency.

Summary

w = k m

( ) cosx t A t