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Chapter 16: Oscillations

Brent Royuk Phys-111

Concordia University

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Periodic Motion •  Periodic Motion is any motion that repeats

itself. •  The Period (T) is the time it takes for one

complete cycle of motion. –  What is the period of rotation of the hour hand on a

clock? •  The Frequency is the number of cycles per

unit of time. –  The period of the reoccurrence of Monday is one

week. What is Monday’s frequency?

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f =1T

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Simple Harmonic Motion •  One particular type of periodic motion is SHM. •  Hooke’s Law

–  F = - kx •  A restoring force

–  A linear restoring force always produces SHM

• Vocabulary – Equilibrium position – Periodic Motion vs. SHM – Displacement (x), Amplitude (A), Period (T), Frequency (f) 4

SHM •  The Strip Chart sine (or cosine) curve

•  Equation of motion:

•  What happens at time (t + T)?

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x = A cos 2πT

t#

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SHM •  Generating the sine (or cosine) curve

Animation courtesy of Dr. Dan Russell, Kettering University

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SHM and Circular Motion •  Casting the Shadow

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Rewrite :

x = A cos 2πT

t#

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Use ω = 2πf ,sox = A cos ωt( )

Example: An oscillating mass on a spring has a period of 3.2 s and an amplitude of 2.4 cm. What is the equation of motion? When is the first time the mass is as x = -2.4 cm?

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Position, Velocity & Acceleration •  Given a position graph for SHM, what would

corresponding velocity and acceleration graphs look like?

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Position, Velocity & Acceleration

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x = A cos ωt( )v = −Aω sin ωt( )a = −Aω2 cos ωt( )

What is vmax?

amax?

Why?

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Graph Animation

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The Period of a Mass on a Spring

•  Are vertical springs different than horizontal springs?

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F = −kxma = −kx

m −Aω2 cos ωt( )[ ] = −k A cos ωt( )[ ]ω2 =

km

ω =km

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Energy of a Mass on a Spring

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K =12

mv2 =12

m −Aω sin(ωt)[ ]2=

12

mA2ω2 sin2(ωt)

U =12

kx2 =12

k A cos(ωt)[ ]2=

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kA2 cos2(ωt)

ω2 = k / m

K =12

kA2 sin2(ωt)

E = U + K

E =12

kA2 sin2(ωt) +12

kA2 cos2(ωt)

E =12

kA2

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Energy of a Mass on a Spring

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Example •  A 240-g object is attached to a

spring with k = 140 N/m and is compressed 12 cm and released. As it oscillates, what is its maximum speed? What is its speed and acceleration when it is at a point 6.0 cm to the left of its equilibrium position? What is its period of oscillation?

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The Pendulum

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Fx = −T sin θ = −TxL

∑Fy = T cos θ − mg∑

cos θ ≈ 0 for small angles, soT ≈ mg

∴ Fx∑ ≈ −mgx

L= max

ax = −gL

x = −ω 2x

ω =gL

T = 2π Lg

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The Pendulum

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= −T = −TxL

∑Fy = T cosθ −F = mg

∑

F = mg sinθsinθ ≈ θ for small angles, and

s = Lθ , or

θ = sL

F = mg sinθ ≈ mgθ =mgL

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)

* + + s.

Comparing to a mass on a spring,

T = 2π mmg

L

= 2π Lg

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The Pendulum •  So a pendulum exhibits SHM for small

angle oscillations. –  What about the mass of the pendulum?

•  What is the length of a simple pendulum with a period of exactly one second?

•  The Physical Pendulum

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T = 2π ImgL

= 2π Lg

ImL2

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Damped Oscillations

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Driven Oscillations •  To increase the amplitude of an

oscillation, you must add energy to the system. The timing is important. If you add energy (drive) the system at its natural (resonant) frequency, you can dramatically increase the amplitude of the oscillations.

•  Example: Pushing a child on a swing

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Driven Oscillations

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Waves •  What is a wave? •  Examples

–  Water, sound, slinky, ER •  Transverse vs. Longitudinal

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Wave Properties •  The magic of waves.

–  Great distances –  What are they made of?

•  Wave Anatomy –  Crest, trough, speed, frequency, wavelength, amplitude,

period. •  The Wave Equation: v = fλ

–  What is the wavelength of a sound wave produced by a violin playing the note A above middle C when the speed of sound is 350 m/s?

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The Four Wave Behaviors 1.  Reflection

–  Waves bounce off obstacles 2.  Refraction

–  Waves bend when entering a new medium at an angle.

3.  Diffraction –  Waves bend around corners and spread out

from small openings.

4.  Superposition (Interference) –  Waves pass through each other, and

their amplitudes add.

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Superposition

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Constructive vs. Destructive Interference

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Interference

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Standing Waves

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Standing Waves •  Nodes, antinodes, fundamental, harmonics

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Standing Waves

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fn = nf1λ1 = 2L

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Example

• A guitar string 60 cm long vibrates with a standing wave that has three antinodes. (a) Which harmonic is this? (b) What is the wavelength of this wave? (c) If this harmonic is excited with a frequency of 600 Hz, what is the frequency of the fundamental?

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Drum Modes Animations courtesy of Dr. Dan Russell, Kettering University

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Beats •  Turn signals analogy •  The beat frequency