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Chapter 13Vibrations and Waves

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Outline1. Hookes Law

2. Elastic Potential Energy

3. Comparing Simple Harmonic Motion

with Uniform Circular Motion.

4. Position, Velocity, and Acceleration as a

Function of Time.

5. Motion of a Pendulum

6. Damped Oscillations

7. Waves

8. Frequency, Amplitude and Wavelength

9. The Speed of Waves on Strings10. Interference of Waves

11. Reflection of Waves

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1-Hookes Law

Restoring Force:

Fis always opposite to

displacement (themass is always beingpulled or pushedtowards the

equilibrium position) Fis alwaysproportional todisplacement

kxF

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SimpleHarmonicMotion

1. Amplitude: Xmax(maximum value for x).

2. Period: Time to execute a cycle. (s)

3. Frequency: Number of cycles per unit of

time (Hz, s-1)

T

F1

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2- Elastic Potential

Energy

2

2

1)( kxPE elastic

mNinconstantspringk

minntdisplacemex

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Conservative system:

Non conservative system:

0)Wpath,closedanyfor(

fi PEKEPEKE

0Wpath,closedanyfor(

forcesveconservatinonbydoneWork

fi

nc

PEKEPEKE

W

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...)()( GravElastic PEPEPE

2

2

1

mvKE

energymechanicalTotalPEKE

Note:

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Concept Test

An object can oscillate around:1. Any equilibrium point

2. Any stable equilibrium point

3. Any unstable equilibrium points

4. Any point, provided the forces exerted on

it obey Hookes law

5. Any point

2- When an object in a stable

equilibrium is disturbed, it tends

to return to its stable equilibriumpoint.

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Velocity as a Function of Position

if EE

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)( 22 xAm

kv

0whenmax xvv

Axv when0

directionindicates

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Concept TestA mass suspended from a spring is oscillating up and

down as indicated.

Consider two possibilities:

(i) At some point during the oscillation, the mass

has zero velocity but is accelerating (+ or -)

(ii) At some point during the oscillation, the mass

has zero velocity and zero acceleration.

1. Both occur during the

oscillation2. Neither occurs during the

oscillation

3. Only (i) occurs

4. Only (ii) occurs

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3 Comparing Simple Harmonic

Motionwith Uniform Circular

Motion.

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0

sinv

v

A

xA 22sin

A

xA

v

v 22

0

22220 xACxA

A

vv

The velocity in the x-direction is related to the displacement x

in exactly the same way as the velocity of an object undergoing

simple harmonic motion.

The shadow moves with simple harmonic motion

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Period

T

Av

20

0

2

v

AT

PQ

At P:2

2

1kAEtotal

At Q:2

02

1mvEtotal

2

0

2

2

1

2

1

mvkA

k

m

v

A

0 k

mT 2

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Frequency

PQ

Tf

1

mkf

21

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Angular frequency

PQ

m

kf 2

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For a simple harmonic motion

)( 22 xA

m

kv

k

m

T 2

m

kf

2

1

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4-Position, Velocity, and Acceleration

as a Function of Time.

cstfrequencyAngular22 fT

t

Ax

cos

)2cos(cos f tAtAx

)sin( tAdtdxv

)cos(

2

2

2

tAdt

dx

a

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C t T t

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A mass attached to a spring oscillates back and forth asindicated in the position vs. time plot below. At P,

the mass has:

1. positive v, positive a

2. positive v, negative a

3. positive v, zero a

4. negative v, positive a

Concept Test

5. negative v, negative a

6. negative v, zero a

7. Zero v, but accelerating

8. Zero v and zero a

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5- Motionof aPendulum

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sin-mgF

radLs but sL

mgF

thatso

L

mgkkxF withmotionharmonicSimple

g

LT

k

mT 22Period

The period is

independent of mass

and amplitude,

rad-mgF For small angles

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Concept TestA person swings on a swing. When the person

sits still, the swing oscillates back andforth at its natural frequency. If instead theperson stands on the swing, the newnatural frequency of the swing is:

1. Greater

2. The same

3. SmallerL is decreased

T decreases

f increases

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6- Damped

Oscillations

Ideal systems:

indefiniteoscillations

Real systems:

frictionreduces the

mechanical energyand oscillation stop.

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7-Waves

A wave is a travelingdisturbance:

Source of

disturbance Medium of

propagation

Physical connection

between the portionsof the medium

Note: no flow of mattertakes place

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A wave can be:

Transverse: displacement is perpendicular to propagation.(guitar)

Longitudinal: displacement along propagation. (sound

waves)

Types ofWaves

8 F A lit d

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8- Frequency, Amplitude

and Wavelength

Any point P oscillates vertically with simple harmonicmotion having an amplitude A.

hWavelengtcrestcrest tofromDistance

fTv

waveaofVelocity

Period)1

(

frequency)(

fT

f

9 Th S d f W

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9- The Speed of Waves on

Strings

mKg

indensityLinearlength

mass

Tensionv

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Concept Test

Two strings, one thick and the other thin areconnected to form one long string. A wave

travels along the string and passes the point

where the two strings are connected. Which of

the following changes will we observe?

1. Frequency

2. Period

3. Propagation speed

h??wavelengttohappensBut what

changes.vandchangesthus

other,thetostringonefromchangesMass

length

mass

changesv3

Tensionv

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10 Interference of Waves

Resultant wave = addition of the two waves.

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Waves can pass

through eachother without

being destroyed.

11 R fl ti f W

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11- Reflection of Waves

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Concept Test

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Concept TestA wave is sent along a long spring by moving the left end rapidly to the

right and keeping it there. The figure shows the wave pulse at QR- part

RS of the long spring is as yet undisturbed. Which of the graphs 1 to 5

correctly shows the relation between the displacement s and the position

x ( displacement to the right is positive)?

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Concept Test

Which of the following is necessary to make

an object oscillate?

1. A stable equilibrium point

2. Little or no friction

3. A disturbance

4. Both 1 and 25. All 1, 2 and 3

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Concept Test

An object is in equilibrium when the net force

on it is zero.

Which one of the statements is / are correct?

1. Any object in equilibrium is at rest.2. Any object in equilibrium need not be at

rest

3. An object at rest must be in equilibrium.

2- An object in equilibrium:

- no net force,

- no acceleration

- The object is at rest or moving at a

constant velocity.