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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.