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11/15/2016
1
Waves and Oscillations Topic 4 Test Monday 11/21
Simple Harmonic Motion
Any vibrating system where the restoring
force is proportional to the negative of
the displacement is in simple harmonic
motion (SHM), and is often called a
simple harmonic oscillator.
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If an object vibrates or oscillates back and forth over the same path, each cycle taking the same amount of time, the motion is called periodic. The mass and spring system is a useful model for a periodic system.
•Period is the time required to complete one cycle
• Displacement is measured from the equilibrium point
•Amplitude is the maximum displacement
•A cycle is a full to-and-fro motion; this figure shows half a cycle
•Frequency is the number of cycles completed per second
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We assume that the surface is frictionless. There is a point where the spring is neither stretched nor compressed; this is the equilibrium position. We measure displacement from that point (x = 0 on the previous figure).
The force exerted by the spring depends on the displacement:
Example The spring constant of the spring is 320 N/m and the bar indicator extends 2.0 cm. What force does the air in the tire apply to the spring?
(320 / )( 0.02 )
6.4
F kx
F N m m
F N
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If the spring is hung vertically, the only change is in the equilibrium position, which is at the point where the spring force equals the gravitational force.
A spring has a length of 15.4 cm and is hanging vertically from a support point above. A weight of 0.200 kg is then attached to the spring, causing it to extend to a length of 28.6 cm. What is the value of the spring constant? How much force is then needed to lift this weight 4.6 cm from that position?
2
.286 .154 0.132
(0.200 )(10 / ) 2.0
2.0 (.132)
15.2 /
(15.2 / )(.046 )
0.7
x m
F kg m s N
N k
k N m
F N m m
F N
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5
(.60)(20 ) 12
12 (50 / )
.24 24
244.8sec
5.0 /
s N
s
f F N N
f kx
N N m x
x m cm
cmtime
cm s
We already know that the potential energy of a spring is given by:
The total mechanical energy is then:
The total mechanical energy will be conserved, as we are assuming the system is frictionless.
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If the mass is at the limits of its motion, the energy is all potential.
If the mass is at the equilibrium point, the energy is all kinetic.
We know what the potential energy is at the turning points:
The total energy is, therefore
And we can write:
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2 2
2 2
22
2 2
1 1
2 2
(10 / )(.1 )
(.01 )
3.2 /
kx mv
kx mv
kxv
m
kx N m mv
m kg
v m s
The Period and Sinusoidal Nature of SHM
Therefore, we can use the period and frequency
of a particle moving in a circle to find the period
and frequency:
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The Period and Sinusoidal Nature of SHM
max
2
max
2 f
v A
a A
The frequency of motion is 1.0 KHz and the amplitude is 0.20 mm.
(a)What is the maximum speed of the diaphragm? (b)Where in the motion does this maximum speed occur?
max
max
max
) 2
(2 )
(.0002 )(6.28)(1000 )
1.3 /
0
a f
v A A f
v m Hz
v m s
occurs
at
x
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The displacement of an object is described by the following equation, where x is in meters and t is in seconds:
x = (0.30m) cos (8.0 t) Determine the oscillating object’s (a) amplitude, (b) frequency, (c) period, (d) max speed, and (e) max acceleration
max
2 2
max
) 0.30
)8 6.28
1.3
1) .77sec
) (.3)(8) 2.4 /
) (.3)(8 ) 19.2 /
a amp m
b f
f Hz
c Tf
d v m s
e a m s
The Simple Pendulum
A simple pendulum consists of a mass at the end of a lightweight cord. We assume that the cord does not stretch, and that its mass is negligible.
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The Simple Pendulum
In order to be in SHM, the restoring force must be proportional to the negative of the displacement. Here we have:
which is proportional to sin θ and not to θ itself.
The period and frequency are:
Determine the length of a simple pendulum that will swing back and forth in simple harmonic motion with a period of 1.00 s.
2
1 6.2810
.159210
.0253610
0.254
LT
g
L
L
L
L m
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Assignment Do pg. 316-318
Questions #2,5
Problems #1,3,5, 9,16,24,28,30,31
Waves and Oscillations Topic 4 Test Monday 11/23
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Damped Harmonic Motion
Damped harmonic motion is harmonic motion with a frictional or drag force. If the damping is small, we can treat it as an “envelope” that modifies the undamped oscillation.
In simple harmonic motion, an object oscillated with a constant amplitude.
In reality, friction or some other energy dissipating mechanism is always present and the amplitude decreases as time passes. This is referred to as damped harmonic motion.
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Damped Harmonic Motion
However, if the damping is large, it no longer resembles SHM at all.
A: underdamping: there are a few small oscillations before the oscillator comes to rest. B: critical damping: this is the fastest way to get to equilibrium. C: overdamping: the system is slowed so much that it takes a long time to get to equilibrium.
1) simple harmonic motion
2&3) underdamped 4) critically damped
5) overdamped
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Forced Vibrations; Resonance Forced vibrations occur when there is a periodic driving force. This force may or may not have the same period as the natural frequency of the system.
RESONANCE Resonance is the condition in which a time-dependent force can transmit large amounts of energy to an oscillating object, leading to a large amplitude motion.
Resonance occurs when the frequency of the force matches a natural frequency at which the object will oscillate.
Forced Vibrations; Resonance
The sharpness of the resonant peak depends on the damping. If the damping is small (A), it can be quite sharp; if the damping is larger (B), it is less sharp.
Like damping, resonance can be wanted or unwanted. Musical instruments and TV/radio receivers depend on it.
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Waves and Oscillations Topic 4 Test Monday 11/23
A wave is a traveling disturbance. A wave carries energy from place to place.
Wave Motion
Study of a single wave pulse shows that it is begun with a vibration and transmitted through internal forces in the medium.
Continuous waves start with vibrations too. If the vibration is SHM, then the wave will be sinusoidal.
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Types of Waves: Transverse and Longitudinal
or parallel to it (longitudinal).
The motion of particles in a wave can either be perpendicular to the wave direction (transverse)
Sound waves are longitudinal waves:
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Water waves are partially transverse and partially longitudinal.
Wave Motion
Wave characteristics:
• Amplitude, A
• Wavelength, λ
• Frequency f and period T
fT
v
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A wave is traveling with a frequency of 200Hz. What speed is the wave traveling at if the wavelength is 20cm?
(.2 )(200 )
40 /
v f
v m Hz
v m s
AM and FM radio waves are transverse waves consisting of electric and magnetic field disturbances traveling at a speed of 3.00x108m/s. A station broadcasts AM radio waves whose frequency is 1230x103Hz and an FM radio wave whose frequency is 91.9x106Hz. Find the distance between adjacent crests in each wave (the wavelength).
8
3
8
6
( )
3.00 10243.9
1230 10
3.00 103.26
91.9 10
AM v f
v xm
f x
FM
v xm
f x
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The speed at which the particles on a wave move to the right depends on how quickly one particle of the string is accelerated upward in response to the net pulling force.
Lm
Fv
tension
length
mass
A 2.0m long string is under 20N of tension. A pulse travels the length of the string in 50ms. What is the mass of the string?
2.040 /
0.05
/
2040
/ 2
401600
400.025
1600
mv m s
s
Fv
m L
m
m
m kg
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Waves Traveling on Guitar Strings Transverse waves travel on each string of an electric guitar after the string is plucked. The length of each string between its two fixed ends is 0.628 m, and the mass is 0.208 g for the highest pitched E string and 3.32 g for the lowest pitched E string. Each string is under a tension of 226 N. Find the speeds of the waves on the two strings.
)(/207005287.0
226
628.0/00332.0
226
)(/8260003312.0
226
628.0/000208.0
226
LowEsmv
mkg
Nv
HighEsmv
mkg
Nv
Lm
Fv
Standing Waves; Resonance Standing waves occur when both ends of a string are fixed. In that case, only waves which are motionless at the ends of the string can persist.
There are nodes, where the amplitude is always zero, and antinodes, where the amplitude varies from zero to the maximum value.
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The velocity of waves on a string fixed at one end is 92 m/s. If the frequency of standing waves is 475 Hz, how far apart are two adjacent nodes?
92 /0.194
475
1from one node to another is
2
1(0.194) 0.097
2
v f
v m sm
f Hz
m
Assignment Do pg. 318-319
Problems #36,38,41,43,44,55
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Waves and Oscillations Topic 4 Test Monday 11/23
1. A wave is a traveling disturbance.
2. A wave carries energy from place to place.
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Types of Waves: Transverse and Longitudinal
The motion of particles in a wave can either be perpendicular to the wave direction (transverse) or parallel to it (longitudinal).
Wave Speed Versus Particle Speed Is the speed of a transverse wave on a string the same as the speed at which a particle on the string moves?
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Reflection and Transmission of Waves
A wave reaching the end of its medium, but where the medium is still free to move, will be reflected (b), and its reflection will be upright.
A wave hitting an obstacle will be reflected (a), and its reflection will be inverted.
Reflection and Transmission of Waves
A wave encountering a denser medium will be partly reflected and partly transmitted; if the wave speed is less in the denser medium, the wavelength will be shorter.
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When the pulses merge, the Slinky assumes a shape that is the sum of the shapes of the individual pulses. The picture gives an example of constructive interference.
When the pulses merge, the Slinky assumes a shape that is the sum of the shapes of the individual pulses. The picture gives an example of destructive interference.
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THE PRINCIPLE OF LINEAR SUPERPOSITION When two or more waves are present simultaneously at the same place, the resultant disturbance is the sum of the disturbances from the individual waves.
Refraction If the wave enters a medium where the wave speed is different, it will be refracted – its wave fronts and rays will change direction.
We can calculate the angle of refraction, which depends on both wave speeds:
(11-20)
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Refraction The law of refraction works both ways – a wave going from a slower medium to a faster one would follow the red line in the other direction.
Refraction An earthquake P wave traveling at 8.0km/s strikes a boundary within the Earth between two kinds of material. If it approaches the boundary at an incident angle of 47o and the angle of refraction is 35o, what is the speed in the second medium?
skmv
skm
v
skm
vo
o
/27.6
/0.87843.0
/0.847sin
35sin
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Diffraction
When waves encounter an obstacle, they bend around it, leaving a “shadow region.” This is called diffraction.
Diffraction The amount of diffraction depends on the size of the obstacle compared to the wavelength. If the obstacle is much smaller than the wavelength, the wave is barely affected (a). If the object is comparable to, or larger than, the wavelength, diffraction is much more significant (b, c, d).
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Assignment Do pg. 319-320
Problems #63,64,66,71,72,74,76
Waves and Oscillations Topic 4 Test Monday 11/23
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LONGITUDINAL SOUND WAVES
Individual air molecules are not carried along with the wave.
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Characteristics of Sound Sound can travel through any kind of matter, but not through a vacuum.
The speed of sound is different in different materials; in general, it is slowest in gases, faster in liquids, and fastest in solids.
The speed depends somewhat on temperature, especially for gases.
Characteristics of Sound
Loudness: related to intensity of the sound wave
Pitch: related to frequency.
Audible range: about 20 Hz to 20,000 Hz; upper limit decreases with age
Ultrasound: above 20,000 Hz
Infrasound: below 20 Hz
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The frequency is the number of cycles per second. A sound with a single frequency is called a pure tone. The brain interprets the frequency in terms of the subjective quality called pitch.
Sound waves carry energy that can be used to do work. The amount of energy transported per second is called the power of the wave. The sound intensity is defined as the power that passes perpendicularly through a surface divided by the area of that surface.
A
PI
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Intensity of Sound: Decibels
The intensity of a wave is the energy transported per unit time across a unit area.
The human ear can detect sounds with an intensity as low as 10-12 W/m2 and as high as 1 W/m2.
Perceived loudness, however, is not proportional to the intensity.
Intensity of Sound: Decibels
The loudness of a sound is much more closely related to the logarithm of the intensity.
Sound level is measured in decibels (dB) and is defined:
I0 is taken to be the threshold of hearing:
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The decibel level of a jackhammer is 130 dB relative to the threshold of hearing. Determine the decibel level if three jackhammers operate side by side.
12 2
12
1 2
2
2
12 2
130 10log1.0 10 /
13 log log1.0 10
13 log ( 12)
1 log
10 10 /
3 times the intensity for 3 jackhammers together
3 30 /
30 /10log 135
1.0 10 /
IdB
x W m
I x
I
I
I W m
I W m
W mdB
x W m
24 r
PI
power of sound source
area of sphere
Uniformly emitted sound
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Two boys are whispering in the library. The radiated sound power from one boy's mouth is 1.2 × 10–9 W; and it spreads out uniformly in all directions. What is the minimum distance the boys must be away from the librarian so that she will not be able to hear them? The threshold of hearing for the librarian is 1.00 × 10–12 W/m2.
mr
r
xrx
r
xx
r
PI
8.9
54.95
102.110256.1
56.12
102.11000.1
4
2
9211
2
912
2
Sources of Sound: Vibrating Strings and Air Columns
Musical instruments produce sounds in various ways – vibrating strings, vibrating membranes, vibrating metal or wood shapes, vibrating air columns.
The vibration may be started by plucking, striking, bowing, or blowing. The vibrations are transmitted to the air and then to our ears.
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The strings on a guitar can be effectively shortened by fingering, raising the fundamental pitch.
The pitch of a string of a given length can also be altered by using a string of different density.
,4,3,2,1 2
n
L
vnfnString fixed at both ends
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A 6.00-m long string sustains a three-loop standing wave pattern as shown. The wave speed is 2.00 × 102 m/s.
What is the frequency of vibration? What is the lowest possible frequency for standing waves on this string?
Hzf
f
Hzm
smf
7.163
50
3
50)00.6(2
)/200(3
31
3
Wind instruments create sound through standing waves in a column of air.
A tube open at both ends (most wind instruments) has pressure nodes, and therefore displacement antinodes, at the ends.
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Example When all the holes are closed on one type of flute, the lowest note it can sound is middle C (261.6 Hz). If the speed of sound is 343 m/s, and the flute is assumed to be a cylinder open at both ends, determine the distance L.
mL
L
L
smHz
656.0
3432.523
2
/3436.261
A tube closed at one end (some organ pipes) has a displacement node (and pressure antinode) at the closed end.
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Determine the shortest length of pipe, open at one end, which will resonate at 256 Hz. The speed of sound is 343 m/s.
mL
L
L
smHz
335.0
3431024
4
/343256
Assignment Do pg. 347 Problems
#2,5,8,11,25
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Waves and Oscillations Topic 4 Test Monday 11/23
THE PRINCIPLE OF LINEAR SUPERPOSITION
When two or more waves are present simultaneously at the same place, the resultant disturbance is the sum of the disturbances from the
individual waves.
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When two waves always meet condensation-to-condensation and rarefaction-to-rarefaction, they are said to be exactly in phase and to exhibit constructive interference.
When two waves always meet condensation-to-rarefaction, they are said to be exactly out of phase and to exhibit destructive interference.
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When two waves always meet condensation-to-rarefaction, they are said to be exactly out of phase and to exhibit destructive interference.
If the wave patters do not shift relative to one another as time passes, the sources are said to be coherent.
For two wave sources vibrating in phase, a difference in path lengths that is zero or an integer number (1, 2, 3, . . ) of wavelengths leads to constructive interference; a difference in path lengths that is a half-integer number (½ , 1 ½, 2 ½, . .) of wavelengths leads to destructive interference.
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Interference of Sound Waves; Beats Waves can also interfere in time, causing a phenomenon called beats. Beats are the slow “envelope” around two waves that are relatively close in frequency.
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Interference of Sound Waves; Beats
Doppler Effect
The Doppler effect occurs when a source of sound is moving with respect to an observer.
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Doppler Effect As can be seen in the previous image, a source moving toward an observer has a higher frequency and shorter wavelength; the opposite is true when a source is moving away from an observer.
Doppler Effect
If the observer is moving with respect to the source, things are a bit different. The wavelength remains the same, but the wave speed is different for the observer.
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If the source is moving away from the stationary observer:
We find, for an observer moving towards a stationary source:
And if it is moving away:
If the source is moving towards the observer:
'
1 source
sound
ff
v
v
'
1 source
sound
ff
v
v
' 1 source
sound
vf f
v
' 1 source
sound
vf f
v
Assignment Do pg. 348 Problems
#35,41,46,50
