Oscillations
1. Different types of motion:
•Uniform motion
•1D motion with constant acceleration
•Projectile motion
•Circular motion
•Oscillations
2. Different types of oscillations:
•Periodic oscillations
A motion is called periodic when the system comes back
to the same physical conditions every time interval T.
•Harmonic oscillations (The simplest, the most
important type of periodic oscillations)
•Chaotic oscillations
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3. Harmonic oscillations
Relation between circular motion and simple harmonic oscillations
y
  t
r=A

 2  T 
2

 2f
T
x  A cos 
x
x  A cost 
v  A sin t 
a   2 A cost 
T – period
f – frequency
ω – angular frequency
A – amplitude
a   2 x
linear equation
Period of SHO is independent from amplitude!
Units:
 f   1s 1  1/ s  1Hz
2
x
T
x  xmax
v0
a  amax
x   xmax
v0
a  amax
x0
v  vmax
a0
A=xmax
t
x0
v  vmax
a0
v
t
a
t
3
4. Hook’s law and simple harmonic motion
Newton’s second law:
F  kx 

F  ma 
Simple harmonic motion:
a   x
Hook’s law:
ma  kx
k
a x
m
2
k
 
m
2
1 2
m
T 
 2
f

k
Question: Two identical masses hang from two identical springs.
In case 1, the mass is pulled down 2 cm and released.
In case 2, the mass is pulled down 4 cm and released.
How do the periods of their motions compare?
A. T1 < T2
B. T1 = T2
C. T1 > T2
Period is independent of amplitude!
This is in fact general to all SHM (not only for springs)!
4
Question: Mass m attached to a spring with a spring constant k. If the
mass m increases by a factor of 4, the frequency of oscillation of the mass
1.
2.
3.
4.
is doubled
is multiplied by a factor of 4
is halved
is multiplied by a factor of 1/4
T
1 2
m

 2
f

k
Question: A 2.0 kg mass attached to a spring with a spring constant of
200 N/m. The angular frequency of oscillation of the mass is __ rad/s.
A. 2
B. 10
C. 60
D. 100
k
 
m
2
k
200N / m


 100rad / s  10rad / s
m
2kg
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5. The simple pendulum
a   g sin x L
ma  F  mgsin 
L

x

L
mg

If
g
a   x   2 x
L
  1 than sin   
x(t )  A cost   0 
g

1
L
; f 
; T   2
L
2
f
g
x
t
Period and frequency
are independent from
amplitude!
Example:
L  0.5m
g
9.8m / s 2


 4.4s 1
L
0.5m

f 
 0.70 Hz
2
1
L
T   2
 1.4s
f
g
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Example: A person swings on a swing. When the person sits still, the swing
moves back and forth at its natural frequency. If, instead, the person stands
on the swing, the new natural frequency of the swing is:

A. Greater
g
L
B. The same
C. Smaller
If the person stands, L becomes smaller
Example: Grandpa decides to move to the Moon, and he naturally takes his
old pendulum clock with him. But gravity on the Moon is approximately g/6...
As a result, his clock is:
A. Too fast
B. Too slow
C. Too fast until noon, too slow after noon
2
L
T
 2

g
The period of the pendulum is longer on the Moon.
So each “second according to this clock” is then
longer than a real second.
To tune the clock you can move the clock’s disk up.
7
6. Damped Harmonic Motion
x(t)
t
7. Resonance
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Waves
1. There are many examples of waves:
•
•
•
•
Seismic waves
Ripples on a pond
Sound
Electromagnetic wave including light
2. Types of Waves
• Wave pulses (waves that have short
duration – just a quick pulse)
•
Periodic waves
•
Special type of periodic waves
– sinusoidal waves
Longitudinal wave: the oscillations are along the direction the wave travels.
Transverse wave: the oscillations are transverse (perpendicular) to the
direction the wave travels.
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3. Sinusoidal waves
y( x, t )  A sin(t  kx)
Wave at fixed x:
y
Wave at fixed t:
y
T
A
t
Amplitude: A
Period:
T
Wave length: 
Frequency: f  1 / T
Angular
frequency: 
Wave
number:
 2 / T  2f
k  2 / 
λ
x
Sinusoidal waves are periodic
a) A periodic wave is periodic in time at
fixed position as the wave passes that
position. The period of the wave is T.
Each particle in the medium oscillates
with the same period and in the same
way as the wave passes the particle.
b) A periodic wave is also periodic in space
at fixed time. The period of the wave in
space is called the wavelength .
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4. Wavelength, period and speed of waves
v   / T  f   / k
λ - wave length
T - period
• The “speed” of a wave is the rate of movement of the disturbance.
• It is not the speed of the individual particles!
• The speed is determined by the properties of the medium.
Question 1: If you double the wavelength  of a periodic wave and the
wave speed does not change, what happens to the wave frequency f ?
1) f is unchanged
2) f is halved
3) f is doubled
Question 2: A sinusoidal wave has a wavelength of 4.00 m and a period
of 2.00s. What is the speed of the wave?
v   / T  4.00m 2.00s   2.00m s
Question 3: A sinusoidal wave has a wavelength of 4.00 m and an
angular frequency of 3.14 rad/s. What is the speed of the wave?


v  f   2  4.00m 3.14s 1 2  2.00m s
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5. Interference – combination of waves
(an interaction of two or more waves arriving at the same place)



Important: principle of superposition  (r , t )  1 (r , t )  2 (r , t )
Valley
Peak
(b) (a)
Valley
Waves source
(b)
(a)
No shift or shift by
r2  r1  m
Shift by
r2  r1  m  12 
m  0,1,2,...
a) Constructive interference: the interfering waves add up so that they reinforce
each other (the total wave is larger)
b) Destructive interference: the interfering waves add up so that they cancel
each other (the total wave is smaller or even zero)
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Example: Two speakers S1 and S2 are driven by the same signal generator
and are different distances from a microphone P as shown.
The minimum frequency for constructive
interference to occur at point P is __ Hz.
(The speed of sound is v = 340 m/s.)
A. 100
r2  r1   n
n  0,1,2,...
f  v /   f min
B. 200
C. 400
D. 800
v
340m / s


 200Hz
r 2 r1 3.40m  1.70m
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6. Diffraction
What it is?
The bending of waves behind obstacles or apertures into the ”shadow region”,
that can be considered as interference of many waves.
Haw to observe?
Diffraction is most pronounced when the wavelength of the wave is similar to
the size of the obstacle or aperture.
For example, the diffraction of sound waves is commonly observed because
the wavelength of sound is similar to the size of doors.
The waves spread out
from the opening!
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Sound
Acoustic waves in the range of frequencies: 20Hz - 20,000Hz
1. Sound waves:
• can travel in any solid, liquid or gas
• in liquids and gases sound waves are longitudinal ONLY!
• longitudinal and transversal sound waves could propagate in in solids
Sound in air is a longitudinal wave that contains regions of low and high pressure
Pressure sensor
Vibrating tuning fork
These pressure variations are usually small – a “loud” sound changes the
pressure by 2.0x10-5 atm
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Speed of Sound in Some Common Substances
Substance
Speed (m/s)
1. Air (20 oC)
344
2. Helium (20 oC)
999
3. Water (0 oC)
1,402
4. Lead
1,200
5. Human tissue
1,540
6. Aluminum
6,420
7. Iron and steel
5,941
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2. Perception of Sound
We use four subjective characteristics to describe how we perceive sound:
pitch, loudness, tone quality and duration
1)
Pitch - how “high” or “low” we perceive sound
(it is directly related to how we perceive frequency)
Combinations of notes that are “pleasing” to the ear have frequencies that
are related by a simple whole-number ratio (Pythagoras)
2)
Loudness - how we perceive the amplitude of the sound wave
Loudness manly depends on amplitude and frequency
3)
Tone quality - how we distinguish sounds of the same pitch and
loudness (how we perceive the qualities of the waveform)
4)
Duration
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3. Doppler Effect
What do you hear when a sound-emitting object (train, car) passes you?
The sound changes in pitch (frequency) as the object goes.
Why does this happen?
v  f
•As the object travels towards you, the distance
between wavefronts is compressed; this makes it
seem like  is smaller and frequency is higher.
•As the object travels away from you, the distance
between wavefronts is extended; this makes it
seem like  is larger and frequency is lower.
Because the speed of a wave in a medium is
constant, change in  will affect change in f.
Velocity of listener (L): vL
Velocity of source (S): vS
Velocity of sound: v
 v  vL
f L  f S 
 v  vS



vL or vS is “+” if in the same
direction as from listener to
source and is “-” otherwise
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4. Slightly mismatched frequencies cause audible “beats”
f beats  f 2  f1
Question: The beat frequency between tones with frequencies f1 and f2 is
2.0 Hz. In order to increase the beat frequency, one must __.
A) Increase f1;
D) Decrease f2;
B) Increase f2;
C) Decrease f1;
E) There is not enough information to choose
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