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PHYSICS 231
Lecture 34: Oscillations & Waves
Period T
6
3
2
Frequency f 1/6 1/3
½
(m/k)
6/(2) 3/(2) 2/(2)

(2)/6 (2)/3 (2)/2
1 2
m
T 
 2
f

k
Remco Zegers
Question hours: Thursday 12:00-13:00 & 17:15-18:15
Helproom
1
PHY 231
v0
Harmonic oscillations vs circular motion
t=0
t=1
t=2
v0=r=A
=t
=t
t=3
t=4
v0
vx 

A
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A
x
xharmonic(t)=Acos(t)
-A
velocity v
time (s)
=2f=2/T=(k/m)
A(k/m)
vharmonic(t)=-Asin(t)
-A(k/m)
kA/m
a
-kA/m
aharmonic(t)=-2Acos(t)
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Another simple harmonic oscillation: the
pendulum
Restoring force: F=-mgsin
The force pushes the mass m
back to the central position.
sin if  is small (<150) radians!!!
F=-mg also =s/L
so: F=-(mg/L)s
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pendulum vs spring
parameter
spring
pendulum
restoring
force F
F=-kx
F=-(mg/L)s
period T
T=2(m/k)
frequency f
f=(k/m)/(2) f=(g/L)/(2)
angular
frequency
=(k/m)
*
T=2(L/g)
=(g/L)
m
L
T  2
 2
mg / L
g
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*
example: a pendulum clock
The machinery in a pendulum clock is kept
in motion by the swinging pendulum.
Does the clock run faster, at the same speed,
or slower if:
a) The mass is hung higher
b) The mass is replaced by a heavier mass
c) The clock is brought to the moon
d) The clock is put in an upward accelerating
elevator?
L
m
moon
elevator
faster
L
T  2
g
same
slower
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example: the height of the lecture room
demo
L
T  2
g
gT 2
2
L

0
.
25
T
4 2
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damped oscillations
In real life, almost all oscillations eventually stop due to
frictional forces. The oscillation is damped. We can also
damp the oscillation on purpose.
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Types of damping
No damping
sine curve
Under damping
sine curve with decreasing
amplitude
Critical damping
Only one oscillations
Over damping
Never goes through zero
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Waves
The wave carries the disturbance, but not the water
position y
position x
Each point makes a simple harmonic vertical oscillation
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Types of waves
wave
oscillation
Transversal: movement is perpendicular to the wave motion
oscillation
Longitudinal: movement is in the direction of the wave motion
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A single pulse
velocity v
time to
time t1
x0
x1
v=(x1-x0)/(t1-t0)
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describing a traveling wave
: wavelength
distance between
two maxima.
While the wave has traveled one
wavelength, each point on the rope
has made one period of oscillation.
v=x/t=/T= f
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2m
2m
example
A traveling wave is seen
to have a horizontal distance
of 2m between a maximum
and the nearest minimum and
vertical height of 2m. If it
moves with 1m/s, what is its:
a) amplitude
b) period
c) frequency
a) amplitude: difference between maximum (or minimum)
and the equilibrium position in the vertical direction
(transversal!) A=2m/2=1m
b) v=1m/s, =2*2m=4m T=/v=4/1=4s
c) f=1/T=0.25 Hz
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sea waves
An anchored fishing boat is going up and down with the
waves. It reaches a maximum height every 5 seconds
and a person on the boat sees that while reaching a
maximum, the previous waves has moves about 40 m away
from the boat. What is the speed of the traveling waves?
Period: 5 seconds (time between reaching two maxima)
Wavelength: 40 m
v= /T=40/5=8 m/s
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Speed of waves on a string
v
F

F tension in the string
 mass of the string per unit length (meter)
 M /L
tension T
screw
example: violin
L M
v= /T= f=(F/)
so f=(1/)(F/) for fixed wavelength the frequency will
go up (higher tone) if the tension is increased.
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example
A wave is traveling through the
wire with v=24 m/s when the
suspended mass M is 3.0 kg.
a) What is the mass per unit length?
b) What is v if M=2.0 kg?
a) Tension F=mg=3*9.8=29.4 N
v=(F/) so =F/v2=0.05 kg/m
b) v=(F/)=(2*9.8/0.05)=19.8 m/s
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bonus ;-)
The block P carries out a simple harmonic motion with f=1.5Hz
Block B rests on it and the surface has a coefficient of
static friction s=0.60. For what amplitude of the motion does
block B slip?
The block starts to slip if Ffriction<Fmovement
sn-maP=0
smg=maP so sg=aP
ap= -2Acos(t) so maximally 2A=2fA
sg=2fA A= sg/2f=0.62 m
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