PHYSICS 231 Lecture 34: Oscillations

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PHYSICS 231Lecture 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

• Remco Zegers
• Question hours Thursday 1200-1300
  1715-1815
• Helproom

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Harmonic oscillations vs circular motion
v0
t0
t1
t2
v0?r?A
??t
??t
t3
t4
v0
?
vx
?
A
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A
xharmonic(t)Acos(?t)
x
time (s)
-A
?2?f2?/T?(k/m)
A?(k/m)
vharmonic(t)-?Asin(?t)
-A?(k/m)
aharmonic(t)-?2Acos(?t)
a
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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 (lt150) 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 T2??(m/k) T2??(L/g)
frequency f f?(k/m)/(2?) f?(g/L)/(2?)
angular frequency ??(k/m) ??(g/L)

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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
• The mass is hung higher
• The mass is replaced by a heavier mass
• The clock is brought to the moon
• The clock is put in an upward accelerating
• elevator?

L? m? moon elevator
faster
same
slower
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example the height of the lecture room
demo
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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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example
2m

• 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
• amplitude
• period
• frequency

2m

• amplitude difference between maximum (or
  minimum)
• and the equilibrium position in the vertical
  direction
• (transversal!) A2m/21m
• v1m/s, ?22m4m T?/v4/14s
• f1/T0.25 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 ?/T40/58 m/s
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Speed of waves on a string
F tension in the string ? mass of the string per
unit length (meter)
screw
tension T
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 v24 m/s when the
• suspended mass M is 3.0 kg.
• What is the mass per unit length?
• What is v if M2.0 kg?

a) Tension Fmg39.829.4 N v?(F/?) so
?F/v20.05 kg/m b) v?(F/?)?(29.8/0.05)19.8
m/s
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bonus -)
The block P carries out a simple harmonic motion
with f1.5Hz Block B rests on it and the surface
has a coefficient of static friction ?s0.60. For
what amplitude of the motion does block B slip?
The block starts to slip if FfrictionltFmovement ?s
n-maP0 ?smgmaP so ?sgaP ap -?2Acos(?t)
so maximally ?2A2?fA ?sg2?fA A
?sg/2?f0.62 m