Welcome back to Physics 211

1
Welcome back to Physics 211

• Todays agenda
• Physical pendulum
• Oscillations -- damping, resonance
• Waves general properties
• Mathematical description
• Reflections at boundary

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Current homework assignments

• WHW12
• From end of chapter 13 in University Physics
• 13.6, 13.46
• due Wednesday, Dec. 6th in recitation
• FHW7
• From end of chapter 15 in University Physics
• 15.20, 15.44
• due Friday, Dec. 8th in recitation

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Final Exam Monday (Dec .11), 715PM

• Exam will be 2 hours long
• Cumulative!
• Material covered
• Textbook chapters 1-16
• Lectures -- all (slides online)
• Tutorials -- all
• Problem Solving Activities -- all
• Homework assignments -- all (select solutions
  online)
• Exams 1, 2 3 (solutions online)
• As with Exams 1, 2 3, Final is closed book, but
  you may bring calculator and one handwritten 8.5
  x 11 sheet of notes -- this may be a different
  sheet from Exams 1, 2 3.
• Practice versions of Final Exam posted online

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Torsion Pendulum (Angular Simple Harmonic
Oscillator)
t - k q
Solution
DEMO
5
(Gravitational) Pendulum Physical Pendulum
Extended Object
tnet d mg sinq

q
For small q
sinq q
tnet d mg q
DEMO
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A person swings on a long swing. When the person
sits still, the swing oscillates back and forth
at its natural frequency. If, instead, the person
stands on the swing, the new natural frequency of
the swing is
1(A). greater 2(B). smaller 3(C). the same
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Damped Harmonic Oscillator
For example air drag for small speeds Fdrag - b
v
Solution
8
Forced Harmonic Oscillator
Differential equation for x(t)
Driving force
Damping force
Natural frequency
Solution yields amplitude of response
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Resonance
A
?d
10
Waves general features

• many examples
• water waves, musical sounds, seismic tremors,
  light, gravity, etc.
• system disturbed from equilibrium can give rise
  to a disturbance which propagates through medium
  (carries energy)
• periodic character both in time and space
• interference

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Wave Motion vs. Particle Motion

• Particle Motion

Net displacement of matter

• Wave Motion

No net displacement of matter
Both particles and waves transport energy from
one place to the other
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Demo -- waves on rods

• Notice
• mean position of rods does not change.
• but energy transported!
• each rod undergoes periodic motion
• speed of wave does not depend on how fast or what
  magnitude of driving force
• example of traveling periodic wave

13
Waves structure

• Wave propagates by making particles in medium
  execute simple harmonic motion
• motion along direction of wave (longitudinal)
  e.g. sound
• perpendicular to direction (transverse) e.g.
  waves on string, light
• Wave motion described by function of both
  position and time

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Transverse and Longitudinal Waves

• Transverse Wave

• Longitudinal Wave

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Mathematical Description of Traveling Wave
t1 0 y f(x)
t2 t y f(x - vwavet)
y f(x - vt)
Wave traveling in x direction
Wave traveling in - x direction
y f(x vt)
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Waves - properties

• waves propagate at constant speed through medium
• speed depends on properties of medium (density
  and elastic forces)
• not same as velocity of particles in medium

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Speed of wave

• wave on string -- how quickly does disturbance
  propagate from one point to another?
• wave speed, v
• expect it depends on tension and mass density of
  string
• dimensional analysis
• v2 T/m -- i.e. (kg m/s2)/(kg/m)m2/s2

18
For waves traveling along the rubber hose
stretched across Stolkin, if I pull on the hose
with a greater tension before plucking the hose,
will the resulting wave speed be

• Greater?
• Smaller?
• Unchanged?
• Cant tell

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Mathematical description

• Describe wave by wave function which tells you
  the size of the wave at each point in space (x)
  and time (t)
• y ?y(x,t)
• Think of sinusoidal waves for simplicity
• At fixed point in space, have SHM
• y?? Acos(wt) Acos(2pft)

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Description continued

• Not full story the amplitude of wave depends on
  position as well as time
• Wave is collection of SHM oscillators where each
  oscillator has different phase f(x)
• y? Acos(wt - f)
• Simplest case f??kx (2p/l)x

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Sinusoidal Waves
Angular frequency
Amplitude
Angular wave number
Initial phase

• Can also use sine in the definition since cos(q)
  sin(q-p/2)

Speed of sinusoidal wave
22
Sinusoidal Waves
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Wave speed

• Notice v f l
• number of wave crests passing by in 1 second
  times distance between crests
• ? speed of wave v
• Also, can define
• w? 2pf and k 2p/l
• w angular frequency, k wave number
• v w/k

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If you double the frequency of a wave what
happens to the speed of the wave?

• doubles
• halves
• stays same
• depends on wavelength

25
Sinusoidal Waves
y
A
0
x
-A
Displacement of waveform at two different times
Speed of harmonic wave
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Rough picture

• as one part of wave wiggles sets off another
  neighboring region
• neighbor wiggles at same frequency but lags the
  first
• lag just depends on how far away it is
• lag is equivalent to different phase

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Another way of thinking

• Consider particle at x0 and t0. Wiggles with
  SHM
• wave travels distance to some point x in time x/v
  (v wave speed)
• So, motion of wave at time t, position x is same
  as initial point at t0, x0 (sinusoidal)
• Thus, replace t ? (t - x/v) in original Y

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Power in a wave

• Consider energy of SHM
• E (1/2)kA2 (1/2)mw2A2
• What is m?
• Also need power energy per unit time delivered
  by wave in time T
• Total mass of excited oscillators is Tvm
• E/T P (1/2)vmw2A2

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Reflection of waves

• Reflection reversal of wave velocity

Fixed end
Free end
Pulse not inverted
f(x - vt) a f(x vt)
Pulse inverted
f(x - vt) a f(x vt)
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Reading assignment

• 16.1 - 16.8 in textbook
• Sound Waves