Physics 2211: Lecture 39 Todays Agenda

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Physics 2211 Lecture 39Todays Agenda

• Wave Power and Energy

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Lecture 39, Act 1Wave Motion

• A boat is moored in a fixed location, and waves
  make it move up and down. If the spacing between
  wave crests is 20 meters and the speed of the
  waves is 5 m/s, how long Dt does it take the boat
  to go from the top of a crest to the bottom of a
  trough?

(a) 2 sec (b) 4 sec (c) 8 sec
t
t Dt
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Wave Power

• A wave propagates because each part of the medium
  communicates its motion to adjacent parts.
• Energy is transferred since work is done!
• How much energy is moving down the string per
  unit time.
• (i.e., how much power is transmitted in the wave?)

P
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Wave Power...

• Think about grabbing the left side of the string
  and pulling it up and down in the y direction.
• You are clearly doing work since
  as your hand moves up and down.
• This energy must be moving away from your hand
  (to the right) since the kinetic energy (motion)
  of the string stays the same.

P
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How is the energy moving?

• Consider any position x on the string. The
  string to the left of x does work on the string
  to the right of x, just as your hand did

x
?
Power
x
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Power along the string

• Since is along the y axis only, to evaluate
  Power we only need to find Fy - F sin?
  ? - F? if ? is small.
• P Fy v - F? v
• We can easily figure out both the velocity v and
  the angle ? at any point on the string
• If

y
?
x
Fy
dy
?
dx
Recall sin ? ? ? cos ? ? 1
for small ?
tan ? ? ?
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Power...

• So

• But and
  .

• Eliminating k in the coefficient,

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Average Power

• We just found that the power flowing past
  location x on the string at time t is given by

• We are often just interested in the average power
  movingdown the string. To find this we recall
  that the averagevalue of the function cos2 (kx -
  ?t) is 1/2 and find that

• It is generally true that wave power is
  proportional to thespeed of the wave v and its
  amplitude squared A2.

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Energy of the Wave

• We have shown that energy flows along the
  string.
• The source of this energy (in our picture) is the
  hand that is shaking the string up and down at
  one end.
• Each segment of string transfers energy to (does
  work on) the next segment by pulling on it, just
  like the hand.

• We found that

So
is the average energy per unit length.
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Power Example

• A rope with a mass of ? 0.2 kg/m lays on a
  frictionless floor. You grab one end and shake
  it from side to side twice per second with an
  amplitude of 0.15 m. You notice that the
  distance between adjacent crests on the wave you
  make is 0.75 m.
• What is the average power you are providing the
  rope?
• What is the average energy per unit length of the
  rope?
• What is the tension in the rope?

f 2 Hz
? 0.75 m
A 0.15 m
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Power Example...

• We know A, ? and ? 2?f. We need to find v !
• Recall v ? f (.75 m)(2 s-1) 1.5 m/s .
• So

Average power
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Power Example...

• So

Average energy per unit length
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Power Example...

• We also know that the tension in the rope is
  related to speed of the wave and the mass
  density

Tension in rope F 0.45 N
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Recap Useful Formulas
y
?
A
x

• Waves on a string

• General harmonic waves

tension
mass / length
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Lecture 39, Act 2Wave Power

• A wave propagates on a string. If both the
  amplitude and the wavelength are doubled, by what
  factor will the average power carried by the wave
  change?

(a) 1 (b) 2 (c) 4
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Lecture 39, Act 2 Problem Setup

• We have shown that the average power

So
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Recap of todays lecture

• Wave Power and Energy
• Read Chapter 16 in Tipler