Vibration and waves

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Vibration and waves
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• A restoring force always pushes or pulls the
  object toward the equilibrium position
• Simple harmonic motion -occurs when the net force
  is proportional to the displacement from the
  equilibrium point and is always directed toward
  the equilibrium point

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• The amplitude A is the maximum distance of the
  object from its equilibrium position
• The period T- is the time it takes the object to
  move through one compete cycle of motion ( from
  xA to x-A and back to xA)
• The frequency f is the number of complete cycles
  or vibrations per unit of time (f1/T)
• The harmonic oscillator equation
• a(-k/m)x
• A ranges over the values kA/m and kA/m

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• Elastic potential energy PEs1/2kx2
• Conservation of energy
• (KE PEgPEs)i (KE PEgPEs)f

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• Velocity as a Function of position
• ½ kA2 1/2mv21/2kx2
• vvk/m(A2-x2)

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• Comparing harmonic motion with circular motion
• v Cv (A2-x2)
• sin?v/vo
• sin ? (vA2-x2)/A
• v/vo(vA2-x2)/A
• v vo(vA2-x2)/A
• C v (A2-x2)

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• Period and frequency
• vo2pA/T
• T2pA/vo
• Conservation of energy1/2kA21/2mv2
• A/vo vm/k
• T2pvm/k
• f (1/2p)vk/m
• The angular frequency
• ?2pf vk/m

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• Position, velocity and acceleration as a fct. of
  time
• xAcos?
• ? ?t
• ???/?t 2p/T 2pf
• x Acos(2pft)
• v-A?sin(2pft)
• ?vk/m
• a-A?2cos(2pft)

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• Motion of a pendulum
• Ft-mg sin? -mg ?
• Ft mg sin(s/L)
• Ft-(mg/L)s
• Ft-kx
• k mg/L
• ?2pf vk/m
• ?vmgL/mvg/L
• T 2pvg/L

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• Simple harmonic motion for an object-spring
  system, and its analogy, the motion of a simple
  pendulum

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• Wave-the motion of a disturbance
• Transverse waves- each segment of the rope that
  is disturbed moves in a direction perpendicular
  to the wave motion
• Longitudinal waves- the elements of the medium
  undergo displacements parallel to the direction
  of wave motion (sound waves)

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• Frequency, Amplitude and wavelength
• v?x/?t
• ?x? ?tT
• v ?/T
• vf ?
• The wavelength ?- is the distance between 2
  successive points that behaves identically
• The speed of waves on strings vvF/µ
• (F-tension, µ- linear density-mas of the
  string/unit length

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• Interference of waves
• Two traveling waves can meet and pass through
  each other without being destroyed or even
  altered
• The superposition principle when 2 or more
  traveling waves encounter other while moving
  through a medium, the resultant wave is found by
  adding togherther the displacements of the
  individual waves point by point (constructive
  interference or destructive interference)