Chapter 11: vibrations and waves

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Chapter 11 vibrations and waves
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Hookes Law

• The spring force and the distance that the spring
  is stretched are proportional but opposite in
  direction.
• Felastic kx
• Felastic is the spring force (N), k is the spring
  constant (N/m), and x is the displacement (m).

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Simple Harmonic Motion

• Simple harmonic motion describes any periodic
  motion that is a result of a restoring force that
  is proportional to displacement.
• In simpler terms, simple harmonic motion
  describes a back and forth motion that occurs
  over the same path (any vibration is an example
  of simple harmonic motion).
• Whenever a back and forth motion occurs and a
  restoring force pushes an object back to its
  equilibrium position, simple harmonic motion
  occurs.

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• A pendulum is an example of simple harmonic
  motion.
• It has a back-and-forth motion
• It has a restoring force (the x component of
  gravity acting on the bob) that pulls the bob
  back to its equilibrium position.
• However, this is only true for small angles of ?.
  (Only when ? lt 15)

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Do Now 3/9/09

• Which of these periodic motions are simple
  harmonic?
• (a) A child swinging on a playground swing (?
  45)
• (b) A CD rotating in a player
• (c) An oscillating clock pendulum (? 10)

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Amplitude, Period, Frequency

• The amplitude is the maximum displacement from an
  equilibrium position.
• The period, T, is the time it takes for a
  complete cycle to occur.
• The frequency, f, is the number of cycles that
  occur in a specific unit of time (usually
  seconds). Frequency is measured in Hertz (Hz).
• If a wave completes one cycle in 2 seconds, how
  many cycles occur in one second?

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• If a wave completes one cycle in 4 seconds, how
  many cycles occur in one second?
• If a wave complete one cycle in 10 seconds, how
  many cycles occur in one second?
• What do you notice about the period and frequency
  in each situation?

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• Period (T) and Frequency (f) are inverses of each
  other
• f 1/T
• T 1/f
• Group Activity
• Decide whether your two pendulums have the same
  period.

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Period of a Pendulum

• The period of a pendulum only depends on the
  pendulums length, not the mass.
• Two pendulums with different masses will have the
  same period. But two pendulums with different
  lengths will have different periods.
• When the length of a pendulum is decreased, so is
  the distance to the equilibrium position, so it
  takes a shorter amount of time for the short
  pendulum to reach equilibrium. The shorter
  pendulum will have a shorter period.

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Do Now 3/10/09

• 1.) A pendulum that moves through its equilibrium
  position once every 1 second is sometimes called
  a seconds pendulum.
• (a)What is the period of any seconds pendulum?
• (b)In Cambridge, England, a seconds pendulum is
  0.9942 m long. What is the free-fall
  acceleration in Cambridge?

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Period of a Mass-Spring System

• Mass affects the period of a mass-spring system.
  
• The heavier the mass that hangs from a spring,
  the more resistance the spring has to going back
  to its equilibrium position. The spring will
  slowly return to its equilibrium position.
• A heavy mass has a small acceleration, which
  means it takes longer to complete one cycle of
  motion.
• If a lighter mass is attached to a spring, there
  is little resistance to restoring to the
  equilibrium position. A lighter mass will cause
  the spring to return to equilibrium much faster.
  
• A light mass has a bigger acceleration, which
  means that it takes a shorter amount of time to
  complete one cycle of motion.
• The period of a heavy mass-spring system will be
  longer than the period of a light mass-spring
  system.

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Period of Mass-Spring System

• Stiffer springs will take less time to return to
  equilibrium, which means that they will take less
  time to complete one cycle of motion (Stiffer
  springs will have shorter periods).
• Flexible springs will take more time to return to
  equilibrium, which means that they will take more
  time to complete one cycle of motion (Flexible
  springs will have longer periods).

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Medium

• Waves are caused by a disturbance in some object
  or material.
• In order to transfer energy, most waves must
  travel through a medium.
• A medium is an object or material that the
  disturbance passes through.
• Once the disturbance passes through a medium, the
  medium will return to its original position or
  shape.

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Types of Waves

• Transverse Waves occur when the motion of the
  medium (such as a slinky) is perpendicular to the
  direction in which the wave travels.
• The slinky moves up and down in a vertical
  direction, but the wave is moving horizontally
  down the slinky.
• Longitudinal Waves occur when the medium moves in
  the same direction in which the wave travels. The
  medium moves along the direction of the wave
  instead of perpendicular to it.

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Parts of a Wave

• Crest
• Trough
• Amplitude
• Wavelength

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• If a train of freight cars, each 10 m long,
  rolls by you at the rate of 2 cars each second,
  what is the speed of the train?

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• Imagine that you fix your eyes at a stationary
  point on the surface of water and observe that
  two crests pass that stationary point each
  second. You also notice that the length of each
  wave is 3 meters. What is the speed of the wave
  in meters per second?

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Wave Speed

• Wave speed frequency x wavelength
• In equation form v f?
• ? Greek letter Lamda represents wavelength
• Wave speed is measured in m/s

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Constructive Interference

• When the crest of one wave interferes with the
  crest of another, their individual effects add
  together.
• The result of two crests interfering is a wave of
  increased amplitude.

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Constructive Interference

• When the trough of one wave interferes with the
  trough of another, their individual effects add
  together.
• The result of two troughs interfering is a wave
  of increased amplitude.

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Destructive Interference

• When the crest of one wave interferes with the
  trough of another wave, their individual effects
  are reduced.
• The high part (which is the crest) of one wave
  fills in the low part (which is the trough) of
  the other wave.

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Destructive Interference
Sometimes a trough and crest will meet and
completely cancel out. Destructive interference
still occurs, but there is no resulting wave.
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Do Now 3/13/09

• Decide whether each labeled position in the
  following diagram is a position where
  constructive or destructive interference occurs.

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Journal 3/13/09

• 1.) What type of wave is associated with sound
  waves? Transverse or longitudinal?
• 2.) Did Bill Nye help you understand certain
  concepts that you were unsure about before you
  watched the video? If so, what concepts?
• 3.) List 5 things that you learned from the
  video.

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Do Now 3/16/09

• 1.) Explain what happens when a pulse is sent
  down a rope that is fixed at one end.
• 2.) Explain what happens when a pulse is sent
  down a rope that is loosely tied at one end.
• 3.) A tuning fork produces a sound with a
  frequency of 245 Hz. What would be the
  wavelength of this sound in water if the speed of
  sound in water is 1500 m/s?
• 4.) What is the difference between a pulse wave
  and a periodic wave?
• 5.) Draw the resultant wave of the two
  interfering waves on the board.

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Standing Wave

• An original wave (called the incidental wave) and
  the reflected wave form a standing wave.

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Nodes and Antinodes

• The parts of the wave that remain stationary are
  called nodes.
• The positions on a standing wave with the largest
  amplitudes are called antinodes. Antinodes occur
  halfway between two nodes.

ANTINODES NODES