Harmonic Motion

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Harmonic Motion
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Describe the motion of a rider on a ferris wheel
relative to the ground.
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Describe the motion of a buoy in an ocean
relative to the ocean floor
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Describe the motion of a vibrating horizontal
guitar string.
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Describe the motion of a piston in a cylinder.
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Describe the motion of the mass in a mass-spring
system
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Harmonic Motion

• simple harmonic motion repetitious motion that
  would continue without stopping if there were no
  outside forces on the object

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Harmonic Motion

• if there were no outside forces on the object

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Harmonic Motion

• if there were no outside forces on the object
• swinging pendulum

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Harmonic Motion

• if there were no outside forces on the object
• swinging pendulum - gravity

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Harmonic Motion

• if there were no outside forces on the object
• swinging pendulum - gravity
• mass in a spring-mass system

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Harmonic Motion

• if there were no outside forces on the object
• swinging pendulum - gravity
• mass in a spring-mass system gravity, friction

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Harmonic Motion

• if there were no outside forces on the object
• swinging pendulum - gravity
• mass in a spring-mass system gravity, friction
• vibrating guitar string

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Harmonic Motion

• if there were no outside forces on the object
• swinging pendulum - gravity
• mass in a spring-mass system gravity, friction
• vibrating guitar string - tension

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Harmonic Motion

• if there were no outside forces on the object
• swinging pendulum - gravity
• mass in a spring-mass system gravity, friction
• vibrating guitar string - tension
• floating buoy

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Harmonic Motion

• if there were no outside forces on the object
• swinging pendulum - gravity
• mass in a spring-mass system gravity, friction
• vibrating guitar string - tension
• floating buoy
• ferris wheel

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Harmonic Motion

• if there were no outside forces on the object
• swinging pendulum - gravity
• mass in a spring-mass system gravity, friction
• vibrating guitar string - tension
• floating buoy
• ferris wheel mechanical/human

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Harmonic Motion

• if there were no outside forces on the object
• swinging pendulum - gravity
• mass in a spring-mass system gravity, friction
• vibrating guitar string - tension
• floating buoy
• ferris wheel mechanical/human
• piston in a cylinder

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Harmonic Motion

• if there were no outside forces on the object
• swinging pendulum - gravity
• mass in a spring-mass system gravity, friction
• vibrating guitar string - tension
• floating buoy
• ferris wheel mechanical/human
• piston in a cylinder friction/ human

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Harmonic Motion

• We know that the motion of objects with simple
  harmonic motion can be represented by a sine or
  cosine function.
• Heres why If you were to trace out the
  objects motion over time (or some other
  x-coordinate quantity), the result would be a
  sine or cosine function.

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Harmonic Motion

• Imagine attaching a pencil to a guitar string,
  setting the string into vibration while sliding a
  sheet of paper to the right underneath the string
  and pencil.
• Or imagine attaching a pencil to the outside edge
  of a circular object and sliding a sheet of paper
  to the right while the circle was turning
  counterclockwise.

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Harmonic Motion

• If we know some information about the objects
  motion, typically the position at some time and
  its frequency, then we can write an equation that
  will model its motion.

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Harmonic Motion

• Problem - solving strategy
• make and label a diagram
• make and label a sine or cosine function graph
• write an equation to represent the graph
• answer the question

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Harmonic Motion

• A buoy floats in a harbor at a location where
  the water depth is 26 ft. As waves pass the buoy
  it rises and falls. The buoy moves a total of 8
  ft from its lowest to its highest point every 12
  seconds. Assume the buoy is at its low point at
  t 0. Find the height of the buoy after 15
  seconds.

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Harmonic Motion

• Sketch a diagram

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Harmonic Motion

• Sketch a graph to represent the motion

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Harmonic Motion

• Write a sine or cosine equation
• Answer the question

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Harmonic Motion

• A mass in a spring-mass system is at rest 5 ft
  from a 10 ft ceiling. The mass is given a push
  up at time, t 0. The mass is 2 ft from the
  ceiling at its highest point and it takes 4
  seconds to complete a cycle and return to its
  starting position. What is the mass height 10
  seconds after being set into motion?

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Harmonic Motion

• Sketch a diagram

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Harmonic Motion

• Sketch a graph to represent the motion

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Harmonic Motion

• Write a sine or cosine equation
• Answer the question