Measuring Simple Harmonic Motion

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Chapter 11

• Section 2

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Measuring Simple Harmonic Motion

• In order to measure simple harmonic motion we
  have to identify a few important terms that we
  use.
• Amplitude is the maximum displacement from
  equilibrium.
• Period is the time that it takes a complete cycle
  to occur.
• Frequency is the number of cycles or vibrations
  per unit of time, which is normally 1 second.

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Period of a Simple Pendulum

• The period of a simple pendulum depends on
  pendulum length and free-fall acceleration.
• The longer the string or pendulum arm the longer
  the period.
• When the amplitude is small the period does not
  depend on the mass or on the amplitude.
• We use the following formula

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Pendulums of Different Lengths

• If we compare two pendulums with different
  lengths and the same amplitude we can see why
  only length matters.
• Since the smaller length has a smaller distance
  to travel, but the same acceleration, it has to
  travel faster.
• The longer length has to travel a longer
  distance.
• This is kind of like the different lanes of a
  track.

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Example Problem 4

• What is the period of a 3.98 m long pendulum?
  What is the period of a 99.4 cm long pendulum?
• 4.00 s 2.00 s

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Example Problem 5

• A desktop toy pendulum swings back and forth once
  every 1.0 s. How long is this pendulum?
• 0.25 m

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Example Problem 6

• What is the free-fall acceleration at a location
  where a 6.00 m long pendulum swings through
  exactly 100 cycles in 492 s?
• 9.79 m/s2

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Example Problem 7 (Book Problem B)

• You need to know the height of a tower, but
  darkness obscures the ceiling. You note that a
  pendulum extending from the ceiling almost
  touches the floor and that its period is 12 s.
  How tall is the tower?
• L 36 m

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Period of a Mass-Spring System

• The period of a mass-spring system depends upon
  the mass and the spring constant.
• This makes sense because the restoring force in
  this system is the spring force.
• Remember that the spring force is dependent upon
  the spring constant.
• Also, since a heavier mass has more inertia then
  it will be harder to accelerate and will have a
  greater period.

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Period of a Mass-Spring System

• The greater the spring constant the more force it
  takes to move the spring hence increasing the
  period also.
• We can find the period of a mass-spring system by
  using the following formula

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Example Problem 8

• A 1.0 kg mass attached to one end of a spring
  completes one oscillation every 2.0 s. Find the
  spring constant.
• 9.9 N/m

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Example Problem 9 (Book Problem C)

• The body of a 1275 kg car is supported on a frame
  by four springs. Two people riding in the car
  have a combined mass of 153 kg. When driven over
  a pothole in the road, the frame vibrates with a
  period of 0.840 s. For the first few seconds,
  the vibration approximates simple harmonic
  motion. Find the spring constant of a single
  spring.
• k 20000 N/m

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Problems to Work

• Pg 379 Practice B
• Pg 381 Practice C
• Pg 381 Section Review