Oscillatory Motion

1
Chapter 13

• Oscillatory Motion

2
Simple Harmonic Motion

• Periodic motionany motion that repeats itself
  regularly
• Orbit and rotation of earth
• Playground swing
• Simple Harmonic Motion (SHM)
• Simplest form of periodic motion
• Described by harmonic functions (sines and
  cosines)
• Springs and simple pendulums

3
Simple Harmonic Motion

• Tendency to return to equilibrium
• restoring force is present
• Ideal spring (with mass attached to the end)
• Hookes Law (F -kx) describes restoring force
• Simple Pendulum
• Gravity is the restoring force
• Mass on the end of a string of negligible mass

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Simple Harmonic Motion

• Terms
• Displacement (x)distance from equilibrium
• Equilibrium at x 0
• Amplitude (A)magnitude of maximum displacement
• Period (T)time needed for one complete cycle
• Frequency (f)number of cycles per second

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Equations of motion

• y displacement
• A amplitude
• ? angular frequency
• t time
• ? phase angle
• Radians

Speed at any point in time, p. 430
6
Simple Harmonic Motion

• Potential Energy (U) of a spring
• Maximum PE of a spring
• Total Mechanical Energy is conserved

Maximum speed of a spring Speed at any point
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Equations of motion

• Period
• Spring
• M mass on end of spring
• k spring constant
• Simple Pendulum
• L Length of pendulum

Grandfather clocks
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Equations of motion

• Frequency of a spring (f)
• Units of Hz (cycles/s)
• Angular frequency of a spring (?)
• Units of rad/s

Period and frequency are independent of Amplitude
9
Damped Oscillations

• Friction messes everything up
• When oscillating objects meet a retarding force,
  theyre called damped oscillators
• Only when the retarding force is much smaller
  than the restoring force is oscillation preserved

10
Damped Oscillations

• Function becomes bound by a decreasing exponential

11
Damped Oscillations

• Natural frequency
• No damping occurs
• Underdamped
• Max retarding force is less than the restoring
  force
• Oscillates, but amplitude decreases over time
• Overdamped
• Max retarding force is more than the restoring
  force
• No oscillation occurs, just slowly returns to
  equilibrium
• Critically damped
• Max retarding force is equal to the restoring
  force
• No oscillation occurs, returns to equilibrium
  very quickly

12
Force Oscillations

• When oscillations are driven by a time variable
  force, they are forced oscillations
• Pushing a swing
• Walking on a suspension bridge
• Sound waves hitting materials
• Natural frequency of vibration is also called the
  resonant frequency
• When driving force matches this frequency, the
  amplitude increases dramatically
• Oscillator is in resonance