Announcements

1
Announcements

• Last class will be Wednesday, December 7
• Will have review session on Friday, December 9
• 900 950, BH 166
• HW 10 is due at end of class
• Homework 11 is due at end of Friday, Dec. 9
  review session
• Final Exam will be Wednesday, December 14
• Note date listed in course information was
  inadvertently listed as Monday, December 14
• Will cover all material in course
• Will focus more heavily on material in HRW
  Chapters 10-13, 15

2
Simple Harmonic Motion

• Will have simple harmonic motion whenever the
  motion is governed by an equation of the form
• Such equations have two independent solutions
• Most general solution is a combination of these
  solutions
• C and S are constants to be determined by initial
  conditions

3
Simple Harmonic Motion

• Can also write solution as an amplitude and a
  phase
• These solutions are related to each other

4
Simple Harmonic Motion

• Simple Harmonic Motion described by 3 parameters
• Amplitude A
• Angular frequency w
• Phase f
• The amplitude describes the magnitude of the
  oscillation
• The frequency tells how many oscillations per
  second
• The phase shifts the periodic motion in time
• Increasing phase shifts the motion backwards
  along the time axis
• Example Oscilloscope with Sine-wave signal
• Example Compare to motion of a mass attached to
  a spring

5
Simple Harmonic Motion

• Velocity and acceleration also vary as
  sine/cosine functions
• Velocity is at its maximum when the object is at
  x0
• Velocity is 0 when position is maximum xxm
• Acceleration is opposite position, always trying
  to bring it back towards x0

6
Energy for Simple Harmonic Motion

• Take the mass attached to a spring as a specific
  example
• We have potential energy in the spring, kinetic
  energy in the mass
• Total energy is constant during motion energy
  sloshes back and forth between potential and
  kinetic energy
• Similar behavior for all types of simple harmonic
  motion

7
Torsion Pendulum

• Torsion pendulum consists of a mass suspended by
  a fine wire
• Twisting of the wire produces a torque that tries
  to bring the mass back to the equilibrium
  position
• Solve for motion of torsion pendulum
• We again have simple harmonic motion, this time
  in twist q

8
Torsion Pendulum

• Example Disk of radius R supported by a wire
• Period inversely proportional to angular
  frequency w

9
Simple Pendulum

• Mass on the end of a string
• Gravity provides force to bring mass back towards
  vertical position
• Look at motion perpendicular to string
• Simple harmonic oscillation for small angles

L
10
Physical Pendulum

• Physical Pendulum consists of a rigid body
  allowed to rotate
• Gravitational force acting on center of gravity
  (e.g., center of mass) creates a torque on the
  object
• We again have simple harmonic motion for small
  angles q

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