Physics 211 lecture 25: Oscillatory Motion

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Physics 211 lecture 25 Oscillatory Motion

• Oscillatory motion is often also called harmonic
  motion  
• Harmonic Motion has two characteristics
• acceleration is proportional to position
• acceleration is opposite to direction of
  displacement (recall F-dU/dx ) 
• Examples of Harmonic Motion
• Spring
• Pendulum
• Electromagnetic waves
• Vibrating string
• Anything with motion fitting a sine or cosine
  shape (sinusoidal motion)  

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The mathematics of harmonic motion

• Equate spring force equation to Newtons law

and
so
Define oscillatory angular frequency (?)
so
The wave equation write the differential
equation for harmonic motion
The wave equation solution position is written
as a function of time

• x(t) displacement from equilibrium at time t
• A amplitude (maximum displacement)
• angular frequency (very similar to angular
  velocity since units are rad/sec)
• t time
• ? phase angle offset from equilibrium at
  starting position

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More on Angular Frequency

• How can we relate angular frequency to
  oscillation frequency and period

Oscillation frequency (f)
and
so
Angular frequency (?)
angle for one rev
one rev time
angle per time
Now we can write
angle for one rev
Position, Velocity and Acceleration for harmonic
motion
Note xmax A vmax ?A a max ?2A
occur when cos or sin 1
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Energy in Harmonic Oscillation
where
Will be all K at equilibrium position (middle of
oscillation)
Will be all U at end points of oscillation
Relating Oscillatory Motion to Circular Motion
Position in SHM is the x part of circular motion
position Velocity in SHM is the vx part of
circular motion velocity Acceleration in SHM is
the ax part of circular motion acceleration
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The Pendulum
Simple Pendulum
Physical Pendulum
O
Both rod and bob have mass and size
light weight string or rod
d
L
c.o.m.
m
and
and
?
?
about O
total mass
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Example Ch 15 1Ball bounces elastically from
4m height. Ignore air resistance. a) show that
motion is periodicb) determine period of
motionc) Explain whether motion is harmonic
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Example Ch 15 8Washing machine vibration
sensor is 1.5cm Al cube mounted on strip of very
thin, long spring steel clamped vertically to
machine. A horizontal force of 1.43N will hold
it 2.75cm away from equilibrium. Find frequency
of oscillation when released.
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Example Ch 15 10Gasoline engine piston
operates at 3600rpm in simple harmonic motion
with maximum displacement of /- 5cm. Find max
velocity and acceleration of piston.
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Example Ch 15 36Light rigid rod 0.5m long
extends out from a meter stick oscillating while
suspended from pivot point at far end. a) find
period of oscillationb) by what does period
differ from 1m long simple pendulum?
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Questions for Groups

• 15.4 Can the following be in the same direction
  for a simple harmonic oscillator?
• Position and velocity
• Velocity and acceleration
• Position and acceleration
• 15.8 A block spring system has simple harmonic
  motion of amplitude A.
• Does total energy change if mass is doubled but
  amplitude is constant? Why?
• Do the potential and kinetic energies depend on
  the mass? Why?
• 15.12 Simple pendulum hangs from ceiling of a
  car coasting freely down a hill
• Is the equilibrium position of the pendulum
  vertical? Why?
• Does the period differ from a pendulum in a
  stationary car? Why?