What is oscillatory motion

1
What is oscillatory motion?

• Oscillatory motion occurs when a force acting on
  a body is proportional to the displacement of the
  body from equilibrium.
• F ??x
• The Force acts towards the equilibrium position
  causing a periodic back and forth motion.

2
What are some examples?

• Pendulum
• Spring-mass system
• Vibrations on a stringed instrument
• Molecules in a solid
• Electromagnetic waves
• AC current
• Many other examples

3
What do these examples have in common?

• Time-period, T. This is the time it takes for one
  oscillation.
• Amplitude, A. This is the maximum displacement
  from equilibrium.
• Period and Amplitude are scalers.

4
Forces

• Consider a mass with two springs attached at
  opposite ends
• We want to find an equation for the motion.
• How should we start?
• Free-body diagram!!

5
Free body diagram
FSpring2
FGravity
FSpring1
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Fnet ma

• Fnet Fg Fs1 Fs2 ma
• Fnet ?Fhorizontal ?Fvertical
• Let us assume the mass does not move up and down
  ? ?Fvertical 0
• So, Fnet ?Fhorizontal FS-horizontal(12)
• Thus, ma m(d2x/dt2) -kx

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Fnet kx
8

• ma m(d2x/dt2) -kx
• Let k/m ??
• (d2x/dt2) ??x 0
• This is the second order differential equation
  for a harmonic oscillator. It is your friend. It
  has a unique solution

9
Simple Harmonic motion

• The displacement for a simple harmonic oscillator
  in one dimension is
• x(t) Acos(?t ??
• ??is the angular frequency. It is constant.
• ??is the phase constant. It depends on the
  initial conditions.
• What is the velocity?
• What is the acceleration?

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• Velocity differentiate x with respect to t.
  dx/dt
• v(t) -?Asin(?t ?)
• Acceleration differentiate v with respect to t.
  dv/dt
• a(t) -??Acos(?t ?)

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X(t)Acos(?t ?)
x
???
Xo
A
t
T
12
a(t) -??Acos(?t ?)
a
t
??A
t
ao
T
13
Using data

• The accelerometer will give us all the
  information we need to confirm our analysis
• We can measure all the parameters of this
  particular system and use them to predict the
  results of the accelerometer.

14
What can we measure without the accelerometer?

• The mass, m
• Hookes constant, k
• Thats all!
• T 2?/?? 2??m/k)1/2 (Recall ??? k/m)
• Everything else depends on the initial
  conditions. What does this tell us?
• The time period, T, is independent of the initial
  conditions!

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Energy

• The system operates at a particular frequency, v,
  regardless of the energy of the system.
• v 1/T 2?(k/m)1/2
• The energy of the system is proportional to the
  square of the amplitude.
• E (1/2)kA2

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Proof of E(1/2)kA2

• Kinetic Energy ? (1/2)mv2
• V ??SIN(?t ?)
• ??? (1/2)M??A?SIN2 (?t ?)
• Elastic potential energy U(1/2)kx2
• x Acos(?t ?)
• U ? (1/2)kA2cos2./(?t ?)

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• E K U
• (1/2)kA2sin2 (?t ?) cos2 (?t ?)
• (1/2)kA2

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Damping

• Simple harmonic motion is really a simplified
  case of oscillatory motion where there is no
  friction (remember our FBD)
• For small to medium data sets this will not
  affect our results noticeably.

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for the rest of class...

• We are going to find k and m and compare to the
  results of the accelerometer

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Some cool oscillatory motion websites

• http//www.phy.ntnu.edu.tw/ntnujava/viewtopic.php?
  t236
• http//www.kettering.edu/drussell/Demos/SHO/mass.
  html
• http//farside.ph.utexas.edu/teaching/301/lectures
  /node136.html
• http//www.physics.uoguelph.ca/tutorials/shm/Q.shm
  .html