Simple Harmonic Motion

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Simple Harmonic Motion (SHM)

SHM is any motion in which the acceleration is directed towards a fixed point (or equilibrium position for 1-D examples) AND is directly proportional to the negative of the displacement.

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  • Mathematically, the definition of SHM can be expressed in a formula:
    • a=ω2xa=-\omega^2 x
      • a is the acceleration.
      • ω=2πf\omega = 2\pi f, where f is the frequency of oscillation
      • x is the displacement.
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Acceleration-displacement graph

  • The acceleration-displacement graph is a straight-line graph that passes through the origin.
  • The angular frequency can be found from the gradient:
    • gradient =ω2= -\omega^2
    • ω=gradient\omega = \sqrt{-gradient}
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Calculating maximum acceleration

  • If we are given the time period, T (or the frequency f), ω² can be calculated.
  • The maximum acceleration is given by:
    • amax=ω2Aa_{max}=\omega^2A
      • where A is the maximum displacement.

Simple Harmonic Motion (SHM) Graphs

SHM can be represented graphically.

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Acceleration and displacement

  • The acceleration can be found graphically, and so the displacement or vice versa from a graph.
    • E.g. if you have a displacement-time graph, you can calculate the period and angular frequency.
  • The displacement at any point can be read off the graph. This means you can calculate the acceleration (and then even the net force) because a=ω2xa=-\omega^2 x.
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Velocity and displacement

  • The velocity at any given time is found from the gradient of a displacement-time graph.
  • By finding the gradient at each point in time, we can produce a velocity-time graph from a displacement-time graph.
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Velocity and acceleration

  • In a similar way, you can use a velocity-time graph to get the corresponding acceleration-time graph. This is because the gradient at a point on any velocity-time graph gives the acceleration at that moment in time.
  • You can then see that the acceleration-time graph is exactly the same shape as the displacement-time graph, reflected in the x-axis.

Simple Harmonic Equations

The condition for simple harmonic motion is that the acceleration is directed towards a fixed point and that the magnitude of the acceleration is proportional to the negative of the displacement.

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  • For SHM the acceleration must be proportional to the negative of the displacement. aαxa{\alpha}-x
  • If an object is under SHM it's displacement is found by x=Acos(ωt)x=A\cos({\omega}t) where A is the amplitude
  • The defining equation of SHM is a=ω2xa=-{\omega}^2x

Maximum points

  • The maximum displacement can be found when cos(ωt)=1\cos({\omega}t)=1
  • The maximum displacement is called the amplitude A
  • The maximum speed is given by v=ωAv={\omega}A
  • The maximum acceleration is given by a=ω2Aa={\omega}^2A

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