Simple harmonic motion

Summary

In the Standard Level waves section we established the definition of simple harmonic oscillations: the acceleration of an oscillating body is proportional to displacement but in the opposite direction.

The AHL Waves requirements are more quantitative - displacement, velocity, acceleration, time period, kinetic energy and potential energy must now be calculated at any time.

We recommend that you revise Oscillations before studying this page.

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Definition

Simple harmonic motion is defined as motion where the acceleration is directly proportional to the displacement from a fixed point and always acts towards that point.

This definition can be represented by an equation:

\(a  = -ω^2x\)

  • \(a\) is acceleration (ms-2)
  • \(\omega\) is angular frequency: \(\omega=2\pi f={2\pi\over T}\) (Hz)
  • \(x\) is displacement (m)

NB: \(f={1\over T}\)

Motion equations

If we assume that the clock starts when the object is released from maximum displacement (the amplitude): 

\(x=x_0\cos{\omega t} \Rightarrow v=-\omega x_0\sin \omega t \)

  • \(x_0\) is amplitude (m)
  • \(t\) is time (s)
  • \(v\) is velocity (ms-1)

This velocity is derived by differentiating the displacement equation. Velocity can be found at any time by finding the gradient of displacement. Here the combination is shown (but the maxima of the three would not, in reality, be the same).

\(\)Alternatively, time might 'commence' when the object passes through equililbrium:

\(x=x_0\sin{\omega t} \Rightarrow v=\omega x_0\cos \omega t\)

Also check out: Equations for SHM

Mass on a spring

A mass oscillating on a spring is an example of simple harmonic motion.

\(T=2\pi \sqrt{m\over k}\)

  • \(T\) is time period (s)
  • \(m\) is mass (kg)
  • \(k\) is spring constant (N m-1)

Also check out: Mass on a spring

Simple pendulum

Simple pendula are another example.

\(T=2\pi\sqrt{l\over g}\)

  • \(T\) is time period (s)
  • \(l\) is the length of the string
  • \(g\) is gravitational field strength (N kg-1)

Also check out: Simple pendulum

Stretch for 7

Energy changes

The energies involved are kinetic and potential energy. The two sum to give the total energy.

\(E_{total}=E_P+E_K\)

To commence our quantitative understanding, we need to be able to calculate velocity at any point.

\(v=\pm \omega\sqrt{{x_0}^2-x^2}\)

Kinetic energy

The usual equation for kinetic energy, \(E_k={1\over 2}mv^2\), still applies. Substituting the velocity from above:

\(E_k={1\over 2}m\omega^2({x_0}^2-x^2)\)

Total energy

Since the total energy is constant, we can find the total energy by maximising the kinetic energy (and thus minimising the potential energy to zero). We do so by considering the body at zero displacement.

\(x=0\)

\(E_T={1\over 2}m\omega^2{x_0}^2\)

Alternatively we could consider the maximum possible potential energy with the displacement as the amplitude.

Also check out: SHM and energy

Potential energy

The potential energy at any point is equal to the total energy minus the kinetic energy.

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