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Simple harmonic motion Cambridge International AS & A Level Physics revision

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In plain words

A child on a swing, a weight bobbing on a spring, a guitar string: each moves to and fro about a middle position, and each is pulled back harder the further it strays. When the pull back is exactly proportional to the distance from the middle, the motion has a beautifully regular form called simple harmonic motion.

5 things to know

  1. Simple harmonic motion: the acceleration is proportional to the displacement from a fixed point and is always directed towards that point: a = −ω²x. That happens whenever the restoring force is F = −kx.
  2. ω is the angular frequency: ω = 2πf = 2π ÷ T. The period does not depend on the amplitude.
  3. The displacement is x = A sin ωt if timing starts at the centre, or x = A cos ωt if it starts at one end. A is the amplitude.
  4. The speed is greatest at the centre, where it is ωA. The acceleration is greatest at the ends, where it is ω²A.
  5. On graphs against time: velocity is the gradient of the displacement graph, and acceleration is the gradient of the velocity graph. Velocity is a quarter of a cycle out of step with displacement, and acceleration is exactly opposite to displacement.

Worked example

An object moves in simple harmonic motion with an amplitude of 0.050 m and a period of 0.40 s. Find its greatest speed and its greatest acceleration.

  1. ω = 2π ÷ T = 2π ÷ 0.40 = 15.7 rad/s.
  2. Greatest speed = ωA = 15.7 × 0.050 = 0.79 m/s.
  3. Greatest acceleration = ω²A = 15.7² × 0.050 = 12 m/s².

Tips and tricks

  • Set your calculator to radians before working out sin ωt or cos ωt.
  • The minus sign in a = −ω²x is the whole point: the acceleration is always directed back towards the centre.
5 questions, about 2 minutes.

It lands in your notebook with its questions as flashcards.

Simple harmonic motion: 5 questions and answers

These are the quiz’s questions. Do the quiz first, then come back here for the ones that got you.

  1. What does the minus sign in a = −ω²x show?
    • The acceleration is in the opposite direction to the displacement. (the answer)
    • The object is slowing down.
    • The object is moving backwards.
    • The displacement is negative.

    The acceleration always points back towards the centre.

  2. Where is the speed of a simple harmonic oscillator greatest?
    • at the centre (the answer)
    • at the ends
    • halfway between the centre and an end
    • it is the same everywhere

    All its energy is kinetic there.

  3. The amplitude of a simple harmonic oscillator is doubled. What happens to its period?
    • It stays the same. (the answer)
    • It doubles.
    • It halves.
    • It becomes four times bigger.

    The period does not depend on the amplitude.

  4. An oscillator has a period of 0.50 s. What is its angular frequency?
    • 0.50 rad/s
    • 2.0 rad/s
    • 3.1 rad/s
    • 12.6 rad/s (the answer)

    ω = 2π ÷ T.

  5. What is the phase difference between the displacement and the acceleration of a simple harmonic oscillator?
    • 0°
    • 90°
    • 180° (the answer)
    • 360°

    They are always in opposite directions.

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