Simple Harmonic Motion (Cambridge (CIE) A Level Physics)

Conditions for simple harmonic motion

• Simple harmonic motion (SHM) is a specific type of oscillation that occurs when acceleration is proportional to displacement from a fixed point and in the opposite direction

• An object is said to perform simple harmonic oscillations when all of the following apply:

  • The oscillations are periodic (repeating)

  • There is a central equilibrium point known as the fixed point

  • The object's displacement, velocity and acceleration change continuously

  • There is a restoring force always directed towards the fixed point

  • The magnitude of the restoring force is proportional to the displacement

  • The restoring force causes the acceleration

• The restoring force and the acceleration must always be:

  • Directed towards the equilibrium position, and hence, in the opposite direction to the displacement

  • Directly proportional to the displacement

a ∝ −x 

• Where:

  • a = acceleration (m s⁻²)

  • x = displacement (m)

Restoring force, acceleration and displacement

Force, acceleration and displacement of a simple pendulum in SHM

Examples of simple harmonic motion

• Examples of oscillators that undergo SHM are:

  • The pendulum of a clock

  • A child on a swing

  • The vibrations of a bowl

  • A bungee jumper reaching the bottom of his fall

  • A mass on a spring

  • Guitar strings vibrating

  • A ruler vibrating off the end of a table

  • The electrons in alternating current flowing through a wire

  • The movement of a swing bridge when someone crosses

  • A marble dropped into a bowl

Examples of simple harmonic motion

A pendulum, bungee jumper, swing bridge, vibrations in a prayer bowl, a swing, a ball rolling up and down the sides of a bowl and a spring are all examples of simple harmonic oscillations.

An example of not SHM

• A person jumping on a trampoline is not an example of simple harmonic motion because:

  • The restoring force on the person is not proportional to their displacement from the equilibrium position and always acts down

  • When the person is not in contact with the trampoline, the restoring force is equal to their weight, which is constant

  • This does not change, even if they jump higher

The restoring force of the person bouncing is equal to their weight and always acts downwards