Simple Harmonic motion – SHM

The full definition of Simple Harmonic Motion (SHM) is below (the most important bit in bold type). But without looking at the examples that might not help much!

There is SHM when:

  • The motion is in a straight line.
  • The motion is symmetrical about a centre point.
  • The force and therefore the acceleration, is directly proportional to the displacement directed towards the centre point.

A good example of SHM is twanging a ruler held firmly at one end on a table.

Example of SHM a vibrating ruler

The examples of SHM most used in schools are an oscillating spring and a pendulum. A light spring with a mass attached bouncing up and down perfectly satisfies the conditions for simple harmonic motion.

Measurements on SHM for a spring

The equation connecting the variables of the motion is: T=2π√m/k

where m is the mass on the end of the spring and k is the spring constant (see below). The equation assumes that the mass of the spring is insignificant compared to the mass on the end.


We can measure the time period directly, using a watch/timer. The spring constant is defined as the force required to extend the spring by 1 metre. Now that sounds a bit daft because most springs will fail if you stretch them that much. However it is just "theoretical. If we test to find the force needed to extend by say 0.20m then the theoretical force to stretch to 1m would be 5 times as much. To get an accurate value of k we go about it like this:

To find the spring constant k we add known weights to the spring and measure the extension (that is the amount it stretches).  When we graph these we can check for any errors in measurement - the graph should be a perfect straight line.

Graph used to find the spring constant

The spring constant is the gradient of the graph, in this example 0.5/0.538  = 0.93 Nm-1

The other pages of notes on SHM may be useful to you:

  Simple Harmonic Motion (SHM) of a spring   

  Simple Harmonic Motion (SHM) of a pendulum   

  SHM graphs for displacement, velocity and acceleration   

   SHM and energy changes

The video below explains the essentials of SHM with examples including pendulum and oscillating spring.