# Features Of A Simple Harmonic Motion

For a simple harmonic motion (SHM) to occur, the following elements need be present:

• The SHM is always stationary at the maximums and minimums. For example, a when a pendulum is swinging at it’s highest, it will always be stationary at that moment.
• The displacement after an infinite amount of time will end up at zero where the equilibrium and displacement=0 is at the middle point.
• The equilibrium will have maximum velocity.
• There is negative and positive displacement from the equilibrium position.
• Velocity is always zero when displacement is at a maximum.
• Kinetic energy –> Potential energy –> Kinetic energy.
• The SHM will always have a fixed time period.

## Graphs Of SHM

For simple harmonic motion, the force always pulls back towards the middle with the force being proportional to the distance from the middle.
So, mathematically…

F $propto !,$ -x

Where x = displacement and F = restoring force. Due to force causing acceleration…

$propto !,$ -x

Where a = acceleration. After long calculations, we can some to the solution as:

x = Acos(2πf)t

Where A = amplitude, f = frequency and t = time.

As you can see, these graphs are sinusoidal in the sense that they follow the patter of a sine graph.

## Summary

For simple harmonic motion to occur, force (and acceleration) must be:
• Proportional to the distance from equilibrium position.
• Pointing towards the equilibrium position.
• The displacement of a SHM is x = Acos(2πf)t.
• The gradient of displacement against time graph produces the velocity against time graph. This is because V = dx/dt (change in displacement / change in time).
• The gradient of velocity against time graph produces the acceleration against time graph. This is because a = dv/dt (change in velocity / change in time).