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