Plotting the displacement, velocity and acceleration on the same axes looks like this.

Print out a copy of the graph and measure the slopes of the x/t and v/t graphs to find instantaneous velocities and accelerations. Check your answers by calculation (the best way to do it – exams sometimes penalize if you try to measure slopes.)

The graph shows that for a body moving with SHM, displacement is proportional to acceleration, but opposite in sign. In an exam, it’s better to say that a is proportional to x and always directed to the zero of displacement.

Since max v = rω, velocity varies with time as:

The slope of the v/t graph gives the acceleration, so:

but x = rsinωt, so

The constant of proportionality is ω^{2 }

f being the frequency of oscillation (Hz) = 1/time period, T

This equationdefines motion which is simple harmonic.

Use the graph to find the time period of the oscillations.

Answer: (careful – vertical and horizontal axes must match – they do here…)