Study this applet of a mass/spring oscillator.
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 ω2
f being the frequency of oscillation (Hz) = 1/time period, T
This equation defines 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…)