It is not obvious that a sinusoidally-varying displacement is a consequence of the condition that the acceleration is proportional to displacement but in the opposite direction!
However, we can verify that the condition for simple harmonic motion is obeyed by differentiating the original function for displacement twice with respect to time.
By analysing the motion of a pendulum we find the displacement is sinusoidally related to time. This means that the displacement equation with respect to time will contain either sin or cos.
The velocity is found from the gradient of the displacement time graph. If, for example, the displacement is a sin curve then velocity will be a cos curve.
Acceleration is the gradient of the velocity time graph. Sticking with velocity as a cos curve, acceleration will be a -sin curve. This is proportional to but in the opposition direction to the displacement time curve.
Imagine watching circular motion from the side, with your eye on the same plane as the circle itself. You would see the object moving from side-to-side or up-and-down repeatedly. The equations of this motion are the same as simple harmonic motion.
This can be used to show that \(a=-\omega^2x\).
The equation for the displacement of a body executing SHM is either:
- \(x = x_o\sin(2πft)\)
- \(x = x_o\cos(2πft)\)
Use quizzes to practise application of theory.