Q7.6 Thinking about SHM (T)
The period of a body oscillating in SHM is independent of the amplitude of the motion. Explain why this might seem surprising and why it is nevertheless true.
A ball released from height h bounces up and down elastically on a flat surface. Sketch its height above the surface as a function of time. What is the amplitude of the motion? What is the period? Is this SHM?
Hint
Reveal

Hint
Isn’t it surprising that it takes the same time to cover a larger distance? While SHM is periodic motion, periodic motion is not necessarily SHM.Solution
Reveal

Solution
The larger the amplitude the bigger is the distance a particle travels in the course of one SHM cycle. One gets used to the idea that ‘bigger distances’ take ‘bigger times’. But that kind of thinking is too sloppy here. In making the amplitude bigger we are also increasing the amount of energy we feed in to the system; so we are ensuring the particle travels its longer route at a bigger average speed. The conspiracy in systems exhibiting SHM is that these two effects (longer route; bigger average speed) precisely compensate for one another to give a period that is independent of amplitude.
A ball dropped from height h follows the equation ![]() until it hits the ground, at time |
This motion is periodic: it comprises a repeating cycle of
duration (period) . But the motion is not simple harmonic.
One can ‘see’ this immediately from the fact that the plot of y against t does not have the familiar sinusoidal form characteristic of SHM. One can also ‘see’ it in the fact that the period of the motion now explicitly depends on its amplitude (one might think of h/2 as being an amplitude). But these are only symptoms; to understand at a deeper level we have to think about forces.
The ball experiences two forces: its weight (gravity) and the reaction force at the floor. The weight is constant; the reaction force is zero when the ball is in the air and grows to very large values as the ball is squashed down on the floor, during its bounce. The net force is thus not a simple linear function of displacement from the floor.