Why doesn't amplitude affect the period in SHM?

Amplitude doesn't affect the period in Simple Harmonic Motion (SHM) because the period depends solely on the mass and restoring force.

In Simple Harmonic Motion (SHM), the period is the time taken for one complete cycle of the motion. It is determined by the properties of the system, specifically the mass of the object in motion and the strength of the restoring force. The restoring force is the force that brings the object back to its equilibrium position. In SHM, this force is proportional to the displacement from the equilibrium position, but it does not depend on the amplitude of the motion.

The amplitude, on the other hand, is the maximum displacement from the equilibrium position. It is a measure of the energy in the system, but it does not affect the time it takes for the object to complete one cycle of motion. This is because the increased energy associated with a larger amplitude is balanced by an increased speed of the object. As the object moves further from the equilibrium position, the restoring force (and therefore the acceleration) increases, causing the object to move faster. This means that even though the object has further to travel when the amplitude is larger, it also travels at a higher speed, so the time taken for one complete cycle remains the same.

This principle is a fundamental characteristic of SHM and is what distinguishes it from other types of oscillatory motion. In other types of motion, the period can depend on the amplitude. For example, in a pendulum, if the amplitude is large (i.e., the pendulum is released from a large angle), the period will be slightly longer than if the amplitude is small. This is because the restoring force is not exactly proportional to the displacement, as it is in SHM. However, for small amplitudes, the period of a pendulum is approximately independent of the amplitude, and the motion is approximately simple harmonic.

In conclusion, the amplitude does not affect the period in SHM because the period is determined by the mass and restoring force, not the energy in the system. The increased speed associated with a larger amplitude balances the increased distance to be travelled, resulting in a constant period.