Question

Show that for a particle in linear SHM the average kinetic energy over a period of oscillation equals the average potential energy over the same period.

Answer 1

The equation of displacement of a particle executing SHM at an instant t is given as:

Where,

A = Amplitude of oscillation

ω = Angular frequency

The velocity of the particle is:

The kinetic energy of the particle is:

The potential energy of the particle is:

For time period T, the average kinetic energy over a single cycle is given as:

And, average potential energy over one cycle is given as:

It can be inferred from equations (i) and (ii) that the average kinetic energy for a given time period is equal to the average potential energy for the same time period.