Question

Showthat for a particle in linear SHM the average kinetic energy over aperiod of oscillation equals the average potential energy over thesame period.

Answer 1

Theequation of displacement of a particle executing SHM at an instant tis givenas:

Where,

A= Amplitude of oscillation

ω= Angular frequency

Thevelocity of the particle is:

Thekinetic energy of the particle is:

Thepotential energy of the particle is:

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

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

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