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. | Learn NCERT solution | Education portal Class 11 Physics | Study online Unit-14 Oscillations



Q.22:- 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:-
The equation of displacement of a particle executing SHM at an instant t is given as:
x = Asin ωt
where,
A = Amplitude of oscillation
ω = Angular frequency = √k/M

The velocity of the particle is:v = dx/dt = Aωcosωt
The kinetic energy of the particle is:
Ek = 1/2 Mv2 = 1/2 MA2 ω2 cos2 ωt
The portential energy of the particle is:
Ep = 1/2 kx2 = 1/2 M2 ω2 A2 sin2 ωt

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.