In a driven harmonic oscillator power absorbed is maximum at the

Explanation:

The external periodic force has to do work against the damping force to continue the oscillations.

In a small interval 'dt' the work done by the driving force 'F' for the displacement 'dx' is given by, Fdx, which will be equal to the energy 'dE' absorbed from the driving force.

Thus, the energy absorbed from the driving force per unit time by the harmonic oscillator will be equal to,

where F = F_o sin pt, 

where

 and 

Thus, the energy absorbed per unit time, i.e. power absorbed by the oscillator is given by.

Hence the average power absorbed by the oscillator is given by;

The average value of sin pt cos pt = 0 and sin² pt = 1/2;

Thus 

Thus substituting A and sin θ 

From the above equation, we can see that the average power will be maximum when p = ω_o and it is equal to

If the velocity amplitude of the oscillator i.e 'Ap' is plotted versus the driving frequency, it is found to be maximum at the driving frequency p = ωo.

Thus, it can be said that the maximum power is absorbed at the frequency of velocity resonance and not at the frequency of amplitude resonance.

The figure given below shows the variation of average power absorbed by the oscillator with the frequency of the driving force.