Question

If ${\omega _0}$ is the natural frequency of damped forced oscillation and p that of driving force, then for amplitude resonance.
(A) ${p_{r}} = {\omega _0}$
(B) ${p_r} < {\omega _0}$
(C) ${p_r} > {\omega _0}$
(D) ${p_r} > = {\omega _0}$

Answer 1

This question uses the concept of damped and forced vibrations. You can solve this question easily if you know about the situation when the amplitude resonance takes place.

Complete step by step answer:
Suppose if a vibrating system is vibrating with its own natural frequency and an external period force is being superimposed over it. Then the driving force tries to impress its own natural frequency over the system. Later, the system due to external force starts to oscillate with a steady amplitude along with the frequency of the force. These oscillations are known as forced oscillation, and the oscillating system is known as a driven harmonic oscillator. Further, the external force is termed as a driving force. Moreover, when the driving frequencies are comparable with that of natural frequency, the phenomenon or situation of amplitude becoming a maximum is called amplitude resonance. The driving frequency of that particular amplitude is known as the resonance frequency.

The condition at which we can say that amplitude resonance took place, is when the natural frequency of the driving force is slightly or somewhat lesser than that of the natural frequency of the damped forced oscillation. This means, the natural frequency of the oscillator must be greater than the frequency of driving force, for amplitude resonance to occur.
Mathematically, it can be written as,
${p_r} < {\omega _0}$
Here, ${\omega _0}$ is the natural frequency of the damped forced oscillator and ${p_r}$ is the frequency of the driving force.

Thus, option (B) is correct, that is ${p_r} < {\omega _0}$.

Note:
You need to know about the driving force, and oscillation in order to answer this question. You do not have to apply the phenomenon of displacement of the forced oscillation here to solve the question, or otherwise you can go wrong.