Question

In forced oscillation of a particle the amplitude is maximum for the frequency \[{\omega _1}\]of the force while the energy is maximum for the frequency \[{\omega _2}\]of the force, then
A. \[{\omega _1} < {\omega _2}\]when damping is small and \[{\omega _1} > {\omega _2}\]when damping is large
B. \[{\omega _1} > {\omega _2}\]
C. \[{\omega _1} = {\omega _2}\]
D. \[{\omega _1} < {\omega _2}\]

Answer 1

Hint: Before moving towards a solution, first we have to know about the forced oscillator. A forced oscillator is an oscillator that oscillates when an external periodic force is applied to it and their oscillation is known as forced oscillation. Those oscillations whose amplitude is decay with time is known as damped oscillation while those oscillations whose amplitude is constant with time is known as undamped oscillation

Complete step by step solution:
Let \[{F_1}\] force be applied to the forced oscillator at the frequency \[{\omega _1}\] to get maximum amplitude. And \[{F_2}\] is the force applied at the frequency \[{\omega _2}\]to get maximum energy. So, it is given that both amplitude and energy are maximum on the application of force.

We know that energy and amplitude are maximum only when a system oscillates with resonating frequency. And this resonating frequency is equivalent to the natural frequency.
i.e. \[{\omega _1} = {\omega _2} = {\omega _0}\]
Here, \[{\omega _0}\] is the natural frequency.
When an oscillator continues its oscillation without any external force, the frequency of that oscillation is known as natural frequency.

Hence, the option C is correct.

Note: Resonating frequency is the frequency of an oscillator at which it acquires the highest value of amplitude. The amplitude of an oscillation is the maximum distance from its mean position. Frequency is the number of oscillations completed in one second.