Question

A particle with restoring force proportional to the displacement and resisting force proportional to velocity is subjected to a force \[F\sin \omega \]. If the amplitude of the particle is maximum for \[\omega = {\omega _1}\]and the energy of the particle is maximum for \[\omega = {\omega _2}\]then (where \[{\omega _0}\]natural frequency of oscillation of particle)
A. \[{\omega _1} = {\omega _0}\]and \[{\omega _2} \ne {\omega _0}\]
B. \[{\omega _1} = {\omega _0}\]and \[{\omega _2} = {\omega _0}\]
C. \[{\omega _1} \ne {\omega _0}\]and \[{\omega _2} = {\omega _0}\]
D. \[{\omega _1} \ne {\omega _0}\]and\[{\omega _2} \ne {\omega _0}\]

Answer 1

Hint:In this question, we are given with a particle whose restoring force is proportional to the displacement and resisting force is proportional to velocity which is subjected to a force \[F\sin \omega \]. In order to find the condition at which the amplitude of the particle is maximum and the energy is maximum, we have to apply the concept of resonance.

Complete step by step solution:
We know, resonance is basically a phenomenon where a body is set into oscillation of high amplitude by the influence of another vibrating body having the same natural frequency.

Given that \[{\omega _0}\] natural frequency of oscillation of the particle. We know that the amplitude and the velocity resonance occur at the same frequency and since restoring force is proportional to the displacement and resisting force is proportional to velocity, we can say
\[{\omega _1} = {\omega _0}\] and \[{\omega _2} = {\omega _0}\]

Therefore option B is correct.

Note: We must note that the restoring force and the resisting force are totally different, in restoring force the particle in motion always tends to come back to its equilibrium position as in case of spring and in case of resisting force the particle in motion do not come back to its equilibrium position, like friction force.