Question
A particle, with restoring force proportional to displacement and resisting force proportional to velocity is subjected to a force $$F\sin \omega \,t.$$ 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
A.
$${\omega _1} = {\omega _0}\,\,{\text{and}}\,\,{\omega _2} \ne {\omega _0}$$
B.
$${\omega _1} = {\omega _0}\,\,{\text{and}}\,\,{\omega _2} = {\omega _0}$$
C.
$${\omega _1} \ne {\omega _0}\,\,{\text{and}}\,\,{\omega _2} = {\omega _0}$$
D.
$${\omega _1} \ne {\omega _0}\,\,{\text{and}}\,\,{\omega _2} \ne {\omega _0}$$
Answer :
$${\omega _1} \ne {\omega _0}\,\,{\text{and}}\,\,{\omega _2} = {\omega _0}$$
Solution :
In harmonic oscillator, the energy is maximum at $${\omega _2} = {\omega _0}$$ and amplitude is maximum at frequency $${\omega _1} < {\omega _0}$$ in the presence of damping, so $${\omega _1} \ne {\omega _0}$$ and $${\omega _2} = {\omega _0}.$$