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 :
At maximum energy of the particle, velocity resonance takes place, which occurs when frequency of external periodic force is equal to natural frequency of undamped vibrations, i.e. $${\omega _2} = {\omega _0}.$$
Further amplitude resonance takes place at a frequency of external force which is less than the frequency of undamped natural vibrations, i.e. $${\omega _1} \ne {\omega _0}.$$