Question

A particle with restoring force proportional to displacement and resisting force proportional to velocity is subjected to force $Fsin$$\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 (where ${\omega }_0$ natural frequency of oscillationn of particle)

Answer 1

As given, energy of the particle is maximum for $\omega ={\omega }_2$
As we know, the energy of the particle is maximum at natural frequency
$\Rightarrow {\omega }_2={\omega }_0$
And amplitude of the particle is maximum for $\omega ={\omega }_1$
For maximum amplitude, frequency of external force will be
$\omega =\sqrt{{\omega }_0^2-{\left(\frac{b}{2m}\right)}^2}$
$\Rightarrow {\omega }_1\ne {\omega }_0$

Here b is damped constant and m is mass of particle.
Final answer :(c)