In the $LCR$ circuit shown in the figure,the ac driving voltage is $V = V_m \sin \omega t$.
$(a)$ Write down the equation of motion for $q(t)$.
$(b)$ At $t = t_0$,the voltage source is removed and $R$ is short-circuited. Write down the energy stored in each of $L$ and $C$ at this instant.
$(c)$ Describe the subsequent motion of charges.

(N/A) Applying Kirchhoff's voltage law to the $LCR$ circuit:
$L \frac{d^2q}{dt^2} + R \frac{dq}{dt} + \frac{q}{C} = V_m \sin \omega t$
$(b)$ At $t = t_0$,the current is $i(t_0) = \frac{dq}{dt}|_{t=t_0}$ and the charge on the capacitor is $q(t_0)$.
The energy stored in the inductor is $U_L = \frac{1}{2} L [i(t_0)]^2$.
The energy stored in the capacitor is $U_C = \frac{1}{2} \frac{[q(t_0)]^2}{C}$.
$(c)$ After $t = t_0$,the circuit becomes an $LC$ circuit with $R=0$. The total energy $U = U_L + U_C$ is conserved. The charge $q(t)$ will oscillate sinusoidally with an angular frequency $\omega_0 = \frac{1}{\sqrt{LC}}$.