= Solution
After switching, <Kirchhoff's voltage law> gives the free <series RLC circuit> equation
$$
L\ddot q+R\dot q+q/C=0.
$$
Its characteristic roots are $-\gamma\pm i\omega_d$, where
$$
\boxed{\gamma=\frac R{2L},\qquad \omega_d=\sqrt{\frac1{LC}-\frac{R^2}{4L^2}}.}
$$
A real, nonzero oscillation <frequency> requires $R^2<4L/C$. For a passive <resistor>,
$$
\boxed{0\leq R<2\sqrt{L/C}\simeq894\ \Omega.}
$$
The endpoint $R=0$ is the undamped limit; strictly decaying oscillations have $0<R<894\ \Omega$. The critical endpoint itself does not have a nonzero $\omega_d$.
Let $Q=CV_0$ denote the initial <electric charge>, so $Q=30\ \mu\mathrm C$ and $\dot q(0)=0$. Distinguish $Q$ from the amplitude coefficient $q_0$ in an exponential-times-cosine representation: imposing zero initial <electric current> generally requires a nonzero phase.
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