c
Torque on rod at displacement \(\theta\) from mean position \(\theta\) is very small. \(x=\frac{L}{2} \theta\)

\(\tau=2 k x \frac{L}{2}=2 k \frac{L^{2}}{4} \theta=\frac{k L^{2}}{2} \theta\)

Now, \(\tau=1 \alpha\)

\(\frac{\mathrm{kL}^{2}}{2} \theta=\frac{\mathrm{mL}^{2}}{12} \alpha \quad ; \quad \alpha=\frac{6 \mathrm{k}}{\mathrm{m}} \theta\)

\(\tau=\frac{\omega}{2 \pi}=\frac{1}{2 \pi} \sqrt{\frac{6 k}{m}}\)