Simple Harmonic Motion#0

Moment arm: $\frac{L}{2} + x$

Force magnitude for small oscillations:

$F = k \, \Big(\frac{L}{2} + x \Big) \, \theta$

Torque:

$\tau = k \, \Big(\frac{L}{2} + x \Big)^2 \, \theta$

Moment of inertia:

$I = \frac{M L^2}{12} + M x^2$

Rotational acceleration:
$k \, \Big(\frac{L}{2} + x \Big)^2 \, \theta = \Big(\frac{M L^2}{12} + M x^2 \Big) \, \ddot{\theta} \\ \ddot{\theta} = \frac{k \, \Big(\frac{L}{2} + x \Big)^2}{\frac{M L^2}{12} + M x^2 } \, \theta \\ \omega^2 = \frac{k \, \Big(\frac{L}{2} + x \Big)^2}{\frac{M L^2}{12} + M x^2 }$

Differentiate to maximize the angular frequency:

$\frac{d \, \omega^2}{d \, x} = 0 \\ \implies \Big(\frac{M L^2}{12} + M x^2 \Big) 2k \, \Big(\frac{L}{2} + x \Big) = 2 M k x \, \Big(\frac{L}{2} + x \Big)^2 \\ \frac{M L^2}{12} + M x^2 = M x \, \Big(\frac{L}{2} + x \Big) \\ \frac{M L^2}{12} = \frac{M x L}{2} \\ x = \frac{L}{6}$