Edge-hinged cube with a central spherical cavity ($\Omega /\omega =\sqrt{(1-\pi /6)/(1-3\pi /20)}$)

For a uniform cube of mass $M$ and side $a$ hinged along an edge, the centre is $a/\sqrt{2}$ from the hinge and $I_h=2M{a}^2/3$. Removing a concentric sphere of radius $a/2$ removes mass $\pi M/6$ and hinge inertia $\pi M{a}^2/10$, without shifting the centre. Combining the reduced weight and reduced inertia in ${\omega }^2=Mgd/I_h$ proves the displayed frequency ratio. Updating only the inertia would miss the change of gravitational restoring torque.