= Edge-hinged cube with a central spherical cavity
{title2=$\Omega/\omega=\sqrt{(1-\pi/6)/(1-3\pi/20)}$}
For a uniform cube of <mass> $M$ and side $a$ hinged along an edge, the centre is $a/\sqrt2$ from the hinge and $I_h=2Ma^2/3$. Removing a concentric <sphere> of <radius> $a/2$ removes <mass> $\pi M/6$ and hinge inertia $\pi Ma^2/10$, 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>.
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