Difficult# Rotational Inertia of a Cube with a Spherical Hole

CLMECH-PSIS45

A uniform cube of mass $M =3.5 \space kg$ and the side length $a =1.2 \space m$ has a spherical hole inside it (see the figure). The radius of the hole is equal to $a/2$. The hole touches the middle of each side of the cube. A vertical axis of rotation is parallel to one of the sides and tangential to the spherical hole, as shown in the figure. Find the rotational inertia of the system around this axis.