Animated Solution for Physics - Rotational Motion: From a solid sphere of mass $M$ and radius $R$, a cube of maximum possible volume is cut. Moment of inertia of cube about an axis passing through its centre and perpendicular to one of its faces is

Imagine a solid sphere of radius $R$. We are tasked with carving out the largest possible cube from this sphere. To maximize the volume of the cube, its vertices must touch the inner surface of the sphere. Geometrically, this means the body diagonal of the cube must be exactly equal to the diameter of the sphere.

Let the side length of the cube be $a$. The body diagonal of a cube is given by $\sqrt{3}a$. Equating this to the diameter of the sphere, we get:

From this relation, we can easily express the side length of the cube in terms of the sphere's radius:

Since the cube is carved out of the uniform solid sphere, both the sphere and the cube share the same uniform density $\rho$. The density of the sphere is its total mass $M$ divided by its total volume:

The mass of the cube, let's call it $m$, is simply its volume $a^3$ multiplied by this density $\rho$. Substituting the values we found:

Let's simplify this expression. Cubing the term gives $\frac{8R^3}{3\sqrt{3}}$. The $R^3$ and $3$ cancel out beautifully, leaving us with the mass of the cube:

Now, we need to find the moment of inertia of this cube about an axis passing through its center and perpendicular to one of its faces. For a uniform cube, the moment of inertia about such an axis is identical to that of a square plate of the same mass and side length. The formula is:

We have both the mass $m$ and the side length $a$. Let's substitute these values into our moment of inertia formula:

Squaring the side gives $\frac{4R^2}{3}$. Multiplying everything together, we arrive at our final answer: