© All Rights reserved @ LearnWithDash
Step-by-Step Solution
Step 1: Identify the side of the largest cube inside the sphere
A cube of maximum volume is inscribed in a sphere of radius $R$ by making the sphere’s diameter coincide with the cube’s body diagonal. For a cube of side $a$, its body diagonal has length $a\sqrt{3}$. This diagonal must equal the sphere’s diameter $2R$. Hence,
$$
a\sqrt{3} = 2R \quad \Longrightarrow \quad a = \frac{2R}{\sqrt{3}}.
$$
Step 2: Determine the mass of the cube
Let the sphere have uniform mass density. The sphere’s mass is given as $M$ and its volume is
$$
V_{\text{sphere}} = \frac{4}{3}\,\pi R^3.
$$
Thus, the density of the sphere is
$$
\rho = \frac{M}{\frac{4}{3}\,\pi R^3} = \frac{3M}{4\pi R^3}.
$$
The volume of the cube is
$$
V_{\text{cube}} = a^3
= \left(\frac{2R}{\sqrt{3}}\right)^3
= \frac{8\,R^3}{3\sqrt{3}}.
$$
Hence the mass of the cube (cut from the same material) is
$$
M_{\text{cube}}
= \rho \times V_{\text{cube}}
= \frac{3M}{4\pi R^3}\;\times\;\frac{8\,R^3}{3\sqrt{3}}
= \frac{2\,M}{\pi \sqrt{3}}.
$$
Step 3: Moment of inertia of a uniform cube about an axis through its center and perpendicular to one face
For a cube of mass $m$ and side $a$, the moment of inertia about an axis passing through its center and perpendicular to one of its faces is
$$
I = \frac{m\,a^2}{6}.
$$
In our case,
$$
m = M_{\text{cube}} = \frac{2\,M}{\pi \sqrt{3}},
\quad
a = \frac{2R}{\sqrt{3}}.
$$
Substitute these into the formula:
$$
I_{\text{cube}}
= \frac{1}{6}\,\biggl(\frac{2\,M}{\pi \sqrt{3}}\biggr)\,\biggl(\frac{2R}{\sqrt{3}}\biggr)^{2}.
$$
Since
$$
\biggl(\frac{2R}{\sqrt{3}}\biggr)^{2} \;=\; \frac{4\,R^{2}}{3},
$$
we get
$$
I_{\text{cube}}
= \frac{1}{6}\;\times\;\frac{2\,M}{\pi \sqrt{3}}\;\times\;\frac{4\,R^{2}}{3}
= \frac{8\,M\,R^{2}}{18\,\pi \sqrt{3}}
= \frac{4\,M\,R^{2}}{9\,\pi \sqrt{3}}.
$$
Step 4: Expressing in the form given by the problem (official answer)
The result is often presented equivalently in the form:
$$
\boxed{\,I_{\text{cube}} \;=\;\frac{4\,M\,R^{2}}{9\,\sqrt{3\,\pi}}\,.}
$$
This matches the answer choice
provided in the question statement.
Reference Image
