Question

A solid sphere of mass $M$ and radius $R$ cannot have moment of inertia
(A) $\dfrac{2}{3}M{R^2}$
(B) $\dfrac{1}{2}M{R^2}$
(C) $\dfrac{2}{5}M{R^2}$
(D) $\dfrac{2}{7}M{R^2}$

Answer 1

Moment of Inertia of a solid sphere along its diameter, which is the ais passing through its center of mass, is $I = \dfrac{2}{5}M{R^2}$. Use the parallel axis theorem to get a relationship between the moment of inertia of a body about the axis passing through the center of mass and other axes.

Complete step by step solution:
We here have a solid sphere with mass $M$ and radius $R$. The moment of inertia of such a sphere about the axis passing through the center can be easily found to be ${I_{CM}} = \dfrac{2}{5}M{R^2}$ .
Now according to the parallel axis theorem, for an axis through any other point $P$ which is at a distance of $d$ from the center of mass of the body, is given by $I = {I_{CM}} + M{d^2}$.
We can now observe that the quantity $M{d^2}$ is always positive and so we have the inequality $I \geqslant {I_{CM}}$ for an axis through any point $P$.
In other words, ${I_{CM}}$ is the least possible inertia for a body. So, any possible inertia of the body should be greater than or equal to ${I_{CM}}$ . As already mentioned, for a solid sphere ${I_{CM}} = \dfrac{2}{5}M{R^2}$. So, we will now check this for each and every option and find the inertia that the body cannot possibly have.
$\dfrac{2}{3}M{R^2} > \dfrac{2}{5}M{R^2} = {I_{CM}}$. So, it can be a moment of inertia of the sphere.
$\dfrac{1}{2}M{R^2} > \dfrac{2}{5}M{R^2} = {I_{CM}}$. So, this value is also a possible value.
$\dfrac{2}{5}M{R^2} = {I_{CM}}$ . Here this value is the least possible moment of inertia for the body and so it surely can have such a value as its moment of inertia.
But here, $\dfrac{2}{7}M{R^2} < \dfrac{2}{5}M{R^2} = {I_{CM}}$. So, this value cannot be a possible value for the moment of inertia of the body.
Hence, the correct option is D. $\dfrac{2}{7}M{R^2}$ .

Note:
Remember that it is easy to confuse the moment of inertia of a solid sphere along its diameter $I = \dfrac{2}{5}M{R^2}$ with that of a hollow sphere $I = \dfrac{2}{3}M{R^2}$ . Always do not forget to take a brief look at the axis mentioned before proceeding with solving the problem.