Physics - Rotation Question with Solution | TestHub

The moment of inertia of a hollow sphere about an axis passing through its center is given by:

$I_{CM}=\frac{2}{3}M{R}^2$

Using the parallel axis theorem, the moment of inertia about the axis $XY$ at a distance $h$ from the center is:

$I_{XY}=I_{CM}+M{h}^2$

$I_{XY}=\frac{2}{3}M{R}^2+M{h}^2$

The radius of gyration $K$ about the axis $XY$ is given as $R$. Therefore, the moment of inertia can also be written as:

$I_{XY}=M{K}^2=M{R}^2$

Equating the two expressions for $I_{XY}$:

$M{R}^2=\frac{2}{3}M{R}^2+M{h}^2$

$h^2=R^2-\frac{2}{3}{R}^2$

$h^2=\frac{1}{3}{R}^2$

$h=\frac{R}{\sqrt{3}}$

Answer: $\frac{R}{\sqrt{3}}$