Two spheres of equal masses, one of which is a thin spherical shell and the other solid sphere, have the same moment of inertia about their respective diameters. The ratio of their radii is

Let the radii of the thin spherical shell and the solid sphere be \(R_1\) and \(R_2\), respectively. Then, the moment of inertia of the shell about its diameter is given by,
\(\mathrm{I}_{\text {shell }}=\frac{2}{3} \mathrm{MR}_1^2\)...(i)
And the moment of inertia of the solid sphere is given by,
\(\mathrm{I}_{\text {sphere }}=\frac{2}{5} \mathrm{MR}_2^2\)...(ii)
Given that, the masses and moment of inertia for both the bodies are equal, then from equations (i) and (ii),
\(\begin{array}{ll}
& \frac{2}{3} M R_1^2=\frac{2}{5} \mathrm{MR}_2^2 \\
\therefore \quad & \frac{\mathrm{R}_1^2}{\mathrm{R}_2^2}=\frac{3}{5} \\
\therefore \quad & \mathrm{R}_1: \mathrm{R}_2=\sqrt{3}: \sqrt{5}
\end{array}\)