Two spheres of equal masses, one of which is a thin spherical shell and the other solid sphere, have the…
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
$3: 5$
$\sqrt{3}: \sqrt{5}$
$\sqrt{3}: \sqrt{7}$
$5: 7$
Solution
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),
$\begin{array}{ll}
& \frac{2}{3} M R_1^2=\frac{2}{5} \mathrm{MR}_2^2 \\
\therefore \quad & \frac{\mathrm{R}_1^2}{\mathrm{R}_2^2}=\frac{3}{5} \\
\therefore \quad & \mathrm{R}_1: \mathrm{R}_2=\sqrt{3}: \sqrt{5}
\end{array}$
.