Question

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

• A. $5:7$
• B. $3:5$
• C. $\sqrt{3}:\sqrt{5}$
• D. $\sqrt{3}:\sqrt{7}$

Answer 1

Answer: (C)

Let the radii of the thin spherical and the solid sphere are $R_{1}$ and $R_{2}$ respectively.
Then the moment of inertia of the spherical shell about their diameter
$I=\frac{2}{3} \, MR_{1}^{2}$ ...(i)
and the moment of inertia of the solid sphere is given by
$I=\frac{2}{5}MR_{2}^{2}$ ...(ii)
Given that the masses and moment of inertia for both the bodies are equal, then from Eqs. (i) and (ii)
$\frac{2}{3}MR_{1}^{2}=\frac{2}{5}MR_{2}^{2}\Rightarrow \frac{R_{1}^{2}}{R_{2}^{2}}=\frac{3}{5}$
$\Rightarrow $ $\frac{R_{1}}{R_{2}}=\sqrt{\frac{3}{5}} \, \Rightarrow R_{1}:R_{2}=\sqrt{3}:\sqrt{5}$