Question

A solid sphere and a hollow sphere of the same mass have the same M.I. about their respective diameters. The ratio of their radii will be:
(A) 1:2
(B) $\sqrt{3}:\sqrt{5}$
(C) $\sqrt{5}:\sqrt{3}$
(D) 5:4

Answer 1

To find the ratio of radii of a solid sphere and a hollow sphere, we consider the moments of inertia of both the bodies.
Moment of inertia for a solid sphere is given by:
${{\text{I}}_{\text{S}}}=\dfrac{2}{5}\text{ }{{\text{m}}_{\text{S}}}{{\text{R}}_{\text{S}}}^{2}$
Moment of inertia for a hollow sphere is given by:
${{\text{I}}_{\text{H}}}=\dfrac{2}{3}\text{ }{{\text{m}}_{\text{H}}}{{\text{R}}_{\text{H}}}^{2}$ .

Complete step by step solution
As per question, we are given that
Mass of solid sphere = Mass of hollow sphere
i.e. ${{\text{m}}_{\text{S}}}={{\text{m}}_{\text{H}}}=\text{m}$
Also,
Their moment of inertia is equal i.e.
${{\text{I}}_{\text{s}}}={{\text{I}}_{\text{H}}}$
As ${{\text{I}}_{\text{S}}}=\dfrac{2}{5}\text{ }{{\text{m}}_{\text{S}}}{{\text{R}}_{\text{S}}}^{2}$
And ${{\text{I}}_{\text{H}}}=\dfrac{2}{3}\text{ }{{\text{m}}_{\text{H}}}{{\text{R}}_{\text{H}}}^{2}$
So
$\dfrac{2}{5}\text{ }{{\text{m}}_{\text{S}}}{{\text{R}}_{\text{S}}}^{2}=\dfrac{2}{3}\text{ }{{\text{m}}_{\text{H}}}{{\text{R}}_{\text{H}}}^{2}$
$\dfrac{3}{5}\text{ }\dfrac{{{\text{R}}_{\text{S}}}^{2}}{{{\text{R}}_{\text{H}}}^{2}}=\dfrac{{{\text{m}}_{\text{H}}}}{{{\text{m}}_{\text{S}}}}$
$\begin{aligned} & \dfrac{{{\text{R}}_{\text{S}}}^{2}}{{{\text{R}}_{\text{H}}}^{2}}=\dfrac{\text{m}}{\text{m}}\text{ }\dfrac{5}{3} \\\ & \dfrac{{{\text{R}}_{\text{S}}}^{2}}{{{\text{R}}_{\text{H}}}^{2}}=\dfrac{5}{3} \\\ & \dfrac{{{\text{R}}_{\text{S}}}}{{{\text{R}}_{\text{H}}}}=\sqrt{\dfrac{5}{3}} \\\ \end{aligned}$
$\therefore \text{ }{{\text{R}}_{\text{S}}}:{{\text{R}}_{\text{H}}}=\sqrt{5}:\sqrt{3}$
The option (C) is the correct answer.

Note
Moment of inertia is the quantity expressing a body’s tendency to resist angular acceleration. It is about the axis of rotation of the body. In the question, it is given that the moment of inertia is about the respective diameters that means it is about their radii and hence, we did not consider any parallel plane to calculate the moments of inertia and so the ratio of their radii.