A solid metallic sphere of radius ' $R$ ' having moment of inertia ' I ' about diameter is melted and recast…
A solid metallic sphere of radius ' $R$ ' having moment of inertia ' I ' about diameter is melted and recast into a solid disc of radius ' $r$ ' of a uniform thickness. The moment of inertia of a disc about an axis passing through its edge and perpendicular to its plane is also equal to ' I '. The ratio $\frac{r}{R}$ is
$\frac{1}{\sqrt{2}}$
$\frac{2}{\sqrt{5}}$
$\frac{2}{\sqrt{10}}$
$\frac{2}{\sqrt{15}}$
Solution
M.I. of the solid sphere about a diameter
$\mathrm{I}=\frac{2}{5} \mathrm{MR}^2...(i)$
M.I. of the disc about an axis through its edge and perpendicular to its plane is
$\begin{array}{ll}
& I=\frac{3}{2} \mathrm{Mr}^2 ...(ii)\\
\therefore \quad & \frac{2}{5} \mathrm{MR}^2=\frac{3}{2} \mathrm{Mr}^2 \\
\therefore \quad & \mathrm{r}=\frac{2}{\sqrt{15}} \mathrm{R}
\end{array}$
...[From(i) and (ii)]