\[\begin {array} {1 1} (a)\;\frac{R}{\sqrt {15}} & \quad (b)\;R \sqrt {\frac{2}{15}} \\ (c)\;\frac{2R}{\sqrt {15}} & \quad (d)\;\frac{R}{4} \end {array}\]

The moment of inertia of a solid spherical shell of mass M and radius R as shown in the figure above is $I_{\text{sphere}} = \large\frac{2}{5}$$MR^2$

Given that a solid sphere or mass M and radius R is recast into a disc of thickness $t$, but the moment of inertia remains the same.

The moment of inertia of a sphere about its central axis and a solid spherical shell of mass M and radius r as is $I = \large\frac{1}{2}$$Mr^2$

Using the theorem of parallel axes, the momentum of inertia of a disc about its edge $I_{\text{disc}} = \large\frac{1}{2} $$Mr^2+Mr^2 = \large\frac{3}{2} $$Mr^2$

Given that $I_{\text{disc}} =I_{\text{sphere}} \rightarrow \large\frac{2}{5}$$MR^2$$ = \large\frac{3}{2} $$Mr^2$

$\Rightarrow r^2 = \large\frac{4}{15}$$R^2$

$\Rightarrow r= \large\frac{2R}{\sqrt {15}}$

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