Question

A solid sphere of mass \(M\), radius \(R\) and having moment of inertia about an axis passing through the center of mass as \(I\) is recast into a disc of thickness t whose moment of inertia about an axis passing through its edge and perpendicular to its plane remains \(I\). Then, the radius of the disc will be

Answer 1

Let the radius of the disc be \(r\).
The radius of the solid sphere is \(R\). The mass
of the sphere is \(M\).

The mass of the solid sphere and the disc will be the same since they are made of the same amount of single material.

For a solid sphere

\(I_1=\dfrac{2}{5} MR^2\)
For a disc

\(I_2 =\left (\dfrac{1}{2} Mr^2+Mr^2 \right )=\dfrac{3}{2} Mr^2\)

\(\therefore~I_1=I_2\)

\(\dfrac{2}{5} MR^2=\dfrac{3}{2} Mr^2\)

\(r=\dfrac{2R}{\sqrt{15}}\)
Final Answer: (a)