c
\(\begin{array}{l}
Kinetic\,energ{y_{\left( {rotational} \right)}}{K_R} = \frac{1}{2}I{\omega ^2}\\
kinetic\,energ{y_{(translational)}}{K_r} = \frac{1}{2}M{v^2}\\
\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\left( {v = R\omega } \right)\\
M.{\mathop{\rm I}\nolimits} {._{(initial)}}{I_{ring}} = M{R^2};{\omega _{initial}} = \omega \\
M.I{._{\left( {new} \right)}}I{'_{\left( {system} \right)}} = M{R^2} + 2m{R^2}\\
\omega {'_{\left( {system} \right)}} = \frac{{m\omega }}{{M + 2m}}
\end{array}\)

Solving we get loss in \(K.E.\)

\( = \frac{{Mm }}{{\left( {M + 2m} \right)}}{\omega ^2}{R^2}\)