Two spheres each of mass ' \(M\) ' and radius \(\frac{R}{2}\) are connected at the ends of massless rod of length ' \(2 \mathrm{R}\) '. What will be the moment of inertia of the system about an axis passing through centre of one of the spheres and perpendicular to the rod?

From parallel axis theorem,
\(\mathrm{I}_{\mathrm{o}}=\mathrm{I}_{\mathrm{c}}+\mathrm{Mh}^2\)
Let the moment of inertia of sphere 1 be
\(\mathrm{I}_1=\frac{2}{5} \mathrm{M}\left(\frac{\mathrm{R}}{2}\right)^2+\mathrm{M}(2 \mathrm{R})^2\)
and,
Let the moment of inertia of sphere 2 be
\(\mathrm{I}_2=\frac{2}{5} \mathrm{M}\left(\frac{\mathrm{R}}{2}\right)^2\)
Moment of inertia of the rod \(\mathrm{I}_3=0\)
\(\therefore \quad\) Moment of inertia of the system,
\(\begin{aligned}
\mathrm{I} & =\mathrm{I}_1+\mathrm{I}_2+\mathrm{I}_3 \\
\mathrm{I} & =\frac{2}{5} \mathrm{M}\left(\frac{\mathrm{R}}{2}\right)^2+\mathrm{M}(2 \mathrm{R})^2+\frac{2}{5} \mathrm{M}\left(\frac{\mathrm{R}}{2}\right)^2 \\
& =\frac{4}{5} \mathrm{M}\left(\frac{\mathrm{R}}{2}\right)^2+4 \mathrm{MR}^2 \\
& =\frac{1}{5} \mathrm{MR}^2+4 \mathrm{MR}^2 \\
& =\frac{21}{5} \mathrm{MR}^2
\end{aligned}\)