Four identical uniform solid spheres each of same mass ' \(M\) ' and radius ' \(R\) ' are placed touching each other as shown in figure, with centres \(A, B, C, D . I_A, I_B, I_C\) and \(I_D\) are the moment of inertia of these spheres respectively about an axis passing through centre and perpendicular to the plane. The difference in \(\mathrm{I}_{\mathrm{A}}\), and \(\mathrm{I}_{\mathrm{B}}\) is

Using the parallel axes theorem, the M.I. of the system about the axis passing through the centre of the sphere \(\mathrm{A}\) is
\(\mathrm{I}_{\mathrm{A}}=\mathrm{I}_{\mathrm{A}}{ }^{\prime}+\mathrm{I}_{\mathrm{B}}{ }^{\prime}+\mathrm{I}_{\mathrm{C}^{\prime}}+\mathrm{I}_{\mathrm{D}}{ }^{\prime} \)
\( \mathrm{I}_{\mathrm{A}}=\frac{2}{5} \mathrm{MR}^2+\left(\frac{2}{5} \mathrm{MR}^2+4 \mathrm{MR}^2\right)~+\) \(\left(\frac{2}{5} \mathrm{MR}^2+16 \mathrm{MR}^2\right) \) \( +\left(\frac{2}{5} \mathrm{MR}^2+36 \mathrm{MR}^2\right)\)
\(\therefore \mathrm{I}_{\mathrm{A}}=57.6 \mathrm{MR}^2\)
The M.I. about sphere B is,
\(\mathrm{I}_{\mathrm{B}}=\frac{2}{5} \mathrm{MR}^2+\left(\frac{2}{5} \mathrm{MR}^2+4 \mathrm{MR}^2\right)~+\) \(\left(\frac{2}{5} \mathrm{MR}^2+16 \mathrm{MR}^2\right) \) \( +\left(\frac{2}{5} \mathrm{MR}^2+4 \mathrm{MR}^2\right) \)
\( \therefore \mathrm{I}_{\mathrm{B}}=25.6 \mathrm{MR}^2 \mathrm{I}_{\mathrm{A}}-\mathrm{I}_{\mathrm{B}}=57.6 \mathrm{MR}^2-25.6 \mathrm{MR}^2\) \(=32 \mathrm{MR}^2\)