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(B) The moment of inertia of a single solid sphere about its own central axis is $I_{cm} = \frac{2}{5} MR^2$. Using the parallel axis theorem,$I = I_{

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$32$

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(B) The moment of inertia of a single solid sphere about its own central axis is $I_{cm} = \frac{2}{5} MR^2$. Using the parallel axis theorem,$I = I_{cm} + Md^2$,where $d$ is the distance from the axis of rotation.For the axis passing through the center of sphere $A$:$I_A = I_A' + I_B' + I_C' + I_D'$$I_A = \frac{2}{5} MR^2 + (\frac{2}{5} MR^2 + M(2R)^2) + (\frac{2}{5} MR^2 + M(4R)^2) + (\frac{2}{5} MR^2 + M(6R)^2)$$I_A = \frac{8}{5} MR^2 + M(4R^2 + 16R^2 + 36R^2) = 1.6 MR^2 + 56 MR^2 = 57.6 MR^2$.For the axis passing through the center of sphere $B$:$I_B = I_A' + I_B' + I_C' + I_D'$$I_B = (\frac{2}{5} MR^2 + M(2R)^2) + \frac{2}{5} MR^2 + (\frac{2}{5} MR^2 + M(2R)^2) + (\frac{2}{5} MR^2 + M(4R)^2)$$I_B = \frac{8}{5} MR^2 + M(4R^2 + 4R^2 + 16R^2) = 1.6 MR^2 + 24 MR^2 = 25.6 MR^2$.The difference is $|I_A - I_B| = 57.6 MR^2 - 25.6 MR^2 = 32 MR^2$.

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