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(A) The system consists of one central ring and six surrounding rings.$1$. For the central ring, the moment of inertia about the axis passing through

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$31$

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(A) The system consists of one central ring and six surrounding rings.$1$. For the central ring, the moment of inertia about the axis passing through its centre and normal to its plane is $I_{central} = MR^2$.$2$. For each of the six surrounding rings, the distance between the centre of the central ring and the centre of the surrounding ring is $d = 2R$.$3$. Using the parallel axis theorem, the moment of inertia of one surrounding ring about the axis passing through the centre of the central ring is $I_{surrounding} = I_{CM} + Md^2 = MR^2 + M(2R)^2 = MR^2 + 4MR^2 = 5MR^2$.$4$. Since there are six such surrounding rings, their total moment of inertia is $6 	imes 5MR^2 = 30MR^2$.$5$. The total moment of inertia of the system is $I_{total} = I_{central} + 30MR^2 = MR^2 + 30MR^2 = 31MR^2$.

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