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

Question

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 $I_{A}$ , and $I_{B}$ is

TestHub · Accepted Answer

Using the parallel axes theorem, the M.I. of the system about the axis passing through the centre of the sphere A is I A ​ = I A ​ ′ + I B ​ ′ + I C ′ ​ + I D ​ ′ I A ​ = 5 2 ​ MR 2 + ( 5 2 ​ MR 2 + 4 MR 2 ) ​ + ( 5 2 ​ MR 2 + 16 MR 2 ) + ( 5 2 ​ MR 2 + 36 MR 2 ) ​ ∴ I A ​ = 57.6 MR 2 The M.I. about sphere B is, I B ​ = 5 2 ​ MR 2 + ( 5 2 ​ MR 2 + 4 MR 2 ) ∴ I B ​ = 25.6 MR 2 ​ + ( 5 2 ​ MR 2 + 16 MR 2 ) + ( 5 2 ​ MR 2 + 4 MR 2 ) I A ​ − I B ​ = 57.6 MR 2 − 25.6 MR 2 = 32 MR 2 ​

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