A uniform solid sphere contains a hollow spherical cavity at its center, the rad

Question

A uniform solid sphere contains a hollow spherical cavity at its center, the radius of the cavity being 1/2 the radius of the sphere. Find the moment of inertia of the sphere about an axis passing through the center of the sphere. For a solid sphere: M = 4/3pi a3 rho I = 2/5M a2 In the case of a composite body, it is clear from the definition of the moment of inertia that we may write I = I1 + I2 + ..., where I1 ,I2, etc, are the moments of inertia of the various parts about the particular axis chosen.

Explanation / Answer

from the given hint in the question,
I = I1 + I2 +................

so for this case, imagine the hollow space to be filled by a sphere of r = R/2 i.e half the radius R of outer sphere.............
inertia of big sphere = inertia of outer shell + inertia of inner small sphere
Inertia of outer shell = inertia of big sphere - inertia of inner small sphere
= 2/5 x 4/3 x x R5 x - 2/5 x 4/3 x x (R/2)5 x   
= 31/60 x x x R5
mass M of shell = 4/3 x x (R3 - (R/2)3) = 7R3/6
ineria of outer shell can also be written as = 31/70 x MR2

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