From a uniform solid sphere of mass M and radius R, a concentric spherical cavity of radius R/2 is created. Find the moment of inertia about an axis passing through the common centre.
Options

Question

From a uniform solid sphere of mass M and radius R, a concentric spherical cavity of radius R/2 is created. Find the moment of inertia about an axis passing through the common centre.

G1 Physics · Accepted Answer

Moment of inertia of complete sphere: I₀ = (2/5)MR². The removed sphere has radius R/2 and, because density is uniform, M r e m o v e d M_removed M r ​ e m o v e d = M/8. Its moment of inertia about the same centre is I r e m o v e d I_removed I r ​ e m o v e d = (2/5)(M/8)(R/2)² = MR²/80. Therefore, I = I₀ - I r e m o v e d I_removed I r ​ e m o v e d = 2MR²/5 - MR²/80 = 31MR²/80.