Moment of inertia of a solid sphere about its diameter is 'I'. It is then casted into 27 small spheres of same diameter. The moment of inertia of each small sphere about its diameter is

Step-by-step solution

Given a solid sphere of mass M and radius R with moment of inertia I = 2 5 M R^2 about its diameter. When recast into 27 smaller spheres of equal diameter, mass conservation gives M = 27m , so m = M 27 . Volume conservation implies 4 3 R^3 = 27 4 3 r^3 , leading to R^3 = 27r^3 and r = R 3 . Each small sphere has moment of inertia I' = 2 5 m r^2 . Substituting m = M 27 and r = R 3 , this becomes I' = 2 5 ( M 27 ) ( R 3 )^2 = 2 5 M R^2 1 243 . Recognizing 2 5 M R^2 = I , we obtain I' = I 243 . The moment of inertia o