A uniform solid sphere of radius R has a moment of inertia I₀ about its diametric axis. A concentric spherical portion of radius R 2 is carved out of the original sphere. If I₁ is the moment of inertia of the remaining hollowed portion and I₂ is the moment of inertia of the carved out portion (both about the same diametric axis), then the ratio I₁ I₂ is __________.

Step-by-step solution

The moment of inertia of a uniform solid sphere of radius R and mass M about its diametric axis is given by I = 2 5 MR^2 . Since mass M = density volume = 4 3 R^3 , we can express the moment of inertia in terms of density and radius as: I = 2 5 ( 4 3 R^3 ) R^2 = 8 15 R^5 This shows that for a uniform sphere of constant density, the moment of inertia is proportional to the fifth power of its radius ( I R^5 ). Let the moment of inertia of the original solid sphere be I₀ = kR^5 , where k = 8 15 . The moment of inertia