A solid sphere and a thin hollow sphere have the same mass and the same radius. The ratio of the radius of gyration of the solid sphere to that of the hollow sphere, both calculated about an axis tangent to their respective surfaces, is:

Step-by-step solution

Using the parallel axis theorem, the moment of inertia of a solid sphere about a tangent axis is: I_ solid = 2 5 MR^2 + MR^2 = 7 5 MR^2 The radius of gyration of the solid sphere about the tangent is: k_ solid = I_ solid M = 7 5 R Similarly, the moment of inertia of a thin hollow sphere about a tangent axis is: I_ hollow = 2 3 MR^2 + MR^2 = 5 3 MR^2 The radius of gyration of the hollow sphere about the tangent is: k_ hollow = I_ hollow M = 5 3 R The required ratio is: k_ solid k_ hollow = 7/5 R 5/3 R = 7 5 3 5 = 21