A solid sphere and a hollow sphere have the same mass and the same radius. Their radii of gyration are measured about axes that are tangential to their respective surfaces. If the ratio of the radius of gyration of the solid sphere to that of the hollow sphere is x 5 , then the value of x is _______.

Step-by-step solution

Let the mass of both spheres be m and their radius be R . Using the parallel axis theorem, the moment of inertia of the solid sphere about a tangential axis is: I_ solid = I_ cm + mR^2 = 2 5 mR^2 + mR^2 = 7 5 mR^2 Its radius of gyration is k_ solid = 7 5 R . Similarly, the moment of inertia of the hollow sphere about a tangential axis is: I_ hollow = I_ cm + mR^2 = 2 3 mR^2 + mR^2 = 5 3 mR^2 Its radius of gyration is k_ hollow = 5 3 R . The ratio of their radii of gyration is: k_ solid k_ hollow = 7/5 R 5/3 R = 7 5