A solid sphere of radius R and a thin hollow cylinder of radius R have the same radius of gyration about their respective axes of rotation. The hollow cylinder rotates about its central geometric axis. The solid sphere rotates about an axis parallel to its diameter, located at a perpendicular distance d from its centre. If d = 3 x R , the value of x is _____.

Step-by-step solution

The moment of inertia of a thin hollow cylinder about its central geometric axis is I_ cyl = mR^2 . Its radius of gyration squared is k_ cyl ^2 = R^2 . For the solid sphere, using the parallel axis theorem, the moment of inertia about the given axis is I_ sph = 2 5 mR^2 + md^2 . Its radius of gyration squared is k_ sph ^2 = 2 5 R^2 + d^2 . Given that the radii of gyration are equal: R^2 = 2 5 R^2 + d^2 d^2 = R^2 - 2 5 R^2 = 3 5 R^2 d = 3 5 R Comparing this with d = 3 x R , we get x = 5 . Answer: 5