A uniform solid sphere of radius R and a uniform thin spherical shell of radius r both rotate about their respective tangential axes. If their radii of gyration about these axes are equal, the ratio R^2 r^2 can be expressed as x 21 . The value of x is ________.

Step-by-step solution

The radius of gyration k of a body about an axis is given by k^2 = I M . For a uniform solid sphere about a tangential axis, using the parallel axis theorem: I_ solid = 2 5 MR^2 + MR^2 = 7 5 MR^2 Thus, k_ solid ^2 = 7 5 R^2 For a uniform thin spherical shell (hollow sphere) about a tangential axis: I_ hollow = 2 3 mr^2 + mr^2 = 5 3 mr^2 Thus, k_ hollow ^2 = 5 3 r^2 Given that their radii of gyration are equal: k_ solid ^2 = k_ hollow ^2 7 5 R^2 = 5 3 r^2 R^2 r^2 = 25 21 Comparing this with x 21 , we get x = 25 . An