A uniform thin rod of length L and a uniform solid sphere of radius R have the same mass. If the radius of gyration of the rod about an axis passing through its center and perpendicular to its length is equal to the radius of gyration of the solid sphere about its diametric axis, then the ratio L R is :

Step-by-step solution

Moment of inertia of a uniform thin rod of mass M and length L about an axis passing through its center and perpendicular to its length is I_ rod = ML^2 12 . The radius of gyration k_ rod is given by k_ rod ^2 = L^2 12 . Moment of inertia of a uniform solid sphere of mass M and radius R about its diametric axis is I_ sphere = 2 5 MR^2 . The radius of gyration k_ sphere is given by k_ sphere ^2 = 2 5 R^2 . Given that k_ rod = k_ sphere , we have: L^2 12 = 2 5 R^2 L^2 R^2 = 24 5 L R = 24 5 Answer: 24 5