A solid cylinder of radius R₁ and a solid sphere of radius R₂ have the same mass. If their radii of gyration about their respective central axes of symmetry are equal, then the ratio R₁ : R₂ is:

Step-by-step solution

The radius of gyration of a solid cylinder about its central axis is given by k_ cyl = I M = 1 2 MR₁^2 M = R₁ 2 . The radius of gyration of a solid sphere about its central axis is given by k_ sph = I M = 2 5 MR₂^2 M = 2 5 R₂ . Given that k_ cyl = k_ sph , we have: R₁ 2 = 2 5 R₂ R₁ R₂ = 2 5 Thus, the ratio R₁ : R₂ is 2 : 5 . Answer: 2 : 5