A solid sphere and a hollow sphere of the same mass have the same radius of gyration about their respective diametric axes. If the ratio of the radius of the solid sphere to the radius of the hollow sphere is x 3 , then the value of x is _______.

Step-by-step solution

Let the mass of both spheres be m . The moment of inertia of the solid sphere about its diameter is I₁ = 2 5 mR₁^2 , so its radius of gyration is k₁ = 2 5 R₁ . The moment of inertia of the hollow sphere about its diameter is I₂ = 2 3 mR₂^2 , so its radius of gyration is k₂ = 2 3 R₂ . Given that k₁ = k₂ , we have: 2 5 R₁ = 2 3 R₂ R₁ R₂ = 5 3 Comparing this with the given ratio x 3 , we get x = 5 . Answer: 5