A uniform solid sphere of mass M and radius R is melted and recast into a uniform solid cylinder of the same radius R . Let k₁ be the radius of gyration of the initial solid sphere about its diameter, and k₂ be the radius of gyration of the recast solid cylinder about an axis passing through its centre and perpendicular to its length. If the ratio k₁ k₂ = x 215 , then the value of x is :

Step-by-step solution

Since the solid sphere is melted and recast into a solid cylinder, the volume remains conserved. Volume of sphere = Volume of cylinder 4 3 R^3 = R^2 L This gives the length of the cylinder as L = 4 3 R . For the initial solid sphere, the moment of inertia about its diameter is: I₁ = 2 5 MR^2 Thus, k₁^2 = 2 5 R^2 . For the recast solid cylinder, the moment of inertia about an axis passing through its centre and perpendicular to its length is: I₂ = M ( R^2 4 + L^2 12 ) Substituting L = 4 3 R into the equation: I₂ = M