A uniform solid sphere of mass M and radius R , and a uniform square plate of mass M and side length 2R are considered. Let k_A be the radius of gyration of the solid sphere about an axis tangent to its surface, and k_B be the radius of gyration of the square plate about an axis passing through one of its corners and perpendicular to its plane. If the ratio k_A k_B = 21 x , then the value of x is :

Step-by-step solution

For the uniform solid sphere, the moment of inertia about its diameter is I_ cm = 2 5 MR^2 . Using the parallel axis theorem, the moment of inertia about a tangent axis is: I_A = I_ cm + MR^2 = 2 5 MR^2 + MR^2 = 7 5 MR^2 Therefore, k_A^2 = 7 5 R^2 . For the uniform square plate of side length a = 2R , the moment of inertia about an axis passing through its centre of mass and perpendicular to its plane is: I_ cm = M(a^2 + a^2) 12 = M((2R)^2 + (2R)^2) 12 = M(8R^2) 12 = 2 3 MR^2 The distance d from the centre of the s