A solid sphere of radius R and a solid cylinder of radius R and length L are made of the same material. If they have equal moments of inertia about their respective central axes of symmetry, the length L can be expressed as L = x 15 R . The value of x is ____.

Step-by-step solution

Let the density of the material be . For the solid sphere, the mass is M_s = ( 4 3 R^3 ) . The moment of inertia about its central axis is: I_s = 2 5 M_s R^2 = 2 5 ( 4 3 R^3 )R^2 = 8 15 R^5 For the solid cylinder, the mass is M_c = ( R^2 L) . The moment of inertia about its central axis is: I_c = 1 2 M_c R^2 = 1 2 ( R^2 L)R^2 = 1 2 R^4 L Given that I_s = I_c : 8 15 R^5 = 1 2 R^4 L Cancelling common terms: 8 15 R = 1 2 L L = 16 15 R Comparing this with L = x 15 R , we get x = 16 . Answer: 16