A uniform solid sphere of mass M and radius R is rotating freely about its diameter with an initial rotational kinetic energy E . Due to internal forces, the sphere collapses symmetrically such that its new radius becomes R 2 , without any loss of mass. If the final rotational kinetic energy of the sphere is xE , the value of x is

Step-by-step solution

The initial moment of inertia of the solid sphere is: I_i = 2 5 MR^2 When the radius becomes R 2 , the final moment of inertia is: I_f = 2 5 M ( R 2 )^2 = 1 4 ( 2 5 MR^2 ) = I_i 4 Since no external torque acts on the sphere during the collapse, its angular momentum L remains conserved. The rotational kinetic energy is given by K = L^2 2I . Since L is constant, K 1 I . Therefore, the final kinetic energy is: K_f = K_i I_i I_f = E I_i I_i 4 = 4E Comparing this with xE , we get x = 4 . Answer: 4