A uniform solid sphere of mass M and radius R is rotating freely about its diameter with an angular velocity ₀ . Due to internal forces, the sphere contracts uniformly such that its radius becomes R 2 . What is the work done by the internal forces during this contraction?

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 there is no external torque acting on the sphere, the angular momentum is conserved: I_i ₀ = I_f _f I_i ₀ = ( I_i 4 ) _f _f = 4 ₀ The initial rotational kinetic energy is: K_i = 1 2 I_i ₀^2 = 1 2 ( 2 5 MR^2 ) ₀^2 = 1 5 MR^2 ₀^2 The final rotational kinetic energy is: K_f = 1 2 I_f _f^2 = 1 2 ( I_i 4 )(4 ₀)^2 = 4 ( 1 2 I_i ₀^2 ) =