A solid sphere of mass 2 kg and radius 0.5 m is initially at rest. It is subjected to a time-varying torque = (0.8 t - t^2) N m about its diameter, where t is in seconds and is a positive constant. The sphere makes exactly 18 rotations before its angular velocity becomes zero for the first time. The value of 10 is ______.

Step-by-step solution

Moment of inertia of the solid sphere about its diameter is I = 2 5 MR^2 = 2 5 (2)(0.5)^2 = 0.2 kg m ^2 . Angular acceleration = I = 0.8 t - t^2 0.2 = 4 t - 5 t^2 . Angular velocity (t) = ₀^t dt = 2 t^2 - 5 3 t^3 . The sphere reverses direction when (t) = 0 . 2 t^2 - 5 3 t^3 = 0 t = 6 5 . Angular displacement (t) = ₀^t dt = 2 3 t^3 - 5 12 t^4 . At the turning point t = 6 5 , = ( 6 5 )^3 [ 2 3 - 5 12 ( 6 5 ) ] = ( 6 5 )^3 ( 2 3 - 2 ) = ( 6 5 )^3 6 . Given the sphere makes 18 rotations, = 18 2 = 36 rad . Equating the