A string is wrapped around a uniform solid cylinder of mass 4 kg and radius 0.5 m which is free to rotate about its fixed central axis. Starting from rest at t = 0 , a time-varying pulling force F = (24t - 12t^2) N is applied to the free end of the string, where t is in seconds. The average power (in W ) delivered by the force over the time interval from t = 0 until the angular acceleration of the cylinder becomes ze

Step-by-step solution

Torque applied on the cylinder is = F R = (24t - 12t^2)(0.5) = 12t - 6t^2 N m . Moment of inertia of the solid cylinder is I = 1 2 MR^2 = 1 2 (4)(0.5)^2 = 0.5 kg m ^2 . Angular acceleration = I = 12t - 6t^2 0.5 = 24t - 12t^2 . The angular acceleration becomes zero when = 0 24t - 12t^2 = 0 t = 2 s . Angular velocity (t) = ₀^t dt = ₀^t (24t - 12t^2) dt = 12t^2 - 4t^3 . At t = 2 s , the angular velocity is (2) = 12(2)^2 - 4(2)^3 = 48 - 32 = 16 rad/s . By the work-energy theorem, the total work done by the force is equ