A pulley of radius 2 m is rotated about its axis by a force = 20 t - 5 t 2 newton (where t is measured in seconds) applied tangentially. If the moment of inertia of the pulley about its axis of rotation is 10 kg m 2 , the number of rotation made by the pulley before its direction of motion is reversed is

Step-by-step solution

We have, Torque = r × F = l α or 2 20 t - 5 t 2 = 10 α α = 4 t - t 2 But α = d ω d t , so d ω d t = 4 t - t 2 d ω = 4 t - t 2 d t Integrating ω = 2 t 2 - t 3 3 ω will be zero at t = 6   s So, ω = d θ d t = 2 t 2 - t 3 3 or d θ = 2 t 2 - t 3 3 d t Again, integrating both sides θ = 2 t 3 3 - t 4 12 since. t = 6   s so, θ = 2 × 6 3 3 - 6 4 12 = 36 Number of turns, n = θ 2 π = 36 2 π = 5 . 73