A pulley of radius 1 . 5 m is rotated about its axis by a force F = 12 t - 3 t 2 N applied tangentially (while t is measured in seconds). If moment of inertia of the pulley about its axis of rotation is 4 . 5 kg m 2 , the number of rotations made by the pulley before its direction of motion is reversed, will be K π . The value of K is _____ .

Step-by-step solution

Torque of pulley about its axis is τ = I α , where, I is moment of inertia and α is angular acceleration. Torque of a force about its axis is τ = F R , here, R is radius of pulley. Then, 12 t - 3 t 2 1 . 5 = 4 . 5 α ⇒ α = 4 t - t 2 Angular acceleration in terms of angular velocity is ⇒ α = d ω d t = 4 t - t 2 ⇒ ω = ∫ 0 t 4 t - t 2 d t ⇒ ω = 2 t 2 - t 3 3 For ω = 0 ,   2 t 2 - t 3 3 = 0 ⇒ t 2 2 - t 3 = 0 ⇒ t = 0 &#1