Question

A pulley of radius 2 m rotates about its axis on applying a tangential force F = (20t - 5t)² N. Where t is measured in seconds. If the moment of inertia of pulley about its axis is 10 kg.m², then on reversing the direction of motion of the pulley, the cycles completed by the pulley is:

(a) less than 3

(b) more than 3 but less than 6

(c) more than 6 but less than 9

(d) more than 9

Answer 1

Correct option is : (b) more than 3 but less than 6

Force F = 20t - 5t²

∵ Torque τ = Iα = F.R

\(\therefore\ \alpha = \frac{F.R}{I} = \frac{(20t - 5t^2)\times2}{10}\)

or α = 4t - t²

or \(\frac{dw}{dt} = 4t^2 - t^2\)

or dω = (4t - t²).dt

On integrating both sides,

When direction is reversed, then ω = 0

or θ = 144 - 108 = 36 rad

or θ = 36 rad

∴ Number of rotation \(n = \frac{\theta}{2\pi} = \frac{36}{2\pi} = \frac{18}{314}\)

or n < 6

Hence option (b) is correct.