A pulley of radius 1.5 m is rotated about its axis by a force F = (12t - 3t²)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 m², the number of rotations made by the pulley before its direction of motion is reversed, will be K/π. The value of K is ___________. Numerical answer type. Work it out, then open the answer.

Answer: 18. τ = FR = 1.5(12t - 3t²) = 18t - 4.5t², α = (τ)/(4.5) = 4t - t². ω = 2t² - t³/3 = 0 at t = 6 s θ = 2t³/3 - t⁴/12|₀⁶ = 144 - 108 = 36 rad Rotations = (36)/(2π) = 18/π, so K = 18.

A pulley of radius 1.5 m is rotated about its axis by a force F =… | JEE Main 2022 (27 Jul, Shift 1)

Q34JEE Main 2022 · 27 JUL · SHIFT 1Medium

Rotational Kinematics & Dynamics

A pulley of radius $1.5\text{ m}$ is rotated about its axis by a force $F = \left(12t - 3t^{2}\right)N$ applied tangentially (while t is measured in seconds). If moment of inertia of the pulley about its axis of rotation is $4.5\text{ kg m}^{2}$ , the number of rotations made by the pulley before its direction of motion is reversed, will be $\frac{K}{\pi}$ . The value of K is ___________.

Numerical answer type. Work it out, then open the answer.

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Answer: 18

$\tau = FR = 1.5(12t - 3t^2) = 18t - 4.5t^2$ , $\alpha = \frac{\tau}{4.5} = 4t - t^2$ .
$\omega = 2t^2 - \frac{t^3}{3} = 0$ at $t = 6$ s
$\theta = \frac{2t^3}{3} - \frac{t^4}{12}\Big|_0^6 = 144 - 108 = 36\text{ rad}$
Rotations $= \frac{36}{2\pi} = \frac{18}{\pi}$ , so $K = 18$ .

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