A pulley of radius 2 m is rotated about its axis by a force F = (20t - 5t²) 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-m² the number of rotation made by the pulley before its direction of motion is reversed, is: (a) more than 3 but less than 6 (b) more than 6 but less than 9 (c) more than 9 (d) less than 3

Answer: (a). τ = FR = 2(20t - 5t²), α = τ/I = 4t - t². ω = 2t² - t³/3; direction reverses when ω = 0: t = 6 s. θ = ∫₀⁶ ω dt = 2t³/3 - t⁴/12 = 144 - 108 = 36 rad Rotations = (36)/(2π) ≈ 5.7 So the answer is (a) more than 3 but less than 6.

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Rotational Kinematics & Dynamics

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

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Answer: (a)

$\tau = FR = 2(20t - 5t^2)$ , $\alpha = \frac{\tau}{I} = 4t - t^2$ .
$\omega = 2t^2 - \frac{t^3}{3}$ ; direction reverses when $\omega = 0$ : $t = 6$ s.
$\theta = \int_0^6 \omega\,dt = \frac{2t^3}{3} - \frac{t^4}{12} = 144 - 108 = 36\text{ rad}$
Rotations $= \frac{36}{2\pi} \approx 5.7$
So the answer is (a) more than 3 but less than 6.

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