A pulley of radius 2m is rotated about its axis by a force $F = 20 t - 5 t^{2}$ newton [where t is measured in second) applied tangentially. If the moment of Inertia of the pulley about its axis of rotation is $10 k g - m^{2}$ .The number of rotations made by the pulley before its direction of motion reversed is

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less than 3

More than 3 but less than 6

More than 6 but less than 9

More than 9

answer is A.

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Detailed Solution

$\begin{array}{l} F = 20 t - 5 t^{2} \\ \Rightarrow α = \frac{F R}{I} = \frac{(20 t - 5 t^{2}) 2}{10} = 4 t - t^{2} \\ \Rightarrow \frac{d ω}{d t} = 4 t - t^{2} \\ {\Rightarrow \int^{}}_{ω}^{0} d ω = \int_{0}^{t} (4 t - t^{2}) d t \\ \Rightarrow ω = \frac{4 t^{2}}{2} - \frac{t^{3}}{3} = 2 t^{2} - \frac{t^{3}}{3} \\ ∴ [ω = 0 at t = 0 or 6 s] \\ {\Rightarrow \int^{}}_{θ}^{0} d θ = \int_{0}^{6}_{}^{} (2 t^{2} - \frac{t^{3}}{3}) d t \\ \Rightarrow θ - 0 = {(\frac{2 t^{3}}{3} - \frac{t^{4}}{3 \times 4})}^{6}_{0} \\ \Rightarrow θ = 36 r a d \\ \Rightarrow n = \frac{36}{2 π} < 6 \end{array}$