A pulley of radius 1.5m 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 rotation made by the pulley before its direction of motion is reversed, will be $\frac{K}{\pi }.$  The value of K is ________.

r = 1.5 m

F = 12t – 3t2 N

l = 4.5 kgm²

$\alpha =\frac{T}{l}=\frac{\frac{2}{3}\left(12t-3t^2\right)}{4.5}=\frac{1}{3}\left(12t-3t^2\right)$

$\Rightarrow \alpha =4t-t^2$

$\Rightarrow \frac{dw}{dt}=ut-t^2={\displaystyle \underset{0}{\overset{0}{\int }}dw={\displaystyle \int {\left(4t-t^2\right)}_0^{t}dt}}$

No. of rev = $\frac{36}{2\pi }=\frac{18}{\pi }=\frac{k}{\pi }\Rightarrow k=18$

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