Explanation

(b) By using equation $\omega^{2} = \omega_{0}^{2} - 2\alpha\theta$
$\left( \frac{\omega_{0}}{2} \right)^{2} = \omega_{0}^{2} - 2\alpha(2\pi n) \Rightarrow \alpha = \frac{3}{4}\frac{\omega_{0}^{2}}{4\pi \times 36},\ (n = 36)$ ..(i)
Now let fan completes total $n'$ revolution from the starting to come to rest
$0 = \omega_{0}^{2} - 2\alpha\left( 2\pi n^{'} \right) \Rightarrow n^{'} = \frac{\omega_{0}^{2}}{4\alpha\pi}$
Substituting the value of $\alpha$ from equation (i)
$n^{'} = \frac{\omega_{0}^{2}}{4\pi}\frac{4 \times 4\pi \times 36}{3\omega_{0}^{2}} = 48$ revolution
Number of rotation $= 48 - 36 = 12$