b
(b) By using equation\({\omega ^2} = \omega _0^2 - 2\alpha \theta \)

\({\left( {\frac{{{\omega _0}}}{2}} \right)^2} = \omega _0^2 - 2\alpha (2\pi n)\)

\(⇒\) \(\alpha = \frac{3}{4}\frac{{\omega _0^2}}{{4\pi \times 36}}\), (n = 36)..\((i)\)

Now let fan completes total \(n' \) revolution from the starting to come to rest

\(0 = \omega _0^2 - 2\alpha (2\pi n')\) \(⇒\) \(n' = \frac{{\omega _0^2}}{{4\alpha \pi }}\)

substituting the value of   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 \(=48 -36 = 12\)