b
\(\alpha \,is\,propotional\,to\,\omega \)

\(Let = \alpha  = k\omega \)                       (\(k\) is constant)

\(\frac{{d\omega }}{{dt}} = k\omega \,\,\,\,\,\,\,\,\,\left[ {also\,\frac{{d\theta }}{{dt}} = \omega  \Rightarrow dt = \frac{{d\theta }}{\omega }} \right]\)

\(\therefore \frac{{\omega d\omega }}{{d\theta }} = k\omega  \Rightarrow d\omega  = kd\theta \)

\(Now\,\int\limits_\omega ^{\omega /2} {d\omega  = k\int {d\theta } } \)

\(\int\limits_{\omega /2}^0 {d\omega  = k\int\limits_0^\theta  {d\theta  \Rightarrow  - \frac{\omega }{2} = k\theta  \Rightarrow  - \frac{\omega }{2} = K{\theta _1}} } \)

\(\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\left( {{\theta _1} = 2\pi n} \right)\)

\(\therefore \theta  = {\theta _1}\,\,or\,\,2\pi {n_1} = 2\pi n\)

\({n_1} = n\)