A stationary horizontal disc is free to rotate about its axis. When a torque is applied on it, its kinetic energy as a function of \(\theta ,\) where \(\theta \) is the angle by which it has rotated, is given as \(k\theta ^2.\) If its moment of inertia is \(I\) then the angular acceleration of the disc is

\( \Rightarrow {\omega ^2} = \frac{{2k{\theta ^2}}}{I} \Rightarrow \omega  = \sqrt {\frac{{2k}}{I}} \theta \,\,\,\,\,\,\,\,\,\,\,...\left( A \right)\)

\(Differentiate\,\left( A \right)\,wrt\,time\, \to \)

\(\frac{{d\omega }}{{dt}} = \alpha  = \sqrt {\frac{{2k}}{I}} \left( {\frac{{d\theta }}{{dt}}} \right)\)

\( \Rightarrow \alpha  = \sqrt {\frac{{2k}}{I}} .\sqrt {\frac{{2k}}{I}} \,\theta \left\{ {by\,\left( 1 \right)} \right\}\)

\( \Rightarrow \alpha  = \frac{{2k}}{I}\theta \)