A ball of mass (m) 0.5 ~kg is attached to the end of a string...

$T \cos \theta$ component will cancel $\mathrm{mg}$. $T \sin \theta$ component will provide necessary centripetal force to the ball towards centre $C$. $\begin{array}{rlrl} & \therefore & T \sin \theta & =m r \omega^2=m(l \sin \theta) \omega^2 \\ & \text{ or } & T & =m l \omega^2 \\ & \therefore & \omega & =\sqrt{\frac{T}{m l}} \\ & \text{ or } & \omega_{\max } & =\sqrt{\frac{T_{\max }}{m l}}=\sqrt{\frac{324}{0.5 \times 0.5}} \\ & = & 36 \mathrm{rad} / \mathrm{s} \end{array}$$\therefore$ Correct option is (d). Analysis of Question (i) Question is simple. (ii) This is called the conical pendulum. (iii) The interesting fact in this problem is that $\omega$ or $T$ is independent of $\theta$. $\omega \propto \sqrt{T}$ If $\omega$ is increased, $T$ will also increase.