A particle $P$ of mass $m$ is attached to one end of a light inextensible string of length $a$. The other end of the string is attached to a fixed point $O$.

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\item 
\includegraphics{figure_5a}

The particle $P$ moves in a horizontal circle with a constant angular speed $\omega$ with the string inclined at $60°$ to the downward vertical through $O$ (see diagram).

Show that $\omega^2 = \frac{2g}{a}$. [4]

\item The particle now hangs at rest and is then projected horizontally so that it begins to move in a vertical circle with centre $O$. When the string makes an angle $\theta$ with the downward vertical through $O$, the angular speed of $P$ is $\sqrt{\frac{2g}{a}}$. The string first goes slack when $OP$ makes an angle $\theta$ with the upward vertical through $O$.

Find the value of $\cos \theta$. [6]
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\hfill \mbox{\textit{CAIE Further Paper 3 2020 Q5 [10]}}