2 A particle P of mass 0.25 kg is connected to a fixed point O by a light inextensible string of length $a$ metres, and is moving in a vertical circle with centre O and radius $a$ metres. When P is vertically below O , its speed is $8.4 \mathrm {~m} \mathrm {~s} ^ { - 1 }$. When OP makes an angle $\theta$ with the downward vertical, and the string is still taut, P has speed $v \mathrm {~m} \mathrm {~s} ^ { - 1 }$ and the tension in the string is $T \mathrm {~N}$, as shown in Fig. 2.

\begin{figure}[h]
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  \includegraphics[alt={},max width=\textwidth]{db60e7d9-bec5-47f7-9e27-38b7d112851e-3_483_551_447_749}
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\caption{Fig. 2}
\end{center}
\end{figure}

(i) Find an expression for $v ^ { 2 }$ in terms of $a$ and $\theta$, and show that

$$T = \frac { 17.64 } { a } + 7.35 \cos \theta - 4.9 .$$

(ii) Given that $a = 0.9$, show that P moves in a complete circle. Find the maximum and minimum magnitudes of the tension in the string.\\
(iii) Find the largest value of $a$ for which P moves in a complete circle.\\
(iv) Given that $a = 1.6$, find the speed of P at the instant when the string first becomes slack.

\hfill \mbox{\textit{OCR MEI M3 2013 Q2 [18]}}