A particle of mass $m$ is suspended from a fixed point $O$ by a light inextensible string of length $l$. The particle hangs in equilibrium at the point $R$ vertically below $O$.

The particle is set into motion with a horizontal velocity $u$ so that it moves in a complete vertical circle with centre $O$.

The point $T$ on the circle is such that angle $ROT$ is $30°$, as shown in the diagram.

\includegraphics{figure_5}

\begin{enumerate}[label=(\alph*)]
\item Find, in terms of $g$, $l$ and $u$, the speed of the particle at the point $T$.
[3 marks]

\item Find, in terms of $g$, $l$, $m$ and $u$, the tension in the string when the particle is at the point $T$.
[3 marks]

\item Find, in terms of $g$, $l$, $m$ and $u$, the tension in the string when the particle returns to the point $R$.
[2 marks]

\item The particle makes complete revolutions.

Find, in terms of $g$ and $l$, the minimum value of $u$.
[4 marks]
\end{enumerate}

\hfill \mbox{\textit{AQA M2 2016 Q5 [12]}}