7.

\begin{figure}[h]
\begin{center}
\captionsetup{labelformat=empty}
\caption{Figure 3}
  \includegraphics[alt={},max width=\textwidth]{67a9cf74-833f-4b4a-9fde-3c62dcc08e8c-5_625_1141_319_424}
\end{center}
\end{figure}

A particle $P$ of mass $m$ is attached to one end of a light inextensible string of length $2 a$. The other end of the string is fixed to a point $A$ which is vertically above the point $O$ on a smooth horizontal table. The particle $P$ remains in contact with the surface of the table and moves in a circle with centre $O$ and with angular speed $\sqrt { \left( \frac { k g } { 3 a } \right) }$, where $k$ is a constant. Throughout the motion the string remains taut and $\angle A P O = 30 ^ { \circ }$, as shown in Figure 3.
\begin{enumerate}[label=(\alph*)]
\item Show that the tension in the string is $\frac { 2 k m g } { 3 }$.
\item Find, in terms of $m , g$ and $k$, the normal reaction between $P$ and the table.
\item Deduce the range of possible values of $k$.

The angular speed of $P$ is changed to $\sqrt { \left( \frac { 2 g } { a } \right) }$. The particle $P$ now moves in a horizontal circle above the table. The centre of this circle is $X$.
\item Show that $X$ is the mid-point of $O A$.
\end{enumerate}

\hfill \mbox{\textit{Edexcel M3 2006 Q7 [15]}}