7.
\begin{figure}[h]
\captionsetup{labelformat=empty}
\caption{Figure 3}
\includegraphics[alt={},max width=\textwidth]{67a9cf74-833f-4b4a-9fde-3c62dcc08e8c-5_625_1141_319_424}
\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.
- Show that the tension in the string is \(\frac { 2 k m g } { 3 }\).
- Find, in terms of \(m , g\) and \(k\), the normal reaction between \(P\) and the table.
- 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 \(X\).
- Show that \(X\) is the mid-point of \(O A\).