7.
\begin{figure}[h]
\captionsetup{labelformat=empty}
\caption{Figure 3}
\includegraphics[alt={},max width=\textwidth]{6d6b6de9-ec06-4b55-8dc6-4923a3554ffa-6_682_553_264_828}
\end{figure}
A particle of mass \(m\) is attached to one end of a light inextensible string of length \(l\). The other end of the string is attached to a fixed point \(O\). The particle is hanging at the point \(A\), which is vertically below \(O\). It is projected horizontally with speed \(u\). When the particle is at the point \(P , \angle A O P = \theta\), as shown in Fig. 3. The string oscillates through an angle \(\alpha\) on either side of \(O A\) where \(\cos \alpha = \frac { 2 } { 3 }\).
- Find \(u\) in terms of g and \(l\).
When \(\angle A O P = \theta\), the tension in the string is \(T\).
- Show that \(T = \frac { m g } { 3 } ( 9 \cos \theta - 4 )\).
- Find the range of values of \(T\).
END