7.
\begin{figure}[h]
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\caption{Figure 5}
\end{figure}
A particle \(P\) of mass \(m\) is attached to one end of a light rod of length \(l\). The other end of the rod is attached to a fixed point \(O\). The rod can turn freely in a vertical plane about \(O\). The particle is projected with speed \(u\) from a point \(A\), where \(O A\) makes an angle \(\alpha\) with the upward vertical through \(O\) and \(0 < \alpha < \frac { \pi } { 2 }\). When \(O P\) makes an angle \(\theta\) with the upward vertical through \(O\) the speed of \(P\) is \(v\) as shown in Figure 5.
- Show that \(v ^ { 2 } = u ^ { 2 } + 2 g l ( \cos \alpha - \cos \theta )\).
It is given that \(\cos \alpha = \frac { 3 } { 5 }\) and that \(P\) moves in a complete vertical circle.
- Show that \(u > 2 \sqrt { } \left( \frac { g l } { 5 } \right)\).
As the rod rotates the least tension in the rod is \(T\) and the greatest tension is \(5 T\).
- Show that \(u ^ { 2 } = \frac { 33 } { 10 } \mathrm { gl }\).