6.
\begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{2cf74ba3-857a-4ce9-ab5b-e6203b279161-18_481_606_246_667}
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\caption{Figure 2}
\end{figure}
A particle \(P\) of mass \(m\) is attached to one end of a light inextensible string of length \(a\). The other end of the string is attached to a fixed point \(O\). The particle is held at the point \(A\), where \(O A = a\) and \(O A\) is horizontal. The particle is projected vertically upwards with speed \(u\), as shown in Figure 2. When the string makes an angle \(\theta\) with the horizontal through \(O\) and the string is still taut, the tension in the string is \(T\).
- Show that \(T = \frac { m } { a } \left( u ^ { 2 } - 3 a g \sin \theta \right)\)
The particle moves in complete circles.
- Find, in terms of \(a\) and \(g\), the minimum value of \(u\).
Given that the least tension in the string is \(S\) and the greatest tension in the string is \(4 S\),
- find, in terms of \(a\) and \(g\), an expression for \(u\).