6.
\begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{826ad8ff-6e5c-4224-88ba-e78b79d1bc21-11_574_540_226_701}
\captionsetup{labelformat=empty}
\caption{Figure 5}
\end{figure}
A particle \(P\) 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 point \(B\) is vertically above \(O\) and the point \(C\) is vertically below \(O\), with \(O B = O C = a\), as shown in Figure 5. The particle is projected vertically upwards with speed \(3 \sqrt { } ( a g )\).
- Show that \(P\) will pass through \(B\).
- Find the speed of \(P\) as it reaches \(C\).
As \(P\) passes through \(C\) it receives an impulse. Immediately after this, the speed of \(P\) is \(\frac { 5 } { 12 } \sqrt { } ( 11 a g )\) and the direction of motion of \(P\) is unchanged.
- Find the angle between the string and the downward vertical when \(P\) comes to instantaneous rest.