6.
\begin{figure}[h]
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\caption{Figure 3}
\end{figure}
A light rod of length \(a\) is free to rotate in a vertical plane about a horizontal axis through one end \(O\). A particle \(P\) of mass \(m\) is attached to the other end of the rod. The particle \(P\) is held at rest with the rod making an angle \(\alpha\) with the upward vertical through \(O\), where \(\tan \alpha = \frac { 3 } { 4 }\)
The particle \(P\) is then projected with speed \(u\) in a direction which is perpendicular to the rod. At the instant when the rod makes an angle \(\theta\) with the upward vertical through \(O\), the speed of \(P\) is \(v\), as shown in Figure 3.
Air resistance is assumed to be negligible.
- Show that \(v ^ { 2 } = u ^ { 2 } + \frac { 2 a g } { 5 } ( 4 - 5 \cos \theta )\)
It is given that \(u ^ { 2 } = \frac { 6 a g } { 5 }\) and \(P\) moves in complete vertical circles.
When \(\theta = \beta\), the force exerted on \(P\) by the rod is zero.
- Find the value of \(\cos \beta\)