8.
\begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{5180a4e0-dafe-4595-a517-e3501f7aed40-5_533_584_292_703}
\captionsetup{labelformat=empty}
\caption{Figure 3}
\end{figure}
A uniform rod \(A B\) has mass \(3 m\) and length \(2 a\). It is free to rotate in a vertical plane about a smooth fixed horizontal axis through the point \(X\) on the rod, where \(A X = \frac { 1 } { 2 } a\). A particle of mass \(m\) is attached to the rod at \(B\). At time \(t = 0\), the rod is vertical, with \(B\) above \(A\), and is given an initial angular speed \(\sqrt { \frac { g } { a } }\). When the rod makes an angle \(\theta\) with the upward vertical, the angular speed of the rod is \(\omega\), as shown in Figure 3.
- By using the principle of the conservation of energy, show that
$$\omega ^ { 2 } = \frac { g } { 2 a } ( 5 - 3 \cos \theta )$$
- Find the angular acceleration of the rod when it makes an angle \(\theta\) with the upward vertical.
When \(\theta = \phi\), the resultant force of the axis on the rod is in a direction perpendicular to the rod.
- Find \(\cos \phi\).