5.
\begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{25c503ad-94c7-4137-83b5-c3e0aea62f0c-09_636_1143_251_468}
\captionsetup{labelformat=empty}
\caption{Figure 2}
\end{figure}
A uniform rod \(A B\), of mass \(m\) and length \(2 a\), is freely hinged to a fixed point \(A\). A particle of mass \(k m\) is fixed to the rod at \(B\). The rod is held in equilibrium, at an angle \(\theta\) to the horizontal, by a force of magnitude \(F\) acting at the point \(C\) on the rod, where \(A C = \frac { 5 } { 4 } a\), as shown in Figure 2. The line of action of the force at \(C\) is at right angles to \(A B\) and in the vertical plane containing \(A B\).
Given that \(\tan \theta = \frac { 3 } { 4 }\)
- show that \(F = \frac { 16 } { 25 } m g ( 1 + 2 k )\),
- find, in terms of \(m , g\) and \(k\),
- the horizontal component of the force exerted by the hinge on the rod at \(A\),
- the vertical component of the force exerted by the hinge on the rod at \(A\).
Given also that the force acting on the rod at \(A\) acts at \(45 ^ { \circ }\) above the horizontal,
- find the value of \(k\).