6.
\begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{3f2bf524-ee27-4eef-8c54-48be61c11677-11_684_1022_114_479}
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\caption{Figure 2}
\end{figure}
A uniform \(\operatorname { rod } A B\) has length \(4 a\) and weight \(W\). A particle of weight \(k W , k < 1\), is attached to the rod at \(B\). The rod rests in equilibrium against a fixed smooth horizontal peg. The end \(A\) of the rod is on rough horizontal ground, as shown in Figure 2. The rod rests on the peg at \(C\), where \(A C = 3 a\), and makes an angle \(\alpha\) with the ground, where \(\tan \alpha = \frac { 1 } { 3 }\). The peg is perpendicular to the vertical plane containing \(A B\).
- Give a reason why the force acting on the rod at \(C\) is perpendicular to the rod.
- Show that the magnitude of the force acting on the rod at \(C\) is
$$\frac { \sqrt { 10 } } { 5 } W ( 1 + 2 k )$$
The coefficient of friction between the rod and the ground is \(\frac { 3 } { 4 }\).
- Show that for the rod to remain in equilibrium \(k \leqslant \frac { 2 } { 11 }\).