7.
\begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{36cced0d-f982-4534-a3fe-13c32fb37f5b-11_513_429_123_762}
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\caption{Figure 2}
\end{figure}
A uniform rod \(A B\) has mass \(m\) and length \(2 a\). The end \(A\) is in contact with rough horizontal ground and the end \(B\) is in contact with a smooth vertical wall. The rod rests in equilibrium in a vertical plane perpendicular to the wall and makes an angle of \(30 ^ { \circ }\) with the wall, as shown in Figure 2. The coefficient of friction between the rod and the ground is \(\mu\).
- Find, in terms of \(m\) and \(g\), the magnitude of the force exerted on the rod by the wall.
- Show that \(\mu \geqslant \frac { \sqrt { 3 } } { 6 }\).
A particle of mass \(k m\) is now attached to the rod at \(B\). Given that \(\mu = \frac { \sqrt { 3 } } { 5 }\) and that the rod is now in limiting equilibrium,
- find the value of \(k\).