3.
\begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{63363c3e-13fc-49a1-8cef-951e6e97e801-08_796_750_242_660}
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\caption{Figure 2}
\end{figure}
A beam \(A B\) has mass \(m\) and length \(2 a\).
The beam rests in equilibrium with \(A\) on rough horizontal ground and with \(B\) against a smooth vertical wall.
The beam is inclined to the horizontal at an angle \(\theta\), as shown in Figure 2.
The coefficient of friction between the beam and the ground is \(\mu\)
The beam is modelled as a uniform rod resting in a vertical plane that is perpendicular to the wall.
Using the model,
- show that \(\mu \geqslant \frac { 1 } { 2 } \cot \theta\)
A horizontal force of magnitude \(k m g\), where \(k\) is a constant, is now applied to the beam at \(A\).
This force acts in a direction that is perpendicular to the wall and towards the wall.
Given that \(\tan \theta = \frac { 5 } { 4 } , \mu = \frac { 1 } { 2 }\) and the beam is now in limiting equilibrium, - use the model to find the value of \(k\).