4.
\begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{f8ece90a-5042-4db1-9855-ffe74333a899-3_407_341_201_635}
\captionsetup{labelformat=empty}
\caption{Fig. 1}
\end{figure}
Figure 1 shows a uniform ladder of mass \(m\) and length \(2 a\) resting against a rough vertical wall with its lower end on rough horizontal ground. The coefficient of friction between the ladder and the wall is \(\frac { 1 } { 2 }\) and the coefficient of friction between the ladder and the ground is \(\frac { 1 } { 3 }\).
Given that the ladder is in limiting equilibrium when it is inclined at an angle \(\theta\) to the horizontal, show that \(\tan \theta = \frac { 5 } { 4 }\).
(9 marks)