3. A uniform ladder \(A B\), of mass \(m\) and length \(2 a\), has one end \(A\) on rough horizontal ground. The coefficient of friction between the ladder and the ground is 0.5 . The other end \(B\) of the ladder rests against a smooth vertical wall. The ladder rests in equilibrium in a vertical plane perpendicular to the wall, and makes an angle of \(30 ^ { \circ }\) with the wall. A man of mass \(5 m\) stands on the ladder which remains in equilibrium. The ladder is modelled as a uniform rod and the man as a particle. The greatest possible distance of the man from \(A\) is \(k a\). Find the value of \(k\).