A uniform rod \(A B\) of length \(2 a\) and weight \(W\) rests against a smooth horizontal peg at a point \(C\) on the rod, where \(A C=x\). The lower end \(A\) of the rod rests on rough horizontal ground. The rod is in equilibrium inclined at an angle of \(45^{\circ}\) to the horizontal (see diagram). The coefficient of friction between the rod and the ground is \(\mu\). The rod is about to slip at \(A\).
(i) Find an expression for \(x\) in terms of \(a\) and \(\mu\).

(ii) Hence show that \(\mu \geqslant \frac{1}{3}\).

(iii) Given that \(x=\frac{3}{2} a\), find the value of \(\mu\) and the magnitude of the resultant force on the rod at \(A\).