9231 P21 - Jun 2018 - Q4 - 11 marks
A uniform \(\operatorname{rod} A B\) has length \(2 a\) and weight \(W\). The end \(A\) rests on rough horizontal ground and the end \(B\) rests against a smooth vertical wall. The rod is in a vertical plane that is perpendicular to the wall. The angle between the rod and the horizontal is \(\theta\). A particle of weight \(5 W\) hangs from the rod at the point \(C\), with \(A C=x a\), where \(0<x<1\).
(i) By taking moments about \(A\), show that the magnitude of the normal reaction at \(B\) is \(\frac{W(5 x+1)}{2 \tan \theta}\).
The particle of weight \(5 W\) is now moved a distance \(a\) up the rod, so that \(A C=(x+1) a\). This results in the magnitude of the normal reaction at \(B\) being double its previous value. The system remains in equilibrium with the rod at angle \(\theta\) with the horizontal.
(ii) Show that \(x=\frac{4}{5}\).
The coefficient of friction between the rod and the ground is \(\frac{2}{3}\).
(iii) Given that the rod is about to slip when the particle of weight 5 W is in its second position, find the value of \(\tan \theta\).
