91Ó°ÊÓ

A uniform ladder of weight \(W\) rests with one end on rough horizontal ground and with the other end against a smooth vertical wall. The ladder is at an angle \(\tan ^{-1} 2\) to the ground and is in a vertical plane perpendicular to the wall. The coefficient of friction between the ladder and the ground is \(1 .\) Find how far up the ladder a boy of weight \(2 W\) can climb without disturbing equilibrium. Find also the least horizontal force which must be applied to the foot of the ladder to enable the boy to climb to the top of the ladder without it slipping.

A uniform rod \(A B\) of weight \(W_{1}\) is attached at \(A\) to a fixed smooth pivot and is freely hinged at \(\mathrm{B}\) to a uniform rod \(\mathrm{BC}\) of weight \(W_{2}\). The system is in equilibrium in a vertical plane with \(\mathrm{AB}\) resting on a smooth peg P below the level of \(\mathrm{A}\) and the end \(\mathrm{C}\) of the rod \(\mathrm{BC}\) on a smooth horizontal plane. The distance \(\mathrm{AP}\) is \(x\), the length \(\mathrm{AB}\) is \(2 a\) and the acute angle which \(\mathrm{AB}\) makes with the horizontal is \(\theta\). Prove that the force between the rods at B is vertical and equal to \(\frac{1}{2} W_{2}\), and find the reaction at the peg. If the reaction at \(\mathrm{A}\) is horizontal, find its magnitude in terms of \(W_{1}, W_{2}\) and \(\theta\), and prove that $$ x=\frac{2 a\left(w_{1}+W_{2}\right) \cos ^{2} \theta}{2 W_{1}+W_{2}} $$