91Ó°ÊÓ

A weight \(W\) is suspended by two ropes which make \(30^{\circ}\) and \(60^{\circ}\) with the horizontal. If the tension in the first rope is \(20 \mathrm{~N}\), find the tension in the other and the value of \(W\).

A uniform rod of length \(l\) rests over the rim of a fixed hemispherical bowl of radius \(r\), with one end in contact with the surface of the bowl. If all contacts are smooth and the inclination of the rod to the horizontal is \(\theta\), prove that the value of \(\theta\) is given by the equation \(8 r \cos ^{2} \theta-l \cos \theta-4 r=0\).

A block rests on a rough inclined plane. Find the coefficient of friction between block and plane: (a) the weight of the block is \(8 \mathrm{~N}\), (b) the elevation of the plane is \(30^{\circ}\), (c) friction is limiting.

A uniform ladder of weight \(W\) rests on rough horizontal ground against a smooth vertical wall. The vertical plane containing the ladder is perpendicular to the wall and the ladder is inclined at an angle \(\alpha\) to the vertical. Prove that, if the ladder is on the point of slipping and \(\mu\) is the coefficient of friction between it and the ground, then \(\tan \alpha=2 \mu\).