Problems

A particle of weight \(W\) is placed on a rough plane inclined at an angle of \(\theta\) to the horizontal. The coefficient of friction between the particle and the plane is \(\mu\). A horizontal force \(X\) acting on the particle is just sufficient to prevent the particle from sliding down the plane; when a horizontal force \(kX\) acts on the particle, the particle is about to slide up the plane. Both horizontal forces act in the vertical plane containing the line of greatest slope. Prove that \[ \left( k-1 \right) \left( 1 + \mu^2 \right) \sin \theta \cos \theta = \mu \left( k + 1 \right) \] and hence that $\displaystyle k \ge \frac{ \left( 1+ \mu \right)^2} { \left( 1 - \mu \right)^2}$ .