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 $\displaystyle k \ge \frac{ \left( 1+ \mu \right)^2} { \left( 1 - \mu \right)^2}$ .

A projectile of unit mass is fired in a northerly direction from a point on a horizontal plain at speed \(u\) and an angle \(\theta\) above the horizontal. It lands at a point \(A\) on the plain. In flight, the projectile experiences two forces: gravity, of magnitude \(g\); and a horizontal force of constant magnitude \(f\) due to a wind blowing from North to South. Derive an expression, in terms of \(u\), \(g\), \(f\) and \(\theta\) for the distance \(OA\).

1. Determine the angle \(\alpha\) such that, for all \(\theta>\alpha\), the wind starts to blow the projectile back towards \(O\) before it lands at \(A\).
2. An identical projectile, which experiences the same forces, is fired from \(O\) in a northerly direction at speed \(u\) and angle \(45^\circ\) above the horizontal and lands at a point \(B\) on the plain. Given that \(\theta\) is chosen to maximise \(OA\), show that \[ \frac{OB}{OA} = \frac{ g-f}{\; \sqrt{g^2+f^2\;}- f \;\;}\;. \] Describe carefully the motion of the second projectile when \(f=g\).