A homogeneous cube is supported, with a face flat against a a rough vertical wall and four edges vertical, by a force \(P\) applied at the middle point of the lowest edge which does not meet the wall, in a plane perpendicular to that edge. Prove that, if \(\mu(=\tan\epsilon)\) is the coefficient of friction, the least value of \(P\) is \[ \text{(1) } \tfrac{1}{2}W\csc\epsilon, \quad \text{(2) } W\cos\epsilon \quad \text{or (3) } W\cos\epsilon\sec\left(\tan^{-1}\frac{1}{\mu+2}-\epsilon\right), \] according as \[ \text{(1) } \epsilon > \tfrac{1}{4}\pi, \quad \text{(2) } \tfrac{1}{4}\pi > \epsilon > \tfrac{1}{8}\pi \quad \text{or (3) } \epsilon < \tfrac{1}{8}\pi. \]
Shew how to reduce a system of coplanar forces to a single force or to a couple. If two forces \(P,Q\) act at fixed points \(A,B\) and have a resultant \(R\), shew that if \(P\) and \(Q\) are turned through the same angle, the resultant passes through a fixed point \(C\), such that the sides of the triangle \(ABC\) are proportional to \(P,Q,R\).