3.04a Calculate moments: about a point

Three forces represented by the vectors \(-4\mathbf{i} + \mathbf{j} + 2\mathbf{j}\) and \(k\mathbf{i} - 2\mathbf{j}\) act at the points with coordinates \((0, 0)\), \((3, 0)\) and \((0, 4)\) respectively.

1. Given that the three forces form a couple, find the value of \(k\). [2]
2. Find the magnitude and direction of the couple. [3]

The vertices of a triangular lamina, which is in the \(x\)–\(y\) plane, are at the origin O and the points A\((2, 3)\) and B\((-2, 1)\). Forces \(2\mathbf{i} + \mathbf{j}\) and \(-3\mathbf{i} + 2\mathbf{j}\) are applied to the lamina at A and B, respectively, and a force \(\mathbf{F}\), whose line of action is in the \(x\)–\(y\) plane, is applied at O. The three forces form a couple.

1. Determine the magnitude and the direction of \(\mathbf{F}\). [4]
2. Determine the magnitude and direction of the additional couple that must be applied to the lamina in order to keep it in equilibrium. [3]

\includegraphics{figure_10} A uniform rod \(AB\), of weight \(W\) N and length \(2a\) m, rests with the end \(A\) on a rough horizontal table. A small object of weight \(2W\) N is attached to the rod at \(B\). The rod is maintained in equilibrium at an angle of \(30°\) to the horizontal by a force acting at \(B\) in a direction perpendicular to the rod in the same vertical plane as the rod (see diagram).

1. Find the least possible value of the coefficient of friction between the rod and the table. [7]
2. Given that the magnitude of the contact force at \(A\) is \(\sqrt{39}\) N, find the value of \(W\). [2]