M5 June 2007 Q5
5. Two forces \(\mathbf{F}_1\) and \(\mathbf{F}_2\) act on a rigid body, where
\(\mathbf{F}_1 = (3\mathbf{i} + 4\mathbf{j} - 6\mathbf{k})\) N and
\(\mathbf{F}_2 = (5\mathbf{i} - \mathbf{j} + 2\mathbf{k})\) N.
The force \(\mathbf{F}_1\) acts at the point with position vector \((\mathbf{i} - 2\mathbf{j})\) m, and the force \(\mathbf{F}_2\) acts at the point with position vector \((3\mathbf{i} - \mathbf{k})\) m. The two forces are equivalent to a single force \(\mathbf{F}\) acting at the point with position vector \((\mathbf{i} - \mathbf{k})\) m, together with a couple \(\mathbf{G}\).
(a) Find \(\mathbf{F}\). (1)
(b) Find the magnitude of \(\mathbf{G}\). (8)
| Scheme | Marks |
|---|---|
| \(\mathbf{F} = \Sigma\mathbf{F}_i = (8\mathbf{i} + 3\mathbf{j} - 4\mathbf{k})\) N | B1 |
| (1) |
| Scheme | Marks |
|---|---|
| \(\Sigma\,\mathbf{r}_i\times\mathbf{F}_i = (\mathbf{i} - 2\mathbf{j})\times(3\mathbf{i} + 4\mathbf{j} - 6\mathbf{k}) + (3\mathbf{i} - \mathbf{k})\times(5\mathbf{i} - \mathbf{j} + 2\mathbf{k})\) | M1 |
| \(= (12\mathbf{i} + 6\mathbf{j} + 10\mathbf{k}) + (-\mathbf{i} - 11\mathbf{j} - 3\mathbf{k}) \quad (= (11\mathbf{i} - 5\mathbf{j} + 7\mathbf{k}))\) | A2, 1, 0 |
| \(\mathbf{G} + (\mathbf{i} - \mathbf{k})\times(8\mathbf{i} + 3\mathbf{j} - 4\mathbf{k}) = (11\mathbf{i} - 5\mathbf{j} + 7\mathbf{k})\) | M1 |
| \((3\mathbf{i} - 4\mathbf{j} + 3\mathbf{k})\), \(\mathbf{G} = (8\mathbf{i} - \mathbf{j} + 4\mathbf{k})\) | B1, A1 |
| \(G = \sqrt{8^2 + (-1)^2 + 4^2} = 9\) Nm. | M1 A1 |
| (8) | |
| (9 marks) |