M5 June 2008 Q3
3. A system of forces consists of two forces \(\mathbf{F}_1\) and \(\mathbf{F}_2\) acting on a rigid body.
\(\mathbf{F}_1 = (-2\mathbf{i} + \mathbf{j} - \mathbf{k})\) N and acts at the point with position vector \(\mathbf{r}_1 = (\mathbf{i} - \mathbf{j} + \mathbf{k})\) m.
\(\mathbf{F}_2 = (3\mathbf{i} - \mathbf{j} + 2\mathbf{k})\) N and acts at the point with position vector \(\mathbf{r}_2 = (4\mathbf{i} - \mathbf{j} - 2\mathbf{k})\) m.
Given that the system is equivalent to a single force \(\mathbf{R}\) N, acting at the point with position vector \((5\mathbf{i} + \mathbf{j} - \mathbf{k})\) m, together with a couple \(\mathbf{G}\) N m, find
(a) \(\mathbf{R}\), (2)
(b) the magnitude of \(\mathbf{G}\). (9)
| Scheme | Marks |
|---|---|
| \(\mathbf{R} = (-2\mathbf{i} + \mathbf{j} - \mathbf{k}) + (3\mathbf{i} - \mathbf{j} + 2\mathbf{k})\) | M1 |
| \(= (\mathbf{i} + \mathbf{k})\) | A1 |
| (2) |
| Scheme | Marks |
|---|---|
| \(\mathbf{G} + (5\mathbf{i} + \mathbf{j} - \mathbf{k})\times(\mathbf{i} + \mathbf{k}) = (\mathbf{i} - \mathbf{j} + \mathbf{k})\times(-2\mathbf{i} + \mathbf{j} - \mathbf{k}) + (4\mathbf{i} - \mathbf{j} - 2\mathbf{k})\times(3\mathbf{i} - \mathbf{j} + 2\mathbf{k})\) | M1 A2 ft |
| \(\mathbf{G} + (\mathbf{i} - 6\mathbf{j} - \mathbf{k}) = (-\mathbf{j} - \mathbf{k}) + (-4\mathbf{i} - 14\mathbf{j} - \mathbf{k})\) | A3 ft |
| \(\mathbf{G} = (-5\mathbf{i} - 9\mathbf{j} - \mathbf{k})\) | A1 |
| \(|\mathbf{G}| = \sqrt{(-5)^2 + (-9)^2 + (-1)^2} = \sqrt{107}\) Nm | M1 A1 |
| (9) | |
| (11 marks) |
Notes
(Corrected from the printed mark scheme: the first term under the square root is printed as \(-5^2\).)