M5 June 2014 (R) Q6
6. Three forces \(\mathbf{F}_1\), \(\mathbf{F}_2\) and \(\mathbf{F}_3\) act on a rigid body at the points with position vectors, \(\mathbf{r}_1\), \(\mathbf{r}_2\) and \(\mathbf{r}_3\) respectively, where
\(\mathbf{F}_1 = (2\mathbf{i} - \mathbf{j} + \mathbf{k})\) N \(\qquad \mathbf{F}_2 = (3\mathbf{i} + \mathbf{j} - 2\mathbf{k})\) N \(\qquad \mathbf{F}_3 = (-\mathbf{i} + 2\mathbf{j} + 2\mathbf{k})\) N
\(\mathbf{r}_1 = (\mathbf{i} - \mathbf{k})\) m \(\qquad \mathbf{r}_2 = (2\mathbf{i} - \mathbf{j} + \mathbf{k})\) m \(\qquad \mathbf{r}_3 = (\mathbf{i} + \mathbf{j} - \mathbf{k})\) m
The system of the three forces is equivalent to a single force \(\mathbf{R}\) acting at the point with position vector \((3\mathbf{i} - \mathbf{j} + \mathbf{k})\) m, together with a couple of moment \(\mathbf{G}\).
| Scheme | Marks |
|---|---|
| \(\mathbf{R} = (2\mathbf{i} - \mathbf{j} + \mathbf{k}) + (3\mathbf{i} + \mathbf{j} - 2\mathbf{k}) + (-\mathbf{i} + 2\mathbf{j} + 2\mathbf{k})\) | M1 |
| \(= (4\mathbf{i} + 2\mathbf{j} + \mathbf{k})\) N | A1 |
| (2) |
| Scheme | Marks |
|---|---|
| \(M(O)\), \((\mathbf{i} - \mathbf{k}) \times (2\mathbf{i} - \mathbf{j} + \mathbf{k}) + (2\mathbf{i} - \mathbf{j} + \mathbf{k}) \times (3\mathbf{i} + \mathbf{j} - 2\mathbf{k}) + (\mathbf{i} + \mathbf{j} - \mathbf{k}) \times (-\mathbf{i} + 2\mathbf{j} + 2\mathbf{k})\) | M1 |
| \(= (-\mathbf{i} - 3\mathbf{j} - \mathbf{k}) + (\mathbf{i} + 7\mathbf{j} + 5\mathbf{k}) + (4\mathbf{i} - \mathbf{j} + 3\mathbf{k})\) | A3 |
| \((3\mathbf{i} - \mathbf{j} + \mathbf{k}) \times (4\mathbf{i} + 2\mathbf{j} + \mathbf{k}) = (-3\mathbf{i} + \mathbf{j} + 10\mathbf{k})\) | M1 A1 |
| so, \(\ (4\mathbf{i} + 3\mathbf{j} + 7\mathbf{k}) = (-3\mathbf{i} + \mathbf{j} + 10\mathbf{k}) + \mathbf{G}\) | M1 A1 |
| \((7\mathbf{i} + 2\mathbf{j} - 3\mathbf{k})\) Nm \(= \mathbf{G}\) | A1 |
| (9) | |
| (11 marks) |