M5 June 2010 Q4
4. Two forces \(\mathbf{F}_1 = (\mathbf{i} + 2\mathbf{j} + 3\mathbf{k})\) N and \(\mathbf{F}_2 = (3\mathbf{i} + \mathbf{j} + 2\mathbf{k})\) N act on a rigid body. The force \(\mathbf{F}_1\) acts through the point with position vector \((2\mathbf{i} + \mathbf{k})\) m and the force \(\mathbf{F}_2\) acts through the point with position vector \((\mathbf{j} + 2\mathbf{k})\) m.
(a) If the two forces are equivalent to a single force \(\mathbf{R}\), find
(i) \(\mathbf{R}\), (2)
(ii) a vector equation of the line of action of \(\mathbf{R}\), in the form \(\mathbf{r} = \mathbf{a} + \lambda\mathbf{b}\). (6)
(b) If the two forces are equivalent to a single force acting through the point with position vector \((\mathbf{i} + 2\mathbf{j} + \mathbf{k})\) m together with a couple of moment \(\mathbf{G}\), find the magnitude of \(\mathbf{G}\). (5)
| Scheme | Marks |
|---|---|
| (i) \(\mathbf{R} = (\mathbf{i} + 2\mathbf{j} + 3\mathbf{k}) + (3\mathbf{i} + \mathbf{j} + 2\mathbf{k})\) | M1 |
| \(= (4\mathbf{i} + 3\mathbf{j} + 5\mathbf{k})\) | A1 |
| (2) | |
| (ii) \((x\mathbf{i} + y\mathbf{j} + z\mathbf{k}) \times (4\mathbf{i} + 3\mathbf{j} + 5\mathbf{k}) = (2\mathbf{i} + \mathbf{k}) \times (\mathbf{i} + 2\mathbf{j} + 3\mathbf{k}) + (\mathbf{j} + 2\mathbf{k}) \times (3\mathbf{i} + \mathbf{j} + 2\mathbf{k})\) | M1 |
| \((5y - 3z)\mathbf{i} + (4z - 5x)\mathbf{j} + (3x - 4y)\mathbf{k} = (-2\mathbf{i} - 5\mathbf{j} + 4\mathbf{k}) + (6\mathbf{j} - 3\mathbf{k})\) \((5y - 3z)\mathbf{i} + (4z - 5x)\mathbf{j} + (3x - 4y)\mathbf{k} = (-2\mathbf{i} + \mathbf{j} + \mathbf{k})\) | A2 |
| a solution is \(x = 0,\ y = -\tfrac{1}{4},\ z = \tfrac{1}{4}\); \(x = \tfrac{1}{3},\ y = 0,\ z = \tfrac{2}{3}\); \(x = -\tfrac{1}{5},\ y = -\tfrac{2}{5},\ z = 0\) | B1 |
| \(\mathbf{r} = -\tfrac{1}{4}\mathbf{j} + \tfrac{1}{4}\mathbf{k} + \lambda(4\mathbf{i} + 3\mathbf{j} + 5\mathbf{k})\) | M1 A1 ft |
| (6) |
| Scheme | Marks |
|---|---|
| \((\mathbf{i} + 2\mathbf{j} + \mathbf{k}) \times (4\mathbf{i} + 3\mathbf{j} + 5\mathbf{k}) + \mathbf{G} = (-2\mathbf{i} + \mathbf{j} + \mathbf{k})\) \((7\mathbf{i} - \mathbf{j} - 5\mathbf{k}) + \mathbf{G} = (-2\mathbf{i} + \mathbf{j} + \mathbf{k})\) | M1 A1 |
| \(\mathbf{G} = (-9\mathbf{i} + 2\mathbf{j} + 6\mathbf{k})\) | A1 |
| \(|\mathbf{G}| = \sqrt{(-9)^2 + 2^2 + 6^2}\) | M1 |
| \(= 11\) (Nm) | A1 ft |
| (5) | |
| (13 marks) |