M5 June 2005 Q1
1. Two constant forces \(\mathbf{F}_1\) and \(\mathbf{F}_2\) are the only forces acting on a particle. \(\mathbf{F}_1\) has magnitude 9 N and acts in the direction of \(2\mathbf{i} + \mathbf{j} + 2\mathbf{k}\). \(\mathbf{F}_2\) has magnitude 18 N and acts in the direction of \(\mathbf{i} + 8\mathbf{j} - 4\mathbf{k}\).
Find the total work done by the two forces in moving the particle from the point with position vector \((\mathbf{i} + \mathbf{j} + \mathbf{k})\) m to the point with position vector \((3\mathbf{i} + 2\mathbf{j} - \mathbf{k})\) m.
| Scheme | Marks |
|---|---|
| \(|2\mathbf{i} + \mathbf{j} + 2\mathbf{k}| = 3 \;\Rightarrow\; \mathbf{F}_1 = 6\mathbf{i} + 3\mathbf{j} + 6\mathbf{k}\) | B1 |
| \(|\mathbf{i} + 8\mathbf{j} - 4\mathbf{k}| = 9 \;\Rightarrow\; \mathbf{F}_2 = 2\mathbf{i} + 16\mathbf{j} - 8\mathbf{k}\) | B1 |
| WD \(= (6\mathbf{i} + 3\mathbf{j} + 6\mathbf{k} + 2\mathbf{i} + 16\mathbf{j} - 8\mathbf{k})\cdot(3\mathbf{i} + 2\mathbf{j} - \mathbf{k} - \mathbf{i} - \mathbf{j} - \mathbf{k})\) | M1 A1ft on \(\mathbf{F}\) |
| \(= (8\mathbf{i} + 19\mathbf{j} - 2\mathbf{k})\cdot(2\mathbf{i} + \mathbf{j} - 2\mathbf{k})\) | M1 |
| \(= 39\) J | A1 |
| (6 marks) |