M5 January 2006 Q1
1. A bead is threaded on a straight wire. The vector equation of the wire is
\[\mathbf{r} = \mathbf{i} - 3\mathbf{j} + \mathbf{k} + t(2\mathbf{i} - \mathbf{j} + 2\mathbf{k}),\]where the unit of length is the metre. The bead is moved from a point \(A\) on the wire through a distance of 6 m along the wire to a point \(B\) by a force \(\mathbf{F} = (7\mathbf{i} + 4\mathbf{j} - 2\mathbf{k})\) N.
Find the magnitude of the work done by \(\mathbf{F}\) in moving the bead from \(A\) to \(B\).
| Scheme | Marks |
|---|---|
| \(\mathbf{AB} = k(2\mathbf{i} - \mathbf{j} + 2\mathbf{k})\); \(AB = 6 \;\Rightarrow\; k\sqrt{(2^2 + 1^2 + 2^2)} = 6\) | M1 |
| \(\Rightarrow\ k = 2 \;\Rightarrow\; \mathbf{AB} = 4\mathbf{i} - 2\mathbf{j} + 4\mathbf{k}\) | A1 |
| Work done \(= (7\mathbf{i} + 4\mathbf{j} - 2\mathbf{k})\cdot(4\mathbf{i} - 2\mathbf{j} + 4\mathbf{k})\) | M1 |
| \(= 12\) J | A1 |
| (4 marks) |