M5 June 2014 Q3
3. 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.
\(\mathbf{F}_1 = (2\mathbf{i} + 3\mathbf{j} - \mathbf{k})\) N and \(\mathbf{r}_1 = (\mathbf{i} + \mathbf{j} - 2\mathbf{k})\) m,
\(\mathbf{F}_2 = (\mathbf{i} - 4\mathbf{j} - 2\mathbf{k})\) N and \(\mathbf{r}_2 = (3\mathbf{i} - \mathbf{j} - \mathbf{k})\) m,
\(\mathbf{F}_3 = (-3\mathbf{i} + \mathbf{j} + 3\mathbf{k})\) N and \(\mathbf{r}_3 = (\mathbf{i} - 2\mathbf{j} + \mathbf{k})\) m.
Show that the system is equivalent to a couple and find the magnitude of the vector moment of this couple. (9)
| Scheme | Marks |
|---|---|
| \((2\mathbf{i} + 3\mathbf{j} - \mathbf{k}) + (\mathbf{i} - 4\mathbf{j} - 2\mathbf{k}) + (-3\mathbf{i} + \mathbf{j} + 3\mathbf{k}) = \mathbf{0}\) | M1 A1 |
| \((\mathbf{i} + \mathbf{j} - 2\mathbf{k}) \times (2\mathbf{i} + 3\mathbf{j} - \mathbf{k}) + (3\mathbf{i} - \mathbf{j} - \mathbf{k}) \times (\mathbf{i} - 4\mathbf{j} - 2\mathbf{k}) + (\mathbf{i} - 2\mathbf{j} + \mathbf{k}) \times (-3\mathbf{i} + \mathbf{j} + 3\mathbf{k})\) (allow \(\sum \mathbf{F} \times \mathbf{r}\)) | M1 |
| \(= (5\mathbf{i} - 3\mathbf{j} + \mathbf{k}) + (-2\mathbf{i} + 5\mathbf{j} - 11\mathbf{k}) + (-7\mathbf{i} - 6\mathbf{j} - 5\mathbf{k})\) | A3 |
| \(= (-4\mathbf{i} - 4\mathbf{j} - 15\mathbf{k})\) | A1 |
| \(\sqrt{(-4)^2 + (-4)^2 + (-15)^2}\) | M1 |
| \(\sqrt{257}\) Nm (2 SF or better) | A1 |
| (9 marks) |
Notes
First M1 for \(\Sigma\mathbf{F}_i\)
First A1 for \(= \mathbf{0}\)
Second M1 for \(\Sigma\,\mathbf{r} \times \mathbf{F}\) or \(\Sigma\,\mathbf{F} \times \mathbf{r}\)
A3 −1 e.e.o.o. (−1 per cross product)
A1 \(\pm(-4\mathbf{i} - 4\mathbf{j} - 15\mathbf{k})\)
Third M1 for \(|\mathbf{G}| = \sqrt{(-4)^2 + (-4)^2 + (-15)^2}\)
A1 for \(\sqrt{257}\) or 2sf or better.