M5 June 2016 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, where
\(\mathbf{F}_1 = (2\mathbf{j} - \mathbf{k})\) N \(\qquad \mathbf{F}_2 = (\mathbf{i} + \mathbf{k})\) N \(\qquad \mathbf{F}_3 = (\mathbf{i} + \mathbf{j})\) N
\(\mathbf{r}_1 = (4\mathbf{j} - \mathbf{k})\) m \(\qquad \mathbf{r}_2 = (2\mathbf{i} + \mathbf{k})\) m \(\qquad \mathbf{r}_3 = (3\mathbf{i} + \mathbf{j} + \mathbf{k})\) m
The system of the three forces is equivalent to a single force \(\mathbf{R}\) acting through the point with position vector \((\mathbf{i} - \mathbf{j} + \mathbf{k})\) m, together with a couple of moment \(\mathbf{G}\).
| Scheme | Marks |
|---|---|
| \(\mathbf{R} = (2\mathbf{j} - \mathbf{k}) + (\mathbf{i} + \mathbf{k}) + (\mathbf{i} + \mathbf{j}) = (2\mathbf{i} + 3\mathbf{j})\) N | M1 A1 |
| (2) |
Notes
M1 for adding the 3 forces together
A1 for \((2\mathbf{i} + 3\mathbf{j})\)
| Scheme | Marks |
|---|---|
| Moments about \(O\): | |
| \((4\mathbf{j} - \mathbf{k}) \times (2\mathbf{j} - \mathbf{k}) + (2\mathbf{i} + \mathbf{k}) \times (\mathbf{i} + \mathbf{k}) + (3\mathbf{i} + \mathbf{j} + \mathbf{k}) \times (\mathbf{i} + \mathbf{j})\) | M1 |
| \(= -2\mathbf{i} + (-\mathbf{j}) + (-\mathbf{i} + \mathbf{j} + 2\mathbf{k})\) | A3 |
| \(= (-3\mathbf{i} + 2\mathbf{k})\) | |
| \((\mathbf{i} - \mathbf{j} + \mathbf{k}) \times (2\mathbf{i} + 3\mathbf{j}) + \mathbf{G} = (-3\mathbf{i} + 2\mathbf{k})\) | M1 A2 ft |
| \((-3\mathbf{i} + 2\mathbf{j} + 5\mathbf{k}) + \mathbf{G} = (-3\mathbf{i} + 2\mathbf{k})\) | A1 |
| \(\mathbf{G} = (-2\mathbf{j} - 3\mathbf{k})\) Nm | A1 |
| (9) | |
| (11 marks) |
Notes
First M1 consistent use of \(\mathbf{r} \times \mathbf{F}\), with correct no. of terms
First A3 for 3 correct vector products, −1 for each incorrect product (A1A1A0)
Second M1 for comparing the rotational effect of the 2 systems about \(O\), to give an equation: Their \(\Sigma\,\mathbf{r} \times \mathbf{F} = \mathbf{G} + (\mathbf{i} - \mathbf{j} + \mathbf{k}) \times\) their \(\mathbf{R}\) with correct terms (M0 if term missing)
Second A2 ft, for the equation ft on their \(\mathbf{R}\) and their \(\Sigma\,\mathbf{r} \times \mathbf{F}\), but no products need to be evaluated.
Sixth A1 for a correct equation with all products evaluated
Seventh A1 for the answer. Units not needed.
3(b) Alt
| Moments about (1, −1, 1): | |
| \((-\mathbf{i} + 5\mathbf{j} - 2\mathbf{k}) \times (2\mathbf{j} - \mathbf{k}) + (\mathbf{i} + \mathbf{j}) \times (\mathbf{i} + \mathbf{k}) + (2\mathbf{i} + 2\mathbf{j}) \times (\mathbf{i} + \mathbf{j})\) | M1 A2 |
| \(= -\mathbf{i} - \mathbf{j} - 2\mathbf{k} + \mathbf{i} - \mathbf{j} - \mathbf{k} + 0\) | A3 |
| \(= -2\mathbf{j} - 3\mathbf{k}\) | |
| Comparing the 2 systems: \(\ \mathbf{0} + \mathbf{G} = (-2\mathbf{j} - 3\mathbf{k})\) Nm | M1 A1 ft |
| \(\mathbf{G} = (-2\mathbf{j} - 3\mathbf{k})\) Nm | A1 |
| (9) |
First M1 consistent use of \(\Sigma\,\mathbf{r} \times \mathbf{F}\), with correct no. of terms, with \(\mathbf{r}\) relative to (1, −1, 1)
First A2 for correct terms, −1 each incorrect vector
First A3 for 3 correct vector products, −1 for each incorrect product
Second M1 for comparing the rotational effect of the 2 systems about (1, −1, 1), to give an equation: \(\mathbf{0} + \mathbf{G} =\) their \(\Sigma\,\mathbf{r} \times \mathbf{F}\)
Sixth A1, ft on their \(\Sigma\,\mathbf{r} \times \mathbf{F}\), for a correct equation
Seventh A1 for the answer. Units not needed.