M5 June 2018 Q2
2. Three forces \(\mathbf{F}_1 = (a\mathbf{i} + b\mathbf{j} - 2\mathbf{k})\) N, \(\mathbf{F}_2 = (-\mathbf{i} + \mathbf{j} - 2\mathbf{k})\) N and \(\mathbf{F}_3 = (-\mathbf{i} - 3\mathbf{j} + \mathbf{k})\) N, where \(a\) and \(b\) are constants, act on a rigid body.
The force \(\mathbf{F}_1\) acts through the point with position vector \(\mathbf{k}\) m, the force \(\mathbf{F}_2\) acts through the point with position vector \((3\mathbf{i} - \mathbf{j} + \mathbf{k})\) m and the force \(\mathbf{F}_3\) acts through the point with position vector \((\mathbf{j} + 2\mathbf{k})\) m.
The system of three forces is equivalent to a single force \(\mathbf{R}\) acting through the origin together with a couple of moment \(\mathbf{G}\). The direction of \(\mathbf{R}\) is parallel to the direction of \(\mathbf{G}\).
Find the value of \(a\) and the value of \(b\). (11)
| Scheme | Marks |
|---|---|
| \(\mathbf{R} = \sum\mathbf{F}_i = \begin{pmatrix}a\\b\\-2\end{pmatrix}+\begin{pmatrix}-1\\1\\-2\end{pmatrix}+\begin{pmatrix}-1\\-3\\1\end{pmatrix}=\begin{pmatrix}a-2\\b-2\\-3\end{pmatrix}\) | M1 A1 |
| Taking moments about \(O\), | |
| \(\mathbf{G} = \begin{pmatrix}0\\0\\1\end{pmatrix}\times\begin{pmatrix}a\\b\\-2\end{pmatrix}+\begin{pmatrix}3\\-1\\1\end{pmatrix}\times\begin{pmatrix}-1\\1\\-2\end{pmatrix}+\begin{pmatrix}0\\1\\2\end{pmatrix}\times\begin{pmatrix}-1\\-3\\1\end{pmatrix}\) | M1 |
| \(= \begin{pmatrix}-b\\a\\0\end{pmatrix}+\begin{pmatrix}1\\5\\2\end{pmatrix}+\begin{pmatrix}7\\-2\\1\end{pmatrix}\) | A3 |
| \(= \begin{pmatrix}8-b\\a+3\\3\end{pmatrix}\) | A1 cao |
| \(8 - b = \lambda(a - 2)\) \(a + 3 = \lambda(b - 2)\) \(3 = \lambda \times -3\) | M1 A1ft |
| \(a = \dfrac{-7}{2};\ b = \dfrac{5}{2}\) | A1; A1 cao |
| (11) |
Notes
First M1 for adding the 3 forces
First A1 for a correct \(\mathbf{R}\)
Second M1 for moments about \(O\); allow \(\mathbf{F} \times \mathbf{r}\)
Second, third, fourth A1 : −1 each error or omission (A1 for each product)
Allow consistent negatives (if using \(\mathbf{F} \times \mathbf{r}\))
Fifth A1 cao
Third M1 for using ‘their \(\mathbf{G}\) parallel with their \(\mathbf{R}\)’ to obtain three equations
Sixth A1 ft Correct ft equations
Seventh A1 −3.5
Eighth A1 2.5