M5 June 2006 Q4
4. A force system consists of three forces \(\mathbf{F}_1\), \(\mathbf{F}_2\) and \(\mathbf{F}_3\) acting on a rigid body.
\(\mathbf{F}_1 = (\mathbf{i} + 2\mathbf{j})\) N and acts at the point with position vector \((-\mathbf{i} + 4\mathbf{j})\) m.
\(\mathbf{F}_2 = (-\mathbf{j} + \mathbf{k})\) N and acts at the point with position vector \((2\mathbf{i} + \mathbf{j} + \mathbf{k})\) m.
\(\mathbf{F}_3 = (3\mathbf{i} - \mathbf{j} + \mathbf{k})\) N and acts at the point with position vector \((\mathbf{i} - \mathbf{j} + 2\mathbf{k})\) m.
It is given that this system can be reduced to a single force \(\mathbf{R}\).
(a) Find \(\mathbf{R}\). (2)
(b) Find a vector equation of the line of action of \(\mathbf{R}\), giving your answer in the form \(\mathbf{r} = \mathbf{a} + \lambda\mathbf{b}\), where \(\mathbf{a}\) and \(\mathbf{b}\) are constant vectors and \(\lambda\) is a parameter. (10)
| Scheme | Marks |
|---|---|
| \(\mathbf{R} = \begin{pmatrix}1\\2\\0\end{pmatrix} + \begin{pmatrix}0\\-1\\1\end{pmatrix} + \begin{pmatrix}3\\-1\\1\end{pmatrix} = \begin{pmatrix}4\\0\\2\end{pmatrix} = (4\mathbf{i} + 2\mathbf{k})\) N | M1 A1 |
| (2) |
| Scheme | Marks |
|---|---|
| \(\begin{pmatrix}-1\\4\\0\end{pmatrix}\times\begin{pmatrix}1\\2\\0\end{pmatrix} + \begin{pmatrix}2\\1\\1\end{pmatrix}\times\begin{pmatrix}0\\-1\\1\end{pmatrix} + \begin{pmatrix}1\\-1\\2\end{pmatrix}\times\begin{pmatrix}3\\-1\\1\end{pmatrix}\) | M1 |
| \(= \begin{pmatrix}0\\0\\-6\end{pmatrix} + \begin{pmatrix}2\\-2\\-2\end{pmatrix} + \begin{pmatrix}1\\5\\2\end{pmatrix}\) | A1 A1 A1 |
| \(= \begin{pmatrix}3\\3\\-6\end{pmatrix}\) | A1 |
| \(\begin{pmatrix}x\\y\\z\end{pmatrix}\times\begin{pmatrix}4\\0\\2\end{pmatrix} = \begin{pmatrix}3\\3\\-6\end{pmatrix}\) | M1 |
| \(\begin{pmatrix}2y\\4z - 2x\\-4y\end{pmatrix} = \begin{pmatrix}3\\3\\-6\end{pmatrix}\) | A1ft |
| e.g. \(x = -\tfrac{3}{2},\ y = \tfrac{3}{2},\ z = 0\) | B1 |
| \(\mathbf{r} = \begin{pmatrix}-\frac{3}{2}\\\frac{3}{2}\\0\end{pmatrix} + \lambda\begin{pmatrix}2\\0\\1\end{pmatrix}\) | M1 A1 |
| (10) | |
| (12 marks) |