M1 June 2009 Q2
2. A particle is acted upon by two forces \(\mathbf{F}_1\) and \(\mathbf{F}_2\), given by
\[\mathbf{F}_1 = (\mathbf{i} - 3\mathbf{j})\text{ N},\]\[\mathbf{F}_2 = (p\mathbf{i} + 2p\mathbf{j})\text{ N, where } p \text{ is a positive constant.}\](a) Find the angle between \(\mathbf{F}_2\) and \(\mathbf{j}\). (2)
The resultant of \(\mathbf{F}_1\) and \(\mathbf{F}_2\) is \(\mathbf{R}\). Given that \(\mathbf{R}\) is parallel to \(\mathbf{i}\),
(b) find the value of \(p\). (4)
| Scheme | Marks |
|---|---|
| \(\tan\theta = \tfrac{p}{2p}\ \Rightarrow\ \theta = 26.6^\circ\) | M1 A1 |
| (2) |
| Scheme | Marks |
|---|---|
| \(\mathbf{R} = (\mathbf{i} - 3\mathbf{j}) + (p\mathbf{i} + 2p\mathbf{j}) = (1 + p)\mathbf{i} + (-3 + 2p)\mathbf{j}\) | M1 A1 |
| \(\mathbf{R}\) is parallel to \(\mathbf{i}\ \ \Rightarrow\ \ (-3 + 2p) = 0\) | DM1 |
| \(\Rightarrow\ p = \tfrac{3}{2}\) | A1 |
| (4) | |
| (6 marks) |