| Exam Board | Edexcel |
|---|---|
| Module | M1 (Mechanics 1) |
| Year | 2009 |
| Session | June |
| Marks | 6 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Forces, equilibrium and resultants |
| Type | Forces in vector form: resultant and acceleration |
| Difficulty | Moderate -0.8 This is a straightforward M1 vectors question requiring basic vector operations. Part (a) uses simple trigonometry (tan θ = 1/2 from components) to find an angle. Part (b) requires setting the j-component of the resultant to zero, giving a simple linear equation 2p - 3 = 0. Both parts are routine applications of standard techniques with no problem-solving insight needed, making this easier than average. |
| Spec | 1.10a Vectors in 2D: i,j notation and column vectors1.10b Vectors in 3D: i,j,k notation1.10c Magnitude and direction: of vectors1.10d Vector operations: addition and scalar multiplication3.03a Force: vector nature and diagrams3.03p Resultant forces: using vectors |
| Answer | Marks |
|---|---|
| (a) \(\tan \theta = \frac{5}{7} \Rightarrow \theta = 26.6°\) | M1 A1 (2) |
| (b) \(\mathbf{R} = (\mathbf{i} - 3\mathbf{j}) + (p\mathbf{i} + 2p\mathbf{j}) = (1 + p)\mathbf{i} + (-3 + 2p)\mathbf{j}\) | M1 A1 |
| \(\mathbf{R}\) is parallel to \(\mathbf{i} \Rightarrow (-3 + 2p) = 0\) | DM1 |
| \(\Rightarrow p = \frac{3}{2}\) | A1 |
| Total: (4) [6] |
**(a)** $\tan \theta = \frac{5}{7} \Rightarrow \theta = 26.6°$ | M1 A1 (2) |
**(b)** $\mathbf{R} = (\mathbf{i} - 3\mathbf{j}) + (p\mathbf{i} + 2p\mathbf{j}) = (1 + p)\mathbf{i} + (-3 + 2p)\mathbf{j}$ | M1 A1 |
$\mathbf{R}$ is parallel to $\mathbf{i} \Rightarrow (-3 + 2p) = 0$ | DM1 |
$\Rightarrow p = \frac{3}{2}$ | A1 |
| **Total: (4) [6]** |A particle is acted upon by two forces $\mathbf{F}_1$ and $\mathbf{F}_2$, given by
$\mathbf{F}_1 = (\mathbf{i} - 3\mathbf{j})$ N,
$\mathbf{F}_2 = (p\mathbf{i} + 2p\mathbf{j})$ N, where $p$ is a positive constant.
\begin{enumerate}[label=(\alph*)]
\item Find the angle between $\mathbf{F}_2$ and $\mathbf{j}$. [2]
\end{enumerate}
The resultant of $\mathbf{F}_1$ and $\mathbf{F}_2$ is $\mathbf{R}$. Given that $\mathbf{R}$ is parallel to $\mathbf{i}$,
\begin{enumerate}[label=(\alph*)]
\setcounter{enumi}{1}
\item find the value of $p$. [4]
\end{enumerate}
\hfill \mbox{\textit{Edexcel M1 2009 Q2 [6]}}