| Exam Board | AQA |
|---|---|
| Module | M1 (Mechanics 1) |
| Year | 2008 |
| Session | June |
| Marks | 7 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Vectors Introduction & 2D |
| Type | Resultant of three coplanar forces |
| Difficulty | Moderate -0.8 This is a standard M1 mechanics question requiring resolution of forces into components, vector addition, and basic trigonometry. The steps are routine: resolve each force into i and j components, sum them, find magnitude using Pythagoras, and find angle using tan^(-1). No novel problem-solving or conceptual insight required—purely algorithmic application of well-practiced techniques. |
| Spec | 3.03a Force: vector nature and diagrams3.03p Resultant forces: using vectors |
| Answer | Marks | Guidance |
|---|---|---|
| \(F = 5j + 8i - 7j = 8i - 2j\) | M1, A1 | Adding the two forces. For incorrect answers, evidence of adding must be seen. Correct resultant. |
| Answer | Marks | Guidance |
|---|---|---|
| \(F = \sqrt{8^2 + 2^2} = \sqrt{68} = 8.25\) N | M1, A1F | Finding magnitude (must see addition and not subtraction). Correct magnitude. Accept \(2\sqrt{17}\), \(\sqrt{68}\) or AWRT 8.25 (eg 8.246) |
| Answer | Marks | Guidance |
|---|---|---|
| Diagram with force in correct quadrant and with correct direction shown by arrow. | B1 | |
| \(\tan \alpha = \frac{2}{8}\) | M1 | Using trig to find angle: if tan, 8 in denominator; if sin or cos, 8.25 or their answer to part (b) in denominator |
| \(\alpha = 14.0°\) | A1 | Correct angle. Accept 14.1 or 14 or AWRT 14.0 (eg 14.04). M1 and A1 not dependent on B1 |
**2(a)**
| $F = 5j + 8i - 7j = 8i - 2j$ | M1, A1 | Adding the two forces. For incorrect answers, evidence of adding must be seen. Correct resultant. |
**2(b)**
| $F = \sqrt{8^2 + 2^2} = \sqrt{68} = 8.25$ N | M1, A1F | Finding magnitude (must see addition and not subtraction). Correct magnitude. Accept $2\sqrt{17}$, $\sqrt{68}$ or AWRT 8.25 (eg 8.246) |
**2(c)**
| Diagram with force in correct quadrant and with correct direction shown by arrow. | B1 |
| $\tan \alpha = \frac{2}{8}$ | M1 | Using trig to find angle: if tan, 8 in denominator; if sin or cos, 8.25 or their answer to part (b) in denominator |
| $\alpha = 14.0°$ | A1 | Correct angle. Accept 14.1 or 14 or AWRT 14.0 (eg 14.04). M1 and A1 not dependent on B1 |2 The diagram shows three forces and the perpendicular unit vectors $\mathbf { i }$ and $\mathbf { j }$, which all lie in the same plane.\\
\includegraphics[max width=\textwidth, alt={}, center]{a381686b-0b1e-41ba-b88f-be1601e42098-2_415_398_1507_605}\\
\includegraphics[max width=\textwidth, alt={}, center]{a381686b-0b1e-41ba-b88f-be1601e42098-2_172_166_1567_1217}
\begin{enumerate}[label=(\alph*)]
\item Express the resultant of the three forces in terms of $\mathbf { i }$ and $\mathbf { j }$.
\item Find the magnitude of the resultant force.
\item Draw a diagram to show the direction of the resultant force, and find the angle that it makes with the unit vector $\mathbf { i }$.
\end{enumerate}
\hfill \mbox{\textit{AQA M1 2008 Q2 [7]}}