| Exam Board | Edexcel |
|---|---|
| Module | M1 (Mechanics 1) |
| Marks | 9 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Vectors Introduction & 2D |
| Type | Forces in equilibrium (find unknowns) |
| Difficulty | Moderate -0.3 This is a standard M1 equilibrium problem requiring conversion of bearings to component form and applying the equilibrium condition (sum of forces = 0). While it involves multiple steps (converting bearings, resolving forces, solving simultaneous equations, finding magnitude/direction), these are all routine procedures taught explicitly in M1 with no novel insight required. Slightly easier than average due to the straightforward nature of the calculations. |
| Spec | 3.03a Force: vector nature and diagrams3.03n Equilibrium in 2D: particle under forces3.03p Resultant forces: using vectors |
| Answer | Marks | Guidance |
|---|---|---|
| (a) Total force to north: \(9\cos 30° - 12\cos 45° = -0.691 \text{ N}\) | M1 A1 | |
| Total force to east: \(9\sin 30° - 12\sin 45° = -3.985 \text{ N}\) | M1 A1 | |
| So \(\mathbf{R} = -3.99\mathbf{i} + 0.69\mathbf{j}\); \(a = 3.99\), \(b = 0.69\) | A1 | |
| (b) \( | \mathbf{R} | = \sqrt{(3.985)^2 + (0.691)^2} = 4.04 \text{ N}\), at \(\tan^{-1} 5.77 = 080.1°\) |
(a) Total force to north: $9\cos 30° - 12\cos 45° = -0.691 \text{ N}$ | M1 A1 |
Total force to east: $9\sin 30° - 12\sin 45° = -3.985 \text{ N}$ | M1 A1 |
So $\mathbf{R} = -3.99\mathbf{i} + 0.69\mathbf{j}$; $a = 3.99$, $b = 0.69$ | A1 |
(b) $|\mathbf{R}| = \sqrt{(3.985)^2 + (0.691)^2} = 4.04 \text{ N}$, at $\tan^{-1} 5.77 = 080.1°$ | M1 A1 M1 A1 | 9 marks3. A particle is in equilibrium under the action of three forces $\mathbf { P } , \mathbf { Q }$ and $\mathbf { R }$ acting in the same horizontal plane. $P$ has magnitude 9 N and acts on a bearing of $030 ^ { \circ } . Q$ has magnitude 12 N . and acts on a bearing of $225 ^ { \circ }$.
\begin{enumerate}[label=(\alph*)]
\item Find the values of $a$ and $b$ such that $\mathbf { R } = ( a \mathbf { i } + b \mathbf { j } ) \mathrm { N }$, where $\mathbf { i }$ and $\mathbf { j }$ are unit vectors in the directions due East and due North respectively.
\item Calculate the magnitude and direction of $\mathbf { R }$
\end{enumerate}
\hfill \mbox{\textit{Edexcel M1 Q3 [9]}}