| Exam Board | Edexcel |
|---|---|
| Module | M1 (Mechanics 1) |
| Marks | 8 |
| 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 vector mechanics question requiring only basic vector addition, Pythagoras for magnitude, and arctangent for angle. All steps are routine with no problem-solving insight needed—easier than the average A-level question which typically requires more conceptual depth or multi-stage reasoning. |
| Spec | 1.10c Magnitude and direction: of vectors1.10d Vector operations: addition and scalar multiplication3.03d Newton's second law: 2D vectors3.03p Resultant forces: using vectors |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Marks | Guidance |
| \(\mathbf{R} = (5\mathbf{i}-3\mathbf{j})+(3\mathbf{i}+2\mathbf{j})+(4\mathbf{i}-5\mathbf{j}) = 12\mathbf{i}-6\mathbf{j}\) | M1 A1 | |
| mag. of \(\mathbf{R} = \sqrt{12^2+(-6)^2} = \sqrt{180} = 6\sqrt{5}\) | M1 A1 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Marks | Guidance |
| \(\mathbf{a} = \frac{\mathbf{F}}{m} = \frac{1}{8}(12\mathbf{i}-6\mathbf{j}) = \frac{3}{2}\mathbf{i} - \frac{3}{4}\mathbf{j}\) | M1 A1 | |
| req'd angle \(= \tan^{-1}\frac{^3/_2}{^3/_4} = \tan^{-1}\frac{1}{2} = 26.6°\) (1dp) | M1 A1 | (8) |
## Question 2:
### Part (a)
| Answer/Working | Marks | Guidance |
|---|---|---|
| $\mathbf{R} = (5\mathbf{i}-3\mathbf{j})+(3\mathbf{i}+2\mathbf{j})+(4\mathbf{i}-5\mathbf{j}) = 12\mathbf{i}-6\mathbf{j}$ | M1 A1 | |
| mag. of $\mathbf{R} = \sqrt{12^2+(-6)^2} = \sqrt{180} = 6\sqrt{5}$ | M1 A1 | |
### Part (b)
| Answer/Working | Marks | Guidance |
|---|---|---|
| $\mathbf{a} = \frac{\mathbf{F}}{m} = \frac{1}{8}(12\mathbf{i}-6\mathbf{j}) = \frac{3}{2}\mathbf{i} - \frac{3}{4}\mathbf{j}$ | M1 A1 | |
| req'd angle $= \tan^{-1}\frac{^3/_2}{^3/_4} = \tan^{-1}\frac{1}{2} = 26.6°$ (1dp) | M1 A1 | **(8)** |
---2. A particle of mass 8 kg moves in a horizontal plane and is acted upon by three forces $\mathbf { F } _ { 1 } = ( 5 \mathbf { i } - 3 \mathbf { j } ) \mathrm { N } , \mathbf { F } _ { 2 } = ( 3 \mathbf { i } + 2 \mathbf { j } ) \mathrm { N }$ and $\mathbf { F } _ { 3 } = ( 4 \mathbf { i } - 5 \mathbf { j } ) \mathrm { N }$, where $\mathbf { i }$ and $\mathbf { j }$ are perpendicular horizontal unit vectors.
\begin{enumerate}[label=(\alph*)]
\item Find the magnitude, in newtons, of the resultant force which acts on the particle, giving your answer in the form $k \sqrt { } 5$.
\item Calculate, giving your answer in degrees correct to 1 decimal place, the angle the acceleration of the particle makes with the vector $\mathbf { i }$.
\end{enumerate}
\hfill \mbox{\textit{Edexcel M1 Q2 [8]}}