| Exam Board | Edexcel |
|---|---|
| Module | M1 (Mechanics 1) |
| Year | 2015 |
| Session | June |
| Marks | 6 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Forces, equilibrium and resultants |
| Type | Forces in vector form: equilibrium (find unknowns) |
| Difficulty | Moderate -0.8 This is a straightforward equilibrium problem requiring students to set the sum of forces equal to zero and solve two simultaneous linear equations. It tests basic vector addition and algebraic manipulation with no conceptual challenges or multi-step reasoning beyond the standard M1 technique. |
| Spec | 3.03m Equilibrium: sum of resolved forces = 0 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Marks | Guidance Notes |
| \((2\mathbf{i}+ 3a\mathbf{j}) + (2a\mathbf{i}+b\mathbf{j}) + (b\mathbf{i}+ 4\mathbf{j}) = \mathbf{0}\) | M1 | Use of resultant force \(= 0\) (seen or implied) |
| \(2a+b+2=0\); \(3a+b+4=0\) | M1 | Equation involving all three forces, compare \(\mathbf{i}\) or \(\mathbf{j}\) components to form equation in \(a\) and \(b\). Allow with \(\mathbf{i}\) or \(\mathbf{j}\). \(\lambda\mathbf{i}=\mu\mathbf{j}\) is M0 |
| A1 | Two correct scalar equations. No i/j | |
| \(a=-2; b=2\) | DM1 | Solve simultaneous equations to find \(a\) or \(b\). Dependent on previous M1 |
| A1 | \(a\) correct | |
| A1 | \(b\) correct | |
| Total: 6 |
# Question 1:
| Answer/Working | Marks | Guidance Notes |
|---|---|---|
| $(2\mathbf{i}+ 3a\mathbf{j}) + (2a\mathbf{i}+b\mathbf{j}) + (b\mathbf{i}+ 4\mathbf{j}) = \mathbf{0}$ | M1 | Use of resultant force $= 0$ (seen or implied) |
| $2a+b+2=0$; $3a+b+4=0$ | M1 | Equation involving all three forces, compare $\mathbf{i}$ or $\mathbf{j}$ components to form equation in $a$ and $b$. Allow with $\mathbf{i}$ or $\mathbf{j}$. $\lambda\mathbf{i}=\mu\mathbf{j}$ is M0 |
| | A1 | Two correct scalar equations. No i/j |
| $a=-2; b=2$ | DM1 | Solve simultaneous equations to find $a$ or $b$. Dependent on previous M1 |
| | A1 | $a$ correct |
| | A1 | $b$ correct |
| **Total: 6** | | |
---\begin{enumerate}
\item Three forces $\mathbf { F } _ { 1 } , \mathbf { F } _ { 2 }$ and $\mathbf { F } _ { 3 }$ act on a particle $P$.
\end{enumerate}
$$\mathbf { F } _ { 1 } = ( 2 \mathbf { i } + 3 a \mathbf { j } ) \mathrm { N } ; \quad \mathbf { F } _ { 2 } = ( 2 a \mathbf { i } + b \mathbf { j } ) \mathrm { N } ; \quad \mathbf { F } _ { 3 } = ( b \mathbf { i } + 4 \mathbf { j } ) \mathrm { N } .$$
The particle $P$ is in equilibrium under the action of these forces.\\
Find the value of $a$ and the value of $b$.\\
\hfill \mbox{\textit{Edexcel M1 2015 Q1 [6]}}