| Exam Board | OCR MEI |
|---|---|
| Module | Paper 1 (Paper 1) |
| Year | 2020 |
| Session | November |
| Marks | 9 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Motion on a slope |
| Type | Equilibrium on slope with force parallel to slope |
| Difficulty | Moderate -0.8 This is a straightforward mechanics question requiring standard resolution of forces on an inclined plane. Students must express weight in component form (routine use of sin/cos 30°), apply equilibrium conditions (ΣF = 0), and solve simultaneous equations. All steps are textbook-standard with no novel problem-solving required, making it easier than average but not trivial due to the vector notation and multi-part structure. |
| Spec | 1.10b Vectors in 3D: i,j,k notation1.10d Vector operations: addition and scalar multiplication3.03b Newton's first law: equilibrium3.03d Newton's second law: 2D vectors3.03f Weight: W=mg |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | AOs |
| \(\mathbf{W} = (-mg\sin30°)\mathbf{i} + (-mg\cos30°)\mathbf{j}\) | M1 | 3.1b |
| \(\mathbf{W} = \left(-\frac{1}{2}mg\right)\mathbf{i} + \left(-\frac{\sqrt{3}}{2}mg\right)\mathbf{j}\) | A1 [2] | 2.5 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | AOs |
| \(\mathbf{W} + \mathbf{R} + \mathbf{F} = \mathbf{0}\) | B1 [1] | 2.5 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | AOs |
| \(\mathbf{R} = R\mathbf{j}\) | B1 | 3.1b |
| \([(-mg\sin30°)\mathbf{i} + (-mg\cos30°)\mathbf{j} + (6\mathbf{i}+8\mathbf{j}) + R\mathbf{j} = \mathbf{0}]\) | ||
| \(\mathbf{i}\) component: \(-mg\sin30° + 6 = 0\) | M1 | 3.1a |
| Correct equation in \(\mathbf{i}\) direction | A1 | 1.1 |
| giving \(m = 1.22\) to 3 sf | A1 | 1.1 |
| \(\mathbf{j}\) component: \(-12\cos30° + 8 + R = 0\) | M1 | 3.1a |
| \(R = 2.39\) so magnitude is 2.39 N | A1 [6] | 3.2a |
## Question 15:
### Part (a):
| Answer | Marks | AOs | Guidance |
|--------|-------|-----|----------|
| $\mathbf{W} = (-mg\sin30°)\mathbf{i} + (-mg\cos30°)\mathbf{j}$ | M1 | 3.1b | Attempting to resolve the weight; Allow sin/cos interchange and sign errors for the method mark; $mg$ must be seen for the method mark |
| $\mathbf{W} = \left(-\frac{1}{2}mg\right)\mathbf{i} + \left(-\frac{\sqrt{3}}{2}mg\right)\mathbf{j}$ | A1 [2] | 2.5 | All correct in this vector form |
### Part (b):
| Answer | Marks | AOs | Guidance |
|--------|-------|-----|----------|
| $\mathbf{W} + \mathbf{R} + \mathbf{F} = \mathbf{0}$ | B1 [1] | 2.5 | Allow any rearrangement of this; Allow if their expression for $\mathbf{W}$ is used instead of $\mathbf{W}$ |
### Part (c):
| Answer | Marks | AOs | Guidance |
|--------|-------|-----|----------|
| $\mathbf{R} = R\mathbf{j}$ | B1 | 3.1b | Allow for any clear indication that $\mathbf{R}$ is a multiple of $\mathbf{j}$ or that it has no component in the $\mathbf{i}$ direction; May be implied with an equation for the i direction with two terms and an equation in the j direction with three terms |
| $[(-mg\sin30°)\mathbf{i} + (-mg\cos30°)\mathbf{j} + (6\mathbf{i}+8\mathbf{j}) + R\mathbf{j} = \mathbf{0}]$ | | | |
| $\mathbf{i}$ component: $-mg\sin30° + 6 = 0$ | M1 | 3.1a | Forming equation from their $\mathbf{i}$ terms, or equivalent by resolving parallel to the plane. FT their $\mathbf{W}$ |
| Correct equation in $\mathbf{i}$ direction | A1 | 1.1 | |
| giving $m = 1.22$ to 3 sf | A1 | 1.1 | AG; $1.22\times9.8 = 11.956$ |
| $\mathbf{j}$ component: $-12\cos30° + 8 + R = 0$ | M1 | 3.1a | Equation from the $\mathbf{j}$ terms (must include all three terms), oe, and using value of $m$; 12 is the value for $mg$ |
| $R = 2.39$ so magnitude is 2.39 N | A1 [6] | 3.2a | Accept arwt 2.4 |15 Fig. 15 shows a particle of mass $m \mathrm {~kg}$ on a smooth plane inclined at $30 ^ { \circ }$ to the horizontal. Unit vectors $\mathbf { i }$ and $\mathbf { j }$ are parallel and perpendicular to the plane, in the directions shown.
\begin{figure}[h]
\begin{center}
\includegraphics[alt={},max width=\textwidth]{7de77679-59c0-4431-a9cb-6ab11d2f9062-09_369_536_349_246}
\captionsetup{labelformat=empty}
\caption{Fig. 15}
\end{center}
\end{figure}
\begin{enumerate}[label=(\alph*)]
\item Express the weight $\mathbf { W }$ of the particle in terms of $m , g , \mathbf { i }$ and $\mathbf { j }$.
The particle is held in equilibrium by a force $\mathbf { F }$, and the normal reaction of the plane on the particle is denoted by $\mathbf { R }$. The units for both $\mathbf { F }$ and $\mathbf { R }$ are newtons.
\item Write down an equation relating $\mathbf { W } , \mathbf { R }$ and $\mathbf { F }$.
\item Given that $\mathbf { F } = 6 \mathbf { i } + 8 \mathbf { j }$,
\begin{itemize}
\item show that $m = 1.22$ correct to 3 significant figures,
\item find the magnitude of $\mathbf { R }$.
\end{itemize}
\end{enumerate}
\hfill \mbox{\textit{OCR MEI Paper 1 2020 Q15 [9]}}