| Exam Board | OCR MEI |
|---|---|
| Module | M1 (Mechanics 1) |
| Year | 2005 |
| Session | January |
| Marks | 7 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Forces, equilibrium and resultants |
| Type | Particle suspended by strings |
| Difficulty | Moderate -0.8 This is a straightforward statics problem requiring a force diagram and resolution of forces in equilibrium. Part (ii) is a standard triangle of forces calculation with given angles, and part (iii) requires recognizing that parallel strings cannot maintain equilibrium—both are routine M1 techniques with no novel problem-solving required. |
| Spec | 3.03n Equilibrium in 2D: particle under forces |
| Answer | Marks | Guidance |
|---|---|---|
| Diagram with \(T_{BA}\), \(T_{BC}\), and 400 N | B1 | Different labels. All forces present with arrows in correct directions. Condone no angles. |
| Total: 1 |
| Answer | Marks | Guidance |
|---|---|---|
| Using triangle of forces | M1 | Attempt at triangle of forces. Ignore angles and arrows. Accept 90, 60, 30 triangle. |
| B1 | Triangle, arrows, labels and angles correct | |
| Triangle isosceles so tension in BC is 400 N. Tension in BA is \(2 \times 400 \times \cos 30° = 400\sqrt{3}\) N (693 N, (3 s. f.)) | A1 | cao |
| F1 | FT BC only | |
| [If resolution used, M1 for 1 eqn; M1 for 2nd eqn + attempt to elim; A1; F1. For M marks all forces present but allow \(s \leftrightarrow c\) and sign errors. No extra forces. If Lami used: M1 first pair of equations in correct format, condone wrong angles. A1. M1 second pair in correct format, with correct angles.F1 FT their first answer if necessary.] | ||
| Total: 4 |
| Answer | Marks | Guidance |
|---|---|---|
| Resolve at B perpendicular to the line ABC | E1 | Attempt to argue unbalanced force |
| Weight has unbalanced component in this direction | E1 | Complete, convincing argument. |
| [or Resolve horiz and establish tensions equal E1. Resolve vert to show inconsistency. E1] | ||
| Total: 2 |
**Part (i)**
| Diagram with $T_{BA}$, $T_{BC}$, and 400 N | B1 | Different labels. All forces present with arrows in correct directions. Condone no angles. |
| | | **Total: 1** |
**Part (ii)**
| Using triangle of forces | M1 | Attempt at triangle of forces. Ignore angles and arrows. Accept 90, 60, 30 triangle. |
| | B1 | Triangle, arrows, labels and angles correct |
| Triangle isosceles so tension in BC is 400 N. Tension in BA is $2 \times 400 \times \cos 30° = 400\sqrt{3}$ N (693 N, (3 s. f.)) | A1 | cao |
| | F1 | FT BC only |
| | | [If resolution used, M1 for 1 eqn; M1 for 2nd eqn + attempt to elim; A1; F1. For M marks all forces present but allow $s \leftrightarrow c$ and sign errors. No extra forces. If Lami used: M1 first pair of equations in correct format, condone wrong angles. A1. M1 second pair in correct format, with **correct** angles.F1 FT their first answer if necessary.] |
| | | **Total: 4** |
**Part (iii)**
| Resolve at B perpendicular to the line ABC | E1 | Attempt to argue unbalanced force |
| Weight has unbalanced component in this direction | E1 | Complete, convincing argument. |
| | | [or Resolve horiz and establish tensions equal E1. Resolve vert to show inconsistency. E1] |
| | | **Total: 2** |
---5 A small box B of weight 400 N is held in equilibrium by two light strings AB and BC . The string BC is fixed at C . The end A of string AB is fixed so that AB is at an angle $\alpha$ to the vertical where $\alpha < 60 ^ { \circ }$. String BC is at $60 ^ { \circ }$ to the vertical. This information is shown in Fig. 5.
\begin{figure}[h]
\begin{center}
\includegraphics[alt={},max width=\textwidth]{c84a748a-a6f4-48c5-b864-fe543569bdf5-3_424_472_1599_774}
\captionsetup{labelformat=empty}
\caption{Fig. 5}
\end{center}
\end{figure}
(i) Draw a labelled diagram showing all the forces acting on the box.\\
(ii) In one situation string AB is fixed so that $\alpha = 30 ^ { \circ }$.
By drawing a triangle of forces, or otherwise, calculate the tension in the string BC and the tension in the string AB .\\
(iii) Show carefully, but briefly, that the box cannot be in equilibrium if $\alpha = 60 ^ { \circ }$ and BC remains at $60 ^ { \circ }$ to the vertical.
\hfill \mbox{\textit{OCR MEI M1 2005 Q5 [7]}}