| Exam Board | OCR |
|---|---|
| Module | M3 (Mechanics 3) |
| Year | 2011 |
| Session | June |
| Marks | 10 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Moments |
| Type | Rod or block on rough surface in limiting equilibrium (no wall) |
| Difficulty | Standard +0.8 This is a challenging M3 statics problem requiring systematic application of equilibrium conditions to a two-rod system with a connecting string. Students must work with multiple bodies, resolve forces in two directions, take moments about strategic points, and handle the geometry carefully (finding angles from the 2.4m height constraint). The 'show that' part provides scaffolding, but parts (ii) and (iii) require independent problem-solving across three interconnected equilibrium equations. This is significantly harder than routine single-body statics problems but remains within the M3 syllabus scope. |
| Spec | 3.04b Equilibrium: zero resultant moment and force6.04e Rigid body equilibrium: coplanar forces |
| Answer | Marks | Guidance |
|---|---|---|
| \((1.8 + 3.2)R_B = (3.2 + 0.9)x300 + 1.6x400\) \(\text{Force exerted on } AB \text{ is } 374 \text{ N}\) \(\text{Force exerted on } AC \text{ is } 326 \text{ N}\) | A1, A1, B1 [4] | For taking moments about \(C\) for the whole for M1 need 3 terms; allow 1 sign error and/or 1 length error and/or still including \(\sin/\cos\) or for taking moments about \(B\) for whole \((1.8 + 3.2)R_C = (1.8 + 1.6)x400 + 0.9x300\) giving force on \(AC\) first: M1A1A1 |
| Answer | Marks | Guidance |
|---|---|---|
| \(0.9x300 + 1.2T = 1.8x374\) \(\text{Tension is } 336 \text{ N}\) | M1, A1, A1 [3] | For taking moments about \(A\) for \(AB\) for M1 need 3 terms, allow 1 sign error and/or 1 length error and/or still including \(\sin/\cos\) or moments about \(A\) for \(AC\) \(1.6x400 + 1.2T = 3.2x326\) |
| Answer | Marks | Guidance |
|---|---|---|
| \(\text{Horizontal component is } 336 \text{ N to the left}\) \([Y = 374 - 300]\) \(\text{Vertical component is } 74 \text{ N downwards\) | B1ft, M1, A1ft [3] | For resolving forces on \(AB\) vertically |
**Part (i):**
$(1.8 + 3.2)R_B = (3.2 + 0.9)x300 + 1.6x400$ $\text{Force exerted on } AB \text{ is } 374 \text{ N}$ $\text{Force exerted on } AC \text{ is } 326 \text{ N}$ | A1, A1, B1 [4] | For taking moments about $C$ for the whole for M1 need 3 terms; allow 1 sign error and/or 1 length error and/or still including $\sin/\cos$ or for taking moments about $B$ for whole $(1.8 + 3.2)R_C = (1.8 + 1.6)x400 + 0.9x300$ giving force on $AC$ first: M1A1A1 |
**Part (ii):**
$0.9x300 + 1.2T = 1.8x374$ $\text{Tension is } 336 \text{ N}$ | M1, A1, A1 [3] | For taking moments about $A$ for $AB$ for M1 need 3 terms, allow 1 sign error and/or 1 length error and/or still including $\sin/\cos$ or moments about $A$ for $AC$ $1.6x400 + 1.2T = 3.2x326$ | M1 |
**Part (iii):**
$\text{Horizontal component is } 336 \text{ N to the left}$ $[Y = 374 - 300]$ $\text{Vertical component is } 74 \text{ N downwards$ | B1ft, M1, A1ft [3] | For resolving forces on $AB$ vertically |
**General note:** Give credit for part (ii) done on the way to part (i) if not contradicted in (ii).\includegraphics{figure_2}
Two uniform rods $AB$ and $AC$, of lengths $3$ m and $4$ m respectively, have weights $300$ N and $400$ N respectively. The rods are freely jointed at $A$. The mid-points of the rods are joined by a light inextensible string. The rods are in equilibrium in a vertical plane with the string taut and $B$ and $C$ in contact with a smooth horizontal surface. The point $A$ is $2.4$ m above the surface (see diagram).
\begin{enumerate}[label=(\roman*)]
\item Show that the force exerted by the surface on $AB$ is $374$ N and find the force exerted by the surface on $AC$. [4]
\item Find the tension in the string. [3]
\item Find the horizontal and vertical components of the force exerted on $AB$ at $A$ and state their directions. [3]
\end{enumerate}
\hfill \mbox{\textit{OCR M3 2011 Q2 [10]}}