| Exam Board | CAIE |
|---|---|
| Module | M2 (Mechanics 2) |
| Year | 2005 |
| Session | November |
| Marks | 6 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Centre of Mass 2 |
| Type | Lamina in equilibrium with applied force |
| Difficulty | Standard +0.8 This is a multi-part centre of mass problem requiring: (i) finding the centroid of a composite trapezium shape using integration or decomposition methods, and (ii) applying equilibrium conditions (moments and forces) with the lamina suspended at an angle. The geometric setup and moment calculation about a point with two unknowns requires systematic problem-solving beyond routine exercises, though the techniques are standard for Further Maths Mechanics. |
| Spec | 6.04c Composite bodies: centre of mass6.04e Rigid body equilibrium: coplanar forces |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| M1 | For obtaining an equation in \(\bar{x}\) by taking moments about, for example, \(BD\) | |
| \(0.6W \times 1 - 0.4W \times (2/3) = W\bar{x}\) or \(\frac{1}{2}(3\times1)\times1 - \frac{1}{2}(2\times1)\times(2/3) = (3/2+1)\bar{x}\) | A1 | Any correct equation in \(\bar{x}\), with or without \(W\) throughout |
| Distance is \(1/3\) m | A1 | [3] |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| \(3T = (8/3)W\) or \(3F_C = (1/3)W\) | M1 | For taking moments about \(C\) or about \(BD\) |
| Tension is \(8W/9\) or force at \(C = W/9\) | A1 ft | ft for \(T = (1 - \bar{x}/3)W\) or \(F_C = (\bar{x}/3)W\) |
| Force at \(C = W/9\) or tension is \(8W/9\) | A1 ft | [3] |
## Question 3:
### Part (i):
| Answer/Working | Mark | Guidance |
|---|---|---|
| | M1 | For obtaining an equation in $\bar{x}$ by taking moments about, for example, $BD$ |
| $0.6W \times 1 - 0.4W \times (2/3) = W\bar{x}$ or $\frac{1}{2}(3\times1)\times1 - \frac{1}{2}(2\times1)\times(2/3) = (3/2+1)\bar{x}$ | A1 | Any correct equation in $\bar{x}$, with or without $W$ throughout |
| Distance is $1/3$ m | A1 | **[3]** |
### Part (ii):
| Answer/Working | Mark | Guidance |
|---|---|---|
| $3T = (8/3)W$ or $3F_C = (1/3)W$ | M1 | For taking moments about $C$ or about $BD$ |
| Tension is $8W/9$ or force at $C = W/9$ | A1 ft | ft for $T = (1 - \bar{x}/3)W$ or $F_C = (\bar{x}/3)W$ |
| Force at $C = W/9$ or tension is $8W/9$ | A1 ft | **[3]** |
---3\\
\includegraphics[max width=\textwidth, alt={}, center]{a20a6641-d771-4c89-b40f-168a0c61f99d-3_293_1045_267_550}
A uniform lamina $A B C D$ is in the form of a trapezium in which $A B$ and $D C$ are parallel and have lengths 2 m and 3 m respectively. $B D$ is perpendicular to the parallel sides and has length 1 m (see diagram).\\
(i) Find the distance of the centre of mass of the lamina from $B D$.
The lamina has weight $W \mathrm {~N}$ and is in equilibrium, suspended by a vertical string attached to the lamina at $B$. The lamina rests on a vertical support at $C$. The lamina is in a vertical plane with $A B$ and $D C$ horizontal.\\
(ii) Find, in terms of $W$, the tension in the string and the magnitude of the force exerted on the lamina at $C$.
\hfill \mbox{\textit{CAIE M2 2005 Q3 [6]}}