| Exam Board | CAIE |
|---|---|
| Module | M2 (Mechanics 2) |
| Year | 2003 |
| Session | November |
| Marks | 10 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Moments |
| Type | Lamina hinged at point with string support |
| Difficulty | Standard +0.3 This is a standard M2 moments problem requiring center of mass calculation for a composite lamina, then resolving forces and taking moments about a hinge. The geometry is straightforward (30° angle given), and the method is routine: find COM, take moments about A to find tension, then resolve vertically for the reaction. Slightly above average due to the multi-part nature and COM calculation, but follows standard textbook procedures with no novel insight required. |
| Spec | 6.04d Integration: for centre of mass of laminas/solids6.05a Angular velocity: definitions |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| e.g. For taking moments about \(BC\) | M1 | |
| Distance of centre of mass of triangular portion is \(9.5 + \frac{1}{3} \times 6\ (= 11.5)\) | B1 | |
| \(8 \times 9.5 \times 4.75 + \frac{1}{2} \times 8 \times 6 \times 11.5 = \left(8 \times 9.5 + \frac{1}{2} \times 8 \times 6\right)\bar{x}\) | A1ft | |
| Distance is 6.37 cm | A1 | |
| Alternative: Moments about axis through \(A\) perpendicular to \(AB\) | M1 | |
| Distance of C.O.M. of triangular piece removed is 2 | B1 | |
| \((8 \times 15.5) \times 7.75 - \left(\frac{1}{2} \times 8 \times 6\right) \times 2 = (124 - 20)\bar{x}_1\) | A1ft | |
| \((\bar{x}_1 = 9.13)\) therefore distance is 6.37 cm | A1 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| For taking moments about \(A\) | M1 | |
| For LHS of \(80(15.5 - 6.37) = T \times 15.5\sin30°\) | A1ft | |
| For RHS of above equation | A1 | |
| Tension is 94.2 N | A1 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| For resolving forces on the lamina vertically (3 term equation) \((V = 80 - 94.2 \times 0.5)\) or taking moments about \(B\) \((15.5V = 8 \times 10 \times 6.37)\) | M1 | |
| Magnitude of vertical component is 32.9 N | A1ft |
# Question 4(i):
| Answer | Mark | Guidance |
|--------|------|----------|
| e.g. For taking moments about $BC$ | M1 | |
| Distance of centre of mass of triangular portion is $9.5 + \frac{1}{3} \times 6\ (= 11.5)$ | B1 | |
| $8 \times 9.5 \times 4.75 + \frac{1}{2} \times 8 \times 6 \times 11.5 = \left(8 \times 9.5 + \frac{1}{2} \times 8 \times 6\right)\bar{x}$ | A1ft | |
| Distance is 6.37 cm | A1 | |
| Alternative: Moments about axis through $A$ perpendicular to $AB$ | M1 | |
| Distance of C.O.M. of triangular piece removed is 2 | B1 | |
| $(8 \times 15.5) \times 7.75 - \left(\frac{1}{2} \times 8 \times 6\right) \times 2 = (124 - 20)\bar{x}_1$ | A1ft | |
| $(\bar{x}_1 = 9.13)$ therefore distance is 6.37 cm | A1 | |
# Question 4(ii):
| Answer | Mark | Guidance |
|--------|------|----------|
| For taking moments about $A$ | M1 | |
| For LHS of $80(15.5 - 6.37) = T \times 15.5\sin30°$ | A1ft | |
| For RHS of above equation | A1 | |
| Tension is 94.2 N | A1 | |
# Question 4(iii):
| Answer | Mark | Guidance |
|--------|------|----------|
| For resolving forces on the lamina vertically (3 term equation) $(V = 80 - 94.2 \times 0.5)$ or taking moments about $B$ $(15.5V = 8 \times 10 \times 6.37)$ | M1 | |
| Magnitude of vertical component is 32.9 N | A1ft | |
---(i) Show that the distance of the centre of mass of the lamina from the side $B C$ is 6.37 cm .
\begin{figure}[h]
\begin{center}
\includegraphics[alt={},max width=\textwidth]{be83d46f-bf5b-4382-b424-bb5067626adc-3_671_608_1050_772}
\captionsetup{labelformat=empty}
\caption{Fig. 2}
\end{center}
\end{figure}
The lamina is smoothly hinged to a wall at $A$ and is supported, with $A B$ horizontal, by a light wire attached at $B$. The other end of the wire is attached to a point on the wall, vertically above $A$, such that the wire makes an angle of $30 ^ { \circ }$ with $A B$ (see Fig. 2). The mass of the lamina is 8 kg . Find\\
(ii) the tension in the wire,\\
(iii) the magnitude of the vertical component of the force acting on the lamina at $A$.
\hfill \mbox{\textit{CAIE M2 2003 Q4 [10]}}