| Exam Board | CAIE |
|---|---|
| Module | M2 (Mechanics 2) |
| Year | 2009 |
| Session | November |
| Marks | 8 |
| 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 moments problem requiring knowledge of the center of mass of a circular sector (given in formula booklet), taking moments about the hinge, and resolving forces. It involves multiple steps but uses routine mechanics techniques without requiring novel insight or complex problem-solving. |
| Spec | 3.02d Constant acceleration: SUVAT formulae3.02i Projectile motion: constant acceleration model6.04d Integration: for centre of mass of laminas/solids |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| (i) \(OG = 2 \times 0.5\sin30° / (3 \times (\pi/6))\) \((= 1/\pi)\) | B1 | |
| M1 | For taking moments about \(O\) | |
| \(3 \times (\sin30°/\pi) = F \times 0.5\) | A1\(\sqrt{}\) | |
| \(F = 0.955\) | A1 | [4] |
| (ii) | M1 | For resolving forces on the lamina horizontally and vertically |
| \(X = F\cos60°\) \((= 0.477)\) | A1 | |
| \(Y = 3 - F\sin60°\) \((= 2.17)\) | A1 | |
| Magnitude is \(2.22\) N | A1\(\sqrt{}\) | ft \((F^2 - 3\sqrt{3}\,F + 9)^{\frac{1}{2}}\) [4] |
## Question 5:
| Answer/Working | Mark | Guidance |
|---|---|---|
| **(i)** $OG = 2 \times 0.5\sin30° / (3 \times (\pi/6))$ $(= 1/\pi)$ | B1 | |
| | M1 | For taking moments about $O$ |
| $3 \times (\sin30°/\pi) = F \times 0.5$ | A1$\sqrt{}$ | |
| $F = 0.955$ | A1 | **[4]** |
| **(ii)** | M1 | For resolving forces on the lamina horizontally and vertically |
| $X = F\cos60°$ $(= 0.477)$ | A1 | |
| $Y = 3 - F\sin60°$ $(= 2.17)$ | A1 | |
| Magnitude is $2.22$ N | A1$\sqrt{}$ | ft $(F^2 - 3\sqrt{3}\,F + 9)^{\frac{1}{2}}$ **[4]** |
---5\\
\includegraphics[max width=\textwidth, alt={}, center]{68acf474-5da2-4949-b3b2-fc42cd73bd4a-3_405_545_630_799}
A uniform lamina $A O B$ is in the shape of a sector of a circle with centre $O$ and radius 0.5 m , and has angle $A O B = \frac { 1 } { 3 } \pi$ radians and weight 3 N . The lamina is freely hinged at $O$ to a fixed point and is held in equilibrium with $A O$ vertical by a force of magnitude $F \mathrm {~N}$ acting at $B$. The direction of this force is at right angles to $O B$ (see diagram). Find\\
(i) the value of $F$,\\
(ii) the magnitude of the force acting on the lamina at $O$.
\hfill \mbox{\textit{CAIE M2 2009 Q5 [8]}}