| Exam Board | CAIE |
|---|---|
| Module | M2 (Mechanics 2) |
| Year | 2003 |
| Session | June |
| Marks | 5 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Centre of Mass 1 |
| Type | Cone stability and toppling conditions |
| Difficulty | Standard +0.3 This is a straightforward centre of mass problem requiring knowledge that the COM of a uniform solid hemisphere is at 3r/8 from the base, then simple geometry to compare this with the vertical height. Part (ii) follows directly from part (i) using the principle that the object topples if COM is beyond the pivot. Standard M2 application with no novel problem-solving required. |
| Spec | 6.04b Find centre of mass: using symmetry |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| \(OQ = 4\tan 20° = 1.456\) | B1 | |
| \(OG = 1.5\) | B1 | |
| \(G\) not between \(O\) and \(Q\) (all calculations correct) | B1 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| Hemisphere does not fall on to its plane face | *B1 ft | |
| Because the moment about \(P\) is clockwise or the centre of mass is to right of \(PQ\) | (dep)* B1 ft |
# Question 2:
## Part (i):
| Answer | Mark | Guidance |
|--------|------|----------|
| $OQ = 4\tan 20° = 1.456$ | B1 | |
| $OG = 1.5$ | B1 | |
| $G$ not between $O$ and $Q$ (all calculations correct) | B1 | |
## Part (ii):
| Answer | Mark | Guidance |
|--------|------|----------|
| Hemisphere does not fall on to its plane face | *B1 ft | |
| Because the moment about $P$ is clockwise or the centre of mass is to right of $PQ$ | (dep)* B1 ft | |
---2\\
\includegraphics[max width=\textwidth, alt={}, center]{7f8646df-a7d8-4ca1-a6ee-3ceab6bb83af-2_439_608_1181_772}
A uniform solid hemisphere, with centre $O$ and radius 4 cm , is held so that a point $P$ of its rim is in contact with a horizontal surface. The plane face of the hemisphere makes an angle of $70 ^ { \circ }$ with the horizontal. $Q$ is the point on the axis of symmetry of the hemisphere which is vertically above $P$. The diagram shows the vertical cross-section of the hemisphere which contains $O , P$ and $Q$.\\
(i) Determine whether or not the centre of mass of the hemisphere is between $O$ and $Q$.
The hemisphere is now released.\\
(ii) State whether or not the hemisphere falls on to its plane face, giving a reason for your answer.
\hfill \mbox{\textit{CAIE M2 2003 Q2 [5]}}