| Exam Board | CAIE |
|---|---|
| Module | M2 (Mechanics 2) |
| Year | 2017 |
| Session | June |
| Marks | 7 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Centre of Mass 1 |
| Type | Solid with removed cylinder or hemisphere from solid |
| Difficulty | Standard +0.3 This is a straightforward centre of mass problem using standard formulas. Part (i) requires applying the composite body formula with given hemisphere volumes and the standard result for hemisphere COM position (3r/8 from base). Part (ii) is a simple application of the COM formula with known positions and masses. The question is slightly easier than average as it provides the volume formula, uses clean numbers, and follows a standard template with no geometric insight or proof required. |
| Spec | 6.04c Composite bodies: centre of mass6.04d Integration: for centre of mass of laminas/solids |
| Answer | Marks |
|---|---|
| 3(i) | 3 3 |
| Answer | Marks |
|---|---|
| 8 8 | B1 |
| Answer | Marks | Guidance |
|---|---|---|
| 8 | M1A1 | Take moments about O |
| X = 0.225 m | A1 | |
| Total: | 4 |
| Answer | Marks | Guidance |
|---|---|---|
| 3(ii) | 24 × 0.225 + W(3 × 0.28 / 8) = (24 + W) × 0.15 | M1A1 |
| Answer | Marks |
|---|---|
| W = 40 N | A1 |
| Total: | 3 |
Question 3:
--- 3(i) ---
3(i) | 3 3
CofM of hemisphere = × 0.56 or × 0.28
8 8 | B1
2 2 2 3 2
[ π × 0.563– π × 0.283]X = π × 0.563 × × 0.56 – π × 0.283×
3 3 3 8 3
3
× 0.28
8 | M1A1 | Take moments about O
X = 0.225 m | A1
Total: | 4
--- 3(ii) ---
3(ii) | 24 × 0.225 + W(3 × 0.28 / 8) = (24 + W) × 0.15 | M1A1 | Attempts to take moments about O
W = weight of uniform hemi-sphere
W = 40 N | A1
Total: | 3\includegraphics{figure_3}
An object is made from a uniform solid hemisphere of radius $0.56$ m and centre $O$ by removing a hemisphere of radius $0.28$ m and centre $O$. The diagram shows a cross-section through $O$ of the object.
\begin{enumerate}[label=(\roman*)]
\item Calculate the distance of the centre of mass of the object from $O$. [4]
[The volume of a hemisphere is $\frac{2}{3}\pi r^3$.]
\item The object has weight $24$ N. A uniform hemisphere $H$ of radius $0.28$ m is placed in the hollow part of the object to create a new uniform hemisphere with centre $O$. The centre of mass of the non-uniform hemisphere is $0.15$ m from $O$.
Calculate the weight of $H$. [3]
\end{enumerate}
\hfill \mbox{\textit{CAIE M2 2017 Q3 [7]}}