| Exam Board | CAIE |
|---|---|
| Module | M2 (Mechanics 2) |
| Year | 2015 |
| Session | November |
| Marks | 9 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Centre of Mass 1 |
| Type | Conical or hemispherical shell composite |
| Difficulty | Challenging +1.2 This is a standard M2 centre of mass problem requiring application of standard formulas (hemisphere shell at r/2, solid cone at h/4 from base), setting up moment equations for composite bodies, and using the toppling condition (centre of mass above contact point). Part (i) involves algebraic manipulation to reach a given result, and part (ii) requires solving for an unknown weight. While it involves multiple steps and careful bookkeeping, it uses well-rehearsed techniques without requiring novel geometric insight or complex problem-solving strategies. |
| Spec | 6.04b Find centre of mass: using symmetry6.04c Composite bodies: centre of mass6.04d Integration: for centre of mass of laminas/solids6.04e Rigid body equilibrium: coplanar forces |
| Answer | Marks | Guidance |
|---|---|---|
| Working | Mark | Guidance |
| \(d\cos\theta = h/\cos\theta\) | M1 | \(\theta\) = semi-vertical angle |
| \(\cos\theta = \frac{h}{\sqrt{(0.2^2 + h^2)}}\) | ||
| \(d = \frac{h}{(h^2/(0.04 + h^2))}\) | M1 | |
| \(d = h + \frac{0.04}{h}\) | A1 AG | [3] |
| Answer | Marks | Guidance |
|---|---|---|
| Working | Mark | Guidance |
| \(0.6 \times 4 + 0.9W = d(4 + W)\) | M1 A1 | Table of moments idea |
| \(d = 0.8 + \frac{0.2^2}{0.8}\) | B1 | \(0.85\) |
| \(2.4 + 0.9W = 0.85(4 + W)\) | M1 | |
| \(0.05W = 1\) | A1 | |
| \(W = 20\) | A1 | [6] |
## Question 6(i):
| Working | Mark | Guidance |
|---------|------|----------|
| $d\cos\theta = h/\cos\theta$ | M1 | $\theta$ = semi-vertical angle |
| $\cos\theta = \frac{h}{\sqrt{(0.2^2 + h^2)}}$ | | |
| $d = \frac{h}{(h^2/(0.04 + h^2))}$ | M1 | |
| $d = h + \frac{0.04}{h}$ | A1 AG | **[3]** |
## Question 6(ii):
| Working | Mark | Guidance |
|---------|------|----------|
| $0.6 \times 4 + 0.9W = d(4 + W)$ | M1 A1 | Table of moments idea |
| $d = 0.8 + \frac{0.2^2}{0.8}$ | B1 | $0.85$ |
| $2.4 + 0.9W = 0.85(4 + W)$ | M1 | |
| $0.05W = 1$ | A1 | |
| $W = 20$ | A1 | **[6]** |
---6\\
\includegraphics[max width=\textwidth, alt={}, center]{727412ec-d783-4392-8b84-e7d5435a3f4e-3_424_953_255_596}
An object is formed by joining a hemispherical shell of radius 0.2 m and a solid cone with base radius 0.2 m and height $h \mathrm {~m}$ along their circumferences. The centre of mass, $G$, of the object is $d \mathrm {~m}$ from the vertex of the cone on the axis of symmetry of the object. The object rests in equilibrium on a horizontal plane, with the curved surface of the cone in contact with the plane (see diagram). The object is on the point of toppling.\\
(i) Show that $d = h + \frac { 0.04 } { h }$.\\
(ii) It is given that the cone is uniform and of weight 4 N , and that the hemispherical shell is uniform and of weight $W \mathrm {~N}$. Given also that $h = 0.8$, find $W$.
\hfill \mbox{\textit{CAIE M2 2015 Q6 [9]}}