| Exam Board | Edexcel |
|---|---|
| Module | M3 (Mechanics 3) |
| Year | 2002 |
| Session | June |
| Marks | 12 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Centre of Mass 1 |
| Type | Composite solid with cone and cylinder |
| Difficulty | Standard +0.3 This is a standard M3 centre of mass question requiring systematic application of formulas for composite bodies (cone and cylinder volumes, standard COM positions) and toppling conditions. Part (a) involves routine calculation with given answer to verify. Part (b) applies the toppling criterion (vertical line through G passes through edge) with basic trigonometry. While multi-step with 12 marks total, it follows predictable M3 patterns without requiring novel insight or complex problem-solving. |
| Spec | 3.04b Equilibrium: zero resultant moment and force6.04d Integration: for centre of mass of laminas/solids6.04e Rigid body equilibrium: coplanar forces |
| Answer | Marks | Guidance |
|---|---|---|
| Cone | Cylinder | Whole |
| Volume | \(\frac{1}{3}\pi(2r)^2h\) | \(\pi r^2 h\) |
| (4) | (3) | (7) |
| Distance from base | \(\frac{1}{4}h\) | \(\frac{1}{2}h\) |
| Moment equation | \(-4 \times \frac{1}{4}h\) | \(+3 \times \frac{1}{2}h\) |
| M1 A1 | ||
| B1 B1 | ||
| M1 A1 | ||
| \(\bar{x} = \frac{1}{14}h\) | M1 A1 cso | (8) |
| Answer | Marks | Guidance |
|---|---|---|
| Use of \(G\) above \(N\) | M1 | |
| \(\tan \alpha = \frac{r}{h - \frac{1}{14}h} = \frac{7}{26}\) | M1 A1 | |
| \(r = \frac{1}{4}h\) | A1 | (4) |
## Part (a)
| | Cone | Cylinder | Whole |
|---|---|---|---|
| Volume | $\frac{1}{3}\pi(2r)^2h$ | $\pi r^2 h$ | $\frac{1}{3}\pi(2r)^2h + \pi r^2h$ |
| | (4) | (3) | (7) |
| Distance from base | $\frac{1}{4}h$ | $\frac{1}{2}h$ | $x$ |
| | | | |
| Moment equation | $-4 \times \frac{1}{4}h$ | $+3 \times \frac{1}{2}h$ | $= 7x$ |
| M1 A1
| B1 B1
| M1 A1
$\bar{x} = \frac{1}{14}h$ | M1 A1 cso | (8)
## Part (b)
Use of $G$ above $N$ | M1
$\tan \alpha = \frac{r}{h - \frac{1}{14}h} = \frac{7}{26}$ | M1 A1
$r = \frac{1}{4}h$ | A1 | (4)
**Total: (12 marks)**
---\includegraphics{figure_2}
A model tree is made by joining a uniform solid cylinder to a uniform solid cone made of the same material. The centre $O$ of the base of the cone is also the centre of one end of the cylinder, as shown in Fig. 2. The radius of the cylinder is $r$ and the radius of the base of the cone is $2r$. The height of the cone and the height of the cylinder are each $h$. The centre of mass of the model is at the point $G$.
\begin{enumerate}[label=(\alph*)]
\item Show that $OG = \frac{1}{14}h$.
[8]
\end{enumerate}
The model stands on a desk top with its plane face in contact with the desk top. The desk top is tilted until it makes an angle $\alpha$ with the horizontal, where $\tan \alpha = \frac{r}{7h}$. The desk top is rough enough to prevent slipping and the model is about to topple.
\begin{enumerate}[label=(\alph*)]
\setcounter{enumi}{1}
\item Find $r$ in terms of $h$.
[4]
\end{enumerate}
\hfill \mbox{\textit{Edexcel M3 2002 Q5 [12]}}