| Exam Board | Edexcel |
|---|---|
| Module | M3 (Mechanics 3) |
| Year | 2005 |
| Session | January |
| Marks | 9 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Centre of Mass 1 |
| Type | Toppling on inclined plane |
| Difficulty | Standard +0.8 This M3 question requires setting up and evaluating a non-trivial centre of mass integral (y-coordinate involves ∫y²dx with sin²x), then applying toppling conditions to a prism on an inclined plane. The integration requires trigonometric identities and the toppling analysis needs geometric insight about the centre of mass position relative to the base, making it moderately challenging but within standard M3 scope. |
| Spec | 6.04d Integration: for centre of mass of laminas/solids6.04e Rigid body equilibrium: coplanar forces |
| Answer | Marks | Guidance |
|---|---|---|
| Content | Marks | Notes |
| \(\int_0^{\pi} \frac{1}{2}e^{2x} dx\) | M1 | |
| \(= \frac{1}{4}\int_0^{\pi} (1-\cos 2x)dx\) | M1 | |
| \(= \frac{1}{4}\left[x - \frac{1}{2}\sin 2x\right]_0^{\pi}\) | A1 | |
| \(= \frac{\pi}{4}\) | A1 | |
| \(J = \frac{\pi/4}{\int_0^{\pi} \sin x \, dx} = \frac{\pi/4}{\pi/2} = \frac{\pi}{8}\) | M1, A1 | (6) |
| Answer | Marks | Guidance |
|---|---|---|
| Content | Marks | Notes |
| \(\tan \theta = \frac{\pi/2}{J} = 4\) | M1 | |
| \(\theta = \pi/8\) (or equivalent) | A1 M1 | |
| \(\theta = 75.96°\) | A1 | (3) |
## Part (c)
| Content | Marks | Notes |
|---------|-------|-------|
| $\int_0^{\pi} \frac{1}{2}e^{2x} dx$ | M1 | |
| $= \frac{1}{4}\int_0^{\pi} (1-\cos 2x)dx$ | M1 | |
| $= \frac{1}{4}\left[x - \frac{1}{2}\sin 2x\right]_0^{\pi}$ | A1 | |
| $= \frac{\pi}{4}$ | A1 | |
| $J = \frac{\pi/4}{\int_0^{\pi} \sin x \, dx} = \frac{\pi/4}{\pi/2} = \frac{\pi}{8}$ | M1, A1 | (6) |
## Part (b)
| Content | Marks | Notes |
|---------|-------|-------|
| $\tan \theta = \frac{\pi/2}{J} = 4$ | M1 | |
| $\theta = \pi/8$ (or equivalent) | A1 M1 | |
| $\theta = 75.96°$ | A1 | (3) |
---\includegraphics{figure_2}
A uniform lamina occupies the region $R$ bounded by the $x$-axis and the curve
$$y = \sin x, \quad 0 \leq x \leq \pi,$$
as shown in Figure 2.
\begin{enumerate}[label=(\alph*)]
\item Show, by integration, that the $y$-coordinate of the centre of mass of the lamina is $\frac{\pi}{8}$. [6]
\end{enumerate}
\includegraphics{figure_3}
A uniform prism $S$ has cross-section $R$. The prism is placed with its rectangular face on a table which is inclined at an angle $\theta^{\circ}$ to the horizontal. The cross-section $R$ lies in a vertical plane as shown in Figure 3. The table is sufficiently rough to prevent $S$ sliding. Given that $S$ does not topple,
\begin{enumerate}[label=(\alph*)]
\setcounter{enumi}{1}
\item find the largest possible value of $\theta$. [3]
\end{enumerate}
\hfill \mbox{\textit{Edexcel M3 2005 Q3 [9]}}