| Exam Board | Edexcel |
|---|---|
| Module | M3 (Mechanics 3) |
| Marks | 12 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Centre of Mass 1 |
| Type | Suspended lamina equilibrium angle |
| Difficulty | Challenging +1.2 This is a standard M3 centre of mass question requiring integration to find the centroid of a lamina (9 marks for showing a given result) followed by a routine suspension problem (3 marks). While the integration involves products of trigonometric functions and requires careful algebraic manipulation, the techniques are well-practiced in M3. The suspension part is straightforward application of equilibrium principles. The 'show that' format and multi-step integration place it above average difficulty, but it remains a typical exam question without requiring novel insight. |
| Spec | 6.04d Integration: for centre of mass of laminas/solids6.04e Rigid body equilibrium: coplanar forces |
| Answer | Marks | Guidance |
|---|---|---|
| (a) \(\bar{x}\int_0^{\pi} y \, dx = \int_0^{\pi} xy \, dx\) | M1 A1 A1 | |
| \(\int_0^{\pi} (1 + \cos x) \, dx = \int_0^{\pi} x + x \cos x \, dx\) | ||
| \(\bar{x}[x + \sin x]_0^{\pi} = \left[\frac{1}{2}x^2 + x \sin x + \cos x\right]_0^{\pi}\) | M1 A1 A1 A1 | |
| (R.H.S. by parts) | ||
| \(\pi x = \frac{\pi^2}{2} - 2\) | M1 A1 | |
| \(\bar{x} = \frac{\pi^2 - 4}{2\pi}\) | ||
| (b) \(\tan \theta = \frac{\pi^2 - 4}{2\pi} \cdot \frac{4}{5} = 0.7473\) | M1 A1 A1 | |
| \(\theta = 36.8°\) | Total: 12 marks |
**(a)** $\bar{x}\int_0^{\pi} y \, dx = \int_0^{\pi} xy \, dx$ | M1 A1 A1 |
$\int_0^{\pi} (1 + \cos x) \, dx = \int_0^{\pi} x + x \cos x \, dx$ | |
$\bar{x}[x + \sin x]_0^{\pi} = \left[\frac{1}{2}x^2 + x \sin x + \cos x\right]_0^{\pi}$ | M1 A1 A1 A1 |
(R.H.S. by parts) | |
$\pi x = \frac{\pi^2}{2} - 2$ | M1 A1 |
$\bar{x} = \frac{\pi^2 - 4}{2\pi}$ | |
**(b)** $\tan \theta = \frac{\pi^2 - 4}{2\pi} \cdot \frac{4}{5} = 0.7473$ | M1 A1 A1 |
$\theta = 36.8°$ | | **Total: 12 marks**A uniform lamina is in the shape of the region enclosed by the coordinate axes and the curve with equation $y = 1 + \cos x$, as shown.
\includegraphics{figure_4}
\begin{enumerate}[label=(\alph*)]
\item Show by integration that the centre of mass of the lamina is at a distance $\frac{\pi^2 - 4}{2\pi}$ from the $y$-axis. [9 marks]
\end{enumerate}
Given that the centre of mass is at a distance 0·75 units from the $x$-axis, and that $P$ is the point $(0, 2)$ and $O$ is the origin $(0, 0)$,
\begin{enumerate}[label=(\alph*)]
\setcounter{enumi}{1}
\item find, to the nearest degree, the angle between the line $OP$ and the vertical when the lamina is freely suspended from $P$. [3 marks]
\end{enumerate}
\hfill \mbox{\textit{Edexcel M3 Q4 [12]}}