| Exam Board | Edexcel |
|---|---|
| Module | M3 (Mechanics 3) |
| Marks | 11 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Centre of Mass 1 |
| Type | Composite solid with hemisphere and cylinder/cone |
| Difficulty | Challenging +1.2 Part (a) is a standard M3 proof requiring integration by discs/shells to find the centre of mass of a hemisphere - a classic bookwork result that appears in most textbooks. Part (b) applies this result to find k using the principle of moments about O, requiring careful setup but straightforward algebra. While the integration in (a) requires competence and the multi-step nature adds some challenge, this is a well-rehearsed M3 question type with no novel insight required. |
| Spec | 6.04d Integration: for centre of mass of laminas/solids |
| Answer | Marks | Guidance |
|---|---|---|
| (a) \(\bar{x}^2 + \bar{y}^2 = r^2\) where \(\pi \int_0^r y^2 \, dx = \int_0^r xy^2 \, dx\) | B1 M1 A1 | |
| \(\pi x \int_0^r r^2 - x^2 \, dx = \pi \int_0^r x^2 - x^3 \, dx\) | ||
| \(\frac{2r^3}{3} - \frac{r^4}{4}\) and \(\bar{x} = \frac{3r}{8}\) | M1 A1 A1 A1 | |
| (b) \(M(O): \frac{2}{3}\pi r^3 \cdot \frac{3r}{8} = \pi\left(\frac{3r}{4}\right)^2 kr \cdot \frac{k}{2}\) where \(k^2 = \frac{8}{9}\) and \(k = \frac{2}{3}\sqrt{2}\) | M1 A1 A1 A1 | |
| (c) \(\tan \theta = \frac{3r}{8} + \frac{2r\sqrt{2}}{3} = \frac{9 + 8r\sqrt{2}}{24} = 0.7955\) so \(\theta = 38.5°\) | M1 A1 A1 A1 | 14 marks |
**(a)** $\bar{x}^2 + \bar{y}^2 = r^2$ where $\pi \int_0^r y^2 \, dx = \int_0^r xy^2 \, dx$ | B1 M1 A1 |
$\pi x \int_0^r r^2 - x^2 \, dx = \pi \int_0^r x^2 - x^3 \, dx$ | |
$\frac{2r^3}{3} - \frac{r^4}{4}$ and $\bar{x} = \frac{3r}{8}$ | M1 A1 A1 A1 |
**(b)** $M(O): \frac{2}{3}\pi r^3 \cdot \frac{3r}{8} = \pi\left(\frac{3r}{4}\right)^2 kr \cdot \frac{k}{2}$ where $k^2 = \frac{8}{9}$ and $k = \frac{2}{3}\sqrt{2}$ | M1 A1 A1 A1 |
**(c)** $\tan \theta = \frac{3r}{8} + \frac{2r\sqrt{2}}{3} = \frac{9 + 8r\sqrt{2}}{24} = 0.7955$ so $\theta = 38.5°$ | M1 A1 A1 A1 | 14 marks\begin{enumerate}[label=(\alph*)]
\item Prove that the centre of mass of a uniform solid hemisphere of radius $r$ is at a distance $\frac{3r}{8}$ from its plane face. [7 marks]
\end{enumerate}
A solid cylinder of radius $\frac{3r}{4}$ and height $kr$, where $k < 1$, is welded to a uniform hemisphere of radius $r$ made of the same material, so that their axes of symmetry coincide. The figure shows the cross section of the resulting solid. If the centre of mass of this solid is at $O$, the centre of the plane face of the hemisphere,
\includegraphics{figure_5}
\begin{enumerate}[label=(\alph*)]
\setcounter{enumi}{1}
\item find the value of $k$. [4 marks]
\end{enumerate}
\hfill \mbox{\textit{Edexcel M3 Q5 [11]}}