| Exam Board | Edexcel |
|---|---|
| Module | M3 (Mechanics 3) |
| Year | 2022 |
| Session | January |
| Marks | 11 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Centre of Mass 2 |
| Type | Composite solid with standard shapes - calculation only |
| Difficulty | Standard +0.8 This is a multi-part Further Maths M3 centre of mass question requiring: (a) proving a standard result using integration, (b) combining centres of mass of composite shapes with algebraic manipulation to reach a given form, and (c) applying equilibrium conditions. While the techniques are standard for M3, the algebraic complexity and the equilibrium condition in part (c) requiring geometric insight elevate this above average difficulty. |
| Spec | 6.04c Composite bodies: centre of mass6.04d Integration: for centre of mass of laminas/solids |
| Answer | Marks | Guidance |
|---|---|---|
| Working/Answer | Mark | Guidance |
| \(\bar{x} = \dfrac{\pi\int_0^r x(r^2-x^2)\,dx}{\dfrac{2\pi r^3}{3}}\) | M1A1 | M1 for use of \(\bar{x} = \dfrac{\pi\int_0^r xy^2\,dx}{\frac{2\pi r^3}{3}}\); A1 for correct integrand |
| \(= \dfrac{\left[\dfrac{r^2x^2}{2} - \dfrac{x^4}{4}\right]_0^r}{\dfrac{2r^3}{3}}\) | A1 | M1 for integrating with powers increasing by 1 |
| \(= \frac{3r}{8}\)* | A1* | Given answer correctly obtained |
| Answer | Marks | Guidance |
|---|---|---|
| Working/Answer | Mark | Guidance |
| Mass ratios: \(\dfrac{2\pi r^3}{3}\), \(\pi r^2 h\), \(\left(\dfrac{2\pi r^3}{3} + \pi r^2 h\right)\) | B1 | Correct mass ratios |
| Distances: \(\dfrac{5r}{8}\), \(\left(r + \frac{1}{2}h\right)\), \(\bar{y}\) | B1 | Correct distances for their parallel axis |
| \(\left(\dfrac{2\pi r^3}{3}\times\dfrac{5r}{8}\right) + \pi r^2 h\left(r+\dfrac{1}{2}h\right) = \left(\dfrac{2\pi r^3}{3}+\pi r^2 h\right)\bar{y}\) | M1A1 | M1 for moments equation with correct terms; A1 correct unsimplified equation |
| \(\bar{y} = \dfrac{5r^2+12rh+6h^2}{8r+12h}\)* | A1* | Given answer correctly obtained |
| Answer | Marks | Guidance |
|---|---|---|
| Working/Answer | Mark | Guidance |
| \(r = \dfrac{5r^2+12rh+6h^2}{8r+12h}\) | M1 | For equating given answer to \(r\) |
| \(r = \sqrt{2}\,h\) | A1 | cao |
# Question 4:
## Part (a):
| Working/Answer | Mark | Guidance |
|---|---|---|
| $\bar{x} = \dfrac{\pi\int_0^r x(r^2-x^2)\,dx}{\dfrac{2\pi r^3}{3}}$ | M1A1 | M1 for use of $\bar{x} = \dfrac{\pi\int_0^r xy^2\,dx}{\frac{2\pi r^3}{3}}$; A1 for correct integrand |
| $= \dfrac{\left[\dfrac{r^2x^2}{2} - \dfrac{x^4}{4}\right]_0^r}{\dfrac{2r^3}{3}}$ | A1 | M1 for integrating with powers increasing by 1 |
| $= \frac{3r}{8}$* | A1* | Given answer correctly obtained |
## Part (b):
| Working/Answer | Mark | Guidance |
|---|---|---|
| Mass ratios: $\dfrac{2\pi r^3}{3}$, $\pi r^2 h$, $\left(\dfrac{2\pi r^3}{3} + \pi r^2 h\right)$ | B1 | Correct mass ratios |
| Distances: $\dfrac{5r}{8}$, $\left(r + \frac{1}{2}h\right)$, $\bar{y}$ | B1 | Correct distances for their parallel axis |
| $\left(\dfrac{2\pi r^3}{3}\times\dfrac{5r}{8}\right) + \pi r^2 h\left(r+\dfrac{1}{2}h\right) = \left(\dfrac{2\pi r^3}{3}+\pi r^2 h\right)\bar{y}$ | M1A1 | M1 for moments equation with correct terms; A1 correct unsimplified equation |
| $\bar{y} = \dfrac{5r^2+12rh+6h^2}{8r+12h}$* | A1* | Given answer correctly obtained |
## Part (c):
| Working/Answer | Mark | Guidance |
|---|---|---|
| $r = \dfrac{5r^2+12rh+6h^2}{8r+12h}$ | M1 | For equating given answer to $r$ |
| $r = \sqrt{2}\,h$ | A1 | cao |
---\begin{enumerate}
\item A uniform solid hemisphere $H$ has radius $r$ and centre $O$\\
(a) Show that the centre of mass of $H$ is $\frac { 3 r } { 8 }$ from $O$
\end{enumerate}
$$\left[ \text { You may assume that the volume of } H \text { is } \frac { 2 \pi r ^ { 3 } } { 3 } \right]$$
A uniform solid $S$, shown below in Figure 3, is formed by attaching a uniform solid right circular cylinder of height $h$ and radius $r$ to $H$, so that one end of the cylinder coincides with the plane face of $H$.
The point $A$ is the point on $H$ such that $O A = r$ and $O A$ is perpendicular to the plane face of $H$
\begin{figure}[h]
\begin{center}
\includegraphics[alt={},max width=\textwidth]{a1365c54-4910-449b-b270-c56c1bc5a751-12_592_791_909_660}
\captionsetup{labelformat=empty}
\caption{Figure 3}
\end{center}
\end{figure}
(b) Show that the distance of the centre of mass of $S$ from $A$ is
$$\frac { 5 r ^ { 2 } + 12 r h + 6 h ^ { 2 } } { 8 r + 12 h }$$
The solid $S$ can rest in equilibrium on a horizontal plane with any point of the curved surface of the hemisphere in contact with the plane.\\
(c) Find $r$ in terms of $h$.
\hfill \mbox{\textit{Edexcel M3 2022 Q4 [11]}}