| Exam Board | Edexcel |
|---|---|
| Module | M3 (Mechanics 3) |
| Year | 2009 |
| Session | June |
| Marks | 9 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Centre of Mass 2 |
| Type | Centre of mass of solid of revolution |
| Difficulty | Standard +0.3 This is a standard M3 centre of mass question using volumes of revolution. Part (a) requires routine integration of π∫(1/x²)dx with given limits. Part (b) applies the standard formula for centre of mass of a solid of revolution, requiring integration of π∫x(1/x²)dx. Both parts follow textbook methods with straightforward algebra and no conceptual challenges beyond applying memorized formulas. |
| Spec | 4.08d Volumes of revolution: about x and y axes6.04d Integration: for centre of mass of laminas/solids |
| Answer | Marks |
|---|---|
| Volume \(= \int_1^4 \pi y^2 dx = \int_1^4 \pi \frac{1}{x} dx\) | M1A1 |
| \(= \left[\pi \times \frac{-1}{3x^3}\right]_1^4\) | A1ft |
| \(= \pi\left(\frac{-1}{3} + \frac{64}{3}\right) = 21\pi\) | A1 |
| Answer | Marks |
|---|---|
| \(21\pi\bar{x} = \rho \int \pi y^2 x dx = \rho \int \pi \frac{1}{x^3} dx\) | M1A1 |
| \(21\pi\bar{x} = \pi \left[\frac{-1}{2x^2}\right]_1^4\) | A1ft |
| \(\bar{x} = \frac{1}{21}\left(\frac{-1}{2} + \frac{16}{2}\right) = \frac{5}{14}\) or awrt 0.36 | A1 |
| \(\bar{y} = 0\) by symmetry | B1 |
## Part (a)
Volume $= \int_1^4 \pi y^2 dx = \int_1^4 \pi \frac{1}{x} dx$ | M1A1 |
$= \left[\pi \times \frac{-1}{3x^3}\right]_1^4$ | A1ft |
$= \pi\left(\frac{-1}{3} + \frac{64}{3}\right) = 21\pi$ | A1 |
## Part (b)
$21\pi\bar{x} = \rho \int \pi y^2 x dx = \rho \int \pi \frac{1}{x^3} dx$ | M1A1 |
$21\pi\bar{x} = \pi \left[\frac{-1}{2x^2}\right]_1^4$ | A1ft |
$\bar{x} = \frac{1}{21}\left(\frac{-1}{2} + \frac{16}{2}\right) = \frac{5}{14}$ or awrt 0.36 | A1 |
$\bar{y} = 0$ by symmetry | B1 |
---The finite region bounded by the $x$-axis, the curve $y = \frac{1}{x}$, the line $x = \frac{1}{4}$ and the line $x = 1$, is rotated through one complete revolution about the $x$-axis to form a uniform solid of revolution.
\begin{enumerate}[label=(\alph*)]
\item Show that the volume of the solid is $21\pi$. [4]
\item Find the coordinates of the centre of mass of the solid. [5]
\end{enumerate}
\hfill \mbox{\textit{Edexcel M3 2009 Q4 [9]}}