| Exam Board | Edexcel |
|---|---|
| Module | M3 (Mechanics 3) |
| Year | 2021 |
| Session | January |
| Marks | 8 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Centre of Mass 2 |
| Type | Centre of mass of solid of revolution |
| Difficulty | Standard +0.8 This is a solid of revolution centre of mass problem requiring volume and moment integrals with the function y=1/x. Part (a) is a standard 'show that' volume calculation, but part (b) requires setting up and evaluating the first moment integral ∫x·πy²dx, then dividing by volume. The integration involves ln terms and algebraic manipulation. This is moderately challenging for M3 level, requiring careful setup and execution of standard techniques rather than novel insight. |
| Spec | 1.08a Fundamental theorem of calculus: integration as reverse of differentiation1.08d Evaluate definite integrals: between limits4.08d Volumes of revolution: about x and y axes6.04d Integration: for centre of mass of laminas/solids |
| Answer | Marks | Guidance |
|---|---|---|
| Working | Mark | Guidance |
| \(V = \pi \int_1^a y^2\,dx\) | ||
| \(V = (\pi)\int_1^a \frac{1}{x^2}\,dx = (\pi)\left[-\frac{1}{x}\right]_1^a\) | M1A1 | M1: Use of \(\int y^2\,dx\) AND attempt at algebraic integration (power increasing by one). A1: Correct integration, \(\pi\) not needed. |
| \(V = \pi\left(1 - \frac{1}{a}\right)\) | A1* | Given result reached from fully correct working. If \(\pi\) not included from the start, its inclusion must now be justified. |
| Answer | Marks | Guidance |
|---|---|---|
| Working | Mark | Guidance |
| \((\pi)\int xy^2\,dx\) | M1 | Must have substituted for \(y\), \(\pi\) not needed. |
| \((\pi)\int_1^a \frac{1}{x}\,dx = (\pi)[\ln x]_1^a = (\pi)\ln a\) | dM1A1 | dM1: Attempt at algebraic integration (\(\ln x\) needs to be seen). A1: Correct result after substitution of limits, \(\pi\) not needed. |
| \((\pi)\left(1-\frac{1}{a}\right)\bar{x} = (\pi)\ln a\) | M1 | Use of \(\frac{\int xy^2\,dx}{\int y^2\,dx}\). If \(\pi\) and/or \(\rho\) appear, they must appear consistently. |
| \(\bar{x} = \dfrac{a\ln a}{a-1}\) | A1 | Correct final answer. Mark lost if \(1-\frac{1}{a}\) left in denominator. |
## Question 1:
### Part (a)
| Working | Mark | Guidance |
|---------|------|----------|
| $V = \pi \int_1^a y^2\,dx$ | | |
| $V = (\pi)\int_1^a \frac{1}{x^2}\,dx = (\pi)\left[-\frac{1}{x}\right]_1^a$ | M1A1 | M1: Use of $\int y^2\,dx$ AND attempt at algebraic integration (power increasing by one). A1: Correct integration, $\pi$ not needed. |
| $V = \pi\left(1 - \frac{1}{a}\right)$ | A1* | Given result reached from fully correct working. If $\pi$ not included from the start, its inclusion must now be justified. |
### Part (b)
| Working | Mark | Guidance |
|---------|------|----------|
| $(\pi)\int xy^2\,dx$ | M1 | Must have substituted for $y$, $\pi$ not needed. |
| $(\pi)\int_1^a \frac{1}{x}\,dx = (\pi)[\ln x]_1^a = (\pi)\ln a$ | dM1A1 | dM1: Attempt at algebraic integration ($\ln x$ needs to be seen). A1: Correct result after substitution of limits, $\pi$ not needed. |
| $(\pi)\left(1-\frac{1}{a}\right)\bar{x} = (\pi)\ln a$ | M1 | Use of $\frac{\int xy^2\,dx}{\int y^2\,dx}$. If $\pi$ and/or $\rho$ appear, they must appear consistently. |
| $\bar{x} = \dfrac{a\ln a}{a-1}$ | A1 | Correct final answer. Mark lost if $1-\frac{1}{a}$ left in denominator. |
---1.
\begin{figure}[h]
\begin{center}
\includegraphics[alt={},max width=\textwidth]{8a687d17-ec7e-463f-84dd-605f5c230db1-02_469_758_251_593}
\captionsetup{labelformat=empty}
\caption{Figure 1}
\end{center}
\end{figure}
The region $R$, shown shaded in Figure 1, is bounded by the curve with equation $y = \frac { 1 } { x }$, the line with equation $x = 1$, the positive $x$-axis and the line with equation $x = a$ where $a > 1$ A uniform solid $S$ is formed by rotating $R$ through $2 \pi$ radians about the $x$-axis.
\begin{enumerate}[label=(\alph*)]
\item Show that the volume of $S$ is
$$\pi \left( 1 - \frac { 1 } { a } \right)$$
\item Find the $x$ coordinate of the centre of mass of $S$.
\end{enumerate}
\hfill \mbox{\textit{Edexcel M3 2021 Q1 [8]}}