| Exam Board | Edexcel |
|---|---|
| Module | M3 (Mechanics 3) |
| Year | 2017 |
| Session | June |
| Marks | 7 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Centre of Mass 2 |
| Type | Centre of mass of lamina by integration |
| Difficulty | Standard +0.3 This is a standard M3 centre of mass question requiring integration of a simple polynomial. Students apply the formula x̄ = ∫xy dx / ∫y dx with y = 4 - x², involving straightforward polynomial integration (x³ and x⁵ terms). While it requires knowing the correct formula and careful algebraic manipulation, it's a routine textbook exercise with no novel problem-solving required, making it slightly easier than average. |
| Spec | 6.04d Integration: for centre of mass of laminas/solids |
Question 1:
Area $= \int_0^2 4x - 2x^2 \, dx$ M1, A1
$= \left[4x - \frac{1}{3}x^3\right]_0^2$ M1
$= 16 - \frac{16}{3}$ dM1, A1
$= \frac{16}{3}$ M1, A1 cso
$\int xy \, dx$ M1
$\int_0^2 2x^2 - x^4 \, dx$ M1, A1
$= \left[2x^2 - \frac{1}{4}x^4\right]_0^2$ M1, A1
$= 4$ A1
$\bar{x} = \frac{16}{3} \div \frac{16}{3} = 4$ oe M1, A1 cso
[7]1.
\begin{figure}[h]
\begin{center}
\includegraphics[alt={},max width=\textwidth]{698b44b5-801c-45ec-b9de-021e44487edb-02_672_732_226_589}
\captionsetup{labelformat=empty}
\caption{Figure 1}
\end{center}
\end{figure}
A uniform lamina is in the shape of the region $R$. Region $R$ is bounded by the curve with equation $y = 4 - x ^ { 2 }$, the positive $x$-axis and the positive $y$-axis, as shown shaded in Figure 1.
Use algebraic integration to find the $x$ coordinate of the centre of mass of the lamina.
\hfill \mbox{\textit{Edexcel M3 2017 Q1 [7]}}