| Exam Board | Edexcel |
|---|---|
| Module | FM2 (Further Mechanics 2) |
| Year | 2022 |
| Session | June |
| Marks | 10 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Centre of Mass 1 |
| Type | Centre of mass with variable parameter |
| Difficulty | Challenging +1.2 This is a standard Further Mechanics 2 centre of mass problem with variable density. Part (a) requires setting up and evaluating standard integrals for mass and moment (∫ρdV), which is routine for FM2 students. Part (b) involves equilibrium of a composite body using the toppling condition, a common textbook exercise. While it requires careful integration and algebraic manipulation, the techniques are well-practiced and the problem structure is familiar to FM2 candidates. |
| Spec | 4.08d Volumes of revolution: about x and y axes6.04c Composite bodies: centre of mass6.04d Integration: for centre of mass of laminas/solids6.04e Rigid body equilibrium: coplanar forces |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| Mass of cone \(= \int_0^9 \pi y^2 \lambda x\, dx = \pi\lambda\int_0^9 \dfrac{x^3}{9}\, dx\) | M1 | Use the model to find the mass of the cone. Allow without limits |
| \(= \pi\lambda\left[\dfrac{x^4}{36}\right]_0^9 \left(= \dfrac{729\pi\lambda}{4}\ \text{(kg)}\right)\) | A1 | Correct integration. Correct limits seen or implied. Substitution not required |
| Moments: \(\int_0^9 \pi y^2 \lambda x \times x\, dx = \pi\lambda\int_0^9 \dfrac{x^4}{9}\, dx\) | M1 | 3.4 |
| \(= \dfrac{\pi\lambda}{45}\left[x^5\right]_0^9 \left(= \dfrac{\pi\lambda}{5}\times 9^4\right)\) | A1 | 1.1b |
| \(\Rightarrow d = \dfrac{\dfrac{\pi\lambda}{5}\times 9^4}{\dfrac{\pi\lambda}{4}\times 9^3}\) | DM1 | 2.1 |
| \(d = \dfrac{36}{5} = 7.2\ \text{(cm)}\) | A1 | 1.1b |
| (6 marks) |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| Remains at rest \(\Rightarrow\) centre of mass at centre of plane surface | B1 | 2.1 |
| Moments about diameter of plane surface: | M1 | 3.1b |
| \((9-d)W \left\{= \left(9-\dfrac{36}{5}\right)W\right\} = \dfrac{3}{8}\times 3\times kW\) | A1ft | 1.1b |
| \(k = \dfrac{8}{5}\) | A1 | 1.1b |
| (4 marks) |
# Question 6(a):
| Answer/Working | Mark | Guidance |
|---|---|---|
| Mass of cone $= \int_0^9 \pi y^2 \lambda x\, dx = \pi\lambda\int_0^9 \dfrac{x^3}{9}\, dx$ | M1 | Use the model to find the mass of the cone. Allow without limits |
| $= \pi\lambda\left[\dfrac{x^4}{36}\right]_0^9 \left(= \dfrac{729\pi\lambda}{4}\ \text{(kg)}\right)$ | A1 | Correct integration. Correct limits seen or implied. Substitution not required |
| Moments: $\int_0^9 \pi y^2 \lambda x \times x\, dx = \pi\lambda\int_0^9 \dfrac{x^4}{9}\, dx$ | M1 | 3.4 |
| $= \dfrac{\pi\lambda}{45}\left[x^5\right]_0^9 \left(= \dfrac{\pi\lambda}{5}\times 9^4\right)$ | A1 | 1.1b |
| $\Rightarrow d = \dfrac{\dfrac{\pi\lambda}{5}\times 9^4}{\dfrac{\pi\lambda}{4}\times 9^3}$ | DM1 | 2.1 |
| $d = \dfrac{36}{5} = 7.2\ \text{(cm)}$ | A1 | 1.1b |
| **(6 marks)** | | |
# Question 6(b):
| Answer/Working | Mark | Guidance |
|---|---|---|
| Remains at rest $\Rightarrow$ centre of mass at centre of plane surface | B1 | 2.1 |
| Moments about diameter of plane surface: | M1 | 3.1b |
| $(9-d)W \left\{= \left(9-\dfrac{36}{5}\right)W\right\} = \dfrac{3}{8}\times 3\times kW$ | A1ft | 1.1b |
| $k = \dfrac{8}{5}$ | A1 | 1.1b |
| **(4 marks)** | | |
**(10 marks total)**6.
\begin{figure}[h]
\begin{center}
\includegraphics[alt={},max width=\textwidth]{1f39620e-c10f-4344-89f1-626fff36d187-20_369_815_255_632}
\captionsetup{labelformat=empty}
\caption{Figure 4}
\end{center}
\end{figure}
The shaded region shown in Figure 4 is bounded by the $x$-axis, the line with equation $x = 9$ and the line with equation $y = \frac { 1 } { 3 } x$. This shaded region is rotated through $360 ^ { \circ }$ about the $x$-axis to form a solid of revolution. This solid of revolution is used to model a solid right circular cone of height 9 cm and base radius 3 cm .
The cone is non-uniform and the mass per unit volume of the cone at the point ( $x , y , z$ ) is $\lambda x \mathrm {~kg} \mathrm {~cm} ^ { - 3 }$, where $0 \leqslant x \leqslant 9$ and $\lambda$ is constant.
\begin{enumerate}[label=(\alph*)]
\item Find the distance of the centre of mass of the cone from its vertex.
A toy is made by joining the circular plane face of the cone to the circular plane face of a uniform solid hemisphere of radius 3 cm , so that the centres of the two plane surfaces coincide.
The weight of the cone is $W$ newtons and the weight of the hemisphere is $k W$ newtons.\\
When the toy is placed on a smooth horizontal plane with any point of the curved surface of the hemisphere in contact with the plane, the toy will remain at rest.
\item Find the value of $k$
\end{enumerate}
\hfill \mbox{\textit{Edexcel FM2 2022 Q6 [10]}}