| Exam Board | Edexcel |
|---|---|
| Module | M2 (Mechanics 2) |
| Year | 2007 |
| Session | January |
| Marks | 10 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Centre of Mass 1 |
| Type | Lamina with removed circle/semicircle |
| Difficulty | Standard +0.8 This is a multi-step centre of mass problem requiring: (a) composite body calculation with negative mass for removed disc, coordinate geometry to find position; (b) moments equilibrium with the template rotating about a point. Requires careful setup of coordinate systems, understanding of composite bodies, and torque balance. More demanding than standard M2 questions due to the geometry and two-part nature, but follows established techniques without requiring novel insight. |
| Spec | 6.04a Centre of mass: gravitational effect6.04b Find centre of mass: using symmetry6.04c Composite bodies: centre of mass6.04e Rigid body equilibrium: coplanar forces |
| Answer | Marks | Guidance |
|---|---|---|
| \(\bar{x} = 25\) (cm) exact | B1, B1ft, M1*, A1, DM1*, A1 | 6 marks |
| Answer | Marks | Guidance |
|---|---|---|
| \(M = \frac{3}{11}m\) (o.e.e.) | M1 †, A1ft, DM1 †, A1 | 4 marks; 10 marks total |
(a) **Mass Ratios:** Large $24^2$, Small $8^2$, Template $512$ (c.1810, c.200, c.1610)
Anything in ratio 9:1:8
$M(A)$ $9 \times 24 = 16 \times 1 + 8x$
$\bar{x} = 25$ (cm) exact | B1, B1ft, M1*, A1, DM1*, A1 | 6 marks
(b) $M(\text{axis})$ $11M = 12 \times \frac{1}{4}m$ fit their $\bar{x}$
$(36 - \bar{x})M = 12 \times \frac{1}{4}m$
$M = \frac{3}{11}m$ (o.e.e.) | M1 †, A1ft, DM1 †, A1 | 4 marks; 10 marks total
---\includegraphics{figure_1}
Figure 1 shows a template $T$ made by removing a circular disc, of centre $X$ and radius 8 cm, from a uniform circular lamina, of centre $O$ and radius 24 cm. The point $X$ lies on the diameter $AOB$ of the lamina and $AX = 16$ cm. The centre of mass of $T$ is at the point $G$.
\begin{enumerate}[label=(\alph*)]
\item Find $AG$. [6]
\end{enumerate}
The template $T$ is free to rotate about a smooth fixed horizontal axis, perpendicular to the plane of $T$, which passes through the mid-point of $OB$. A small stud of mass $\frac{1}{4}m$ is fixed at $B$, and $T$ and the stud are in equilibrium with $AB$ horizontal. Modelling the stud as a particle,
\begin{enumerate}[label=(\alph*)]
\setcounter{enumi}{1}
\item find the mass of $T$ in terms of $m$. [4]
\end{enumerate}
\hfill \mbox{\textit{Edexcel M2 2007 Q3 [10]}}