| Exam Board | Edexcel |
|---|---|
| Module | M3 (Mechanics 3) |
| Year | 2007 |
| Session | January |
| Marks | 5 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Moments |
| Type | Centre of mass of composite shapes |
| Difficulty | Challenging +1.3 This is a standard M3 non-uniform body equilibrium problem requiring knowledge that the center of mass of a cone is at h/4 from the base, then taking moments about the suspension point. While it involves 3D geometry and trigonometry, it follows a well-established method taught in M3 with no novel insight required—harder than average due to the spatial reasoning and calculation involved, but routine for Further Maths students. |
| Spec | 6.04b Find centre of mass: using symmetry6.04e Rigid body equilibrium: coplanar forces |
| Answer | Marks | Guidance |
|---|---|---|
| Height of cone = \(\frac{a}{\tan \alpha} = 3a\) | M1 A1 ↓ M1 | |
| Hence \(h = \frac{4}{3}a\) | ||
| \(\tan \theta = \frac{a}{\frac{4}{3}a} = \frac{4}{3} \Rightarrow \theta = 53.1°\) | M1 A1 (5) | |
| 1st M1 (generous) allow any trig ratio to get height of cone (e.g. using sin) | Guidance note | |
| 3rd M1 For correct trig ratio on a suitable triangle to get \(\theta\) or complement (even if they call the angle by another name – hence if they are aware or not that they are getting the required angle) | Guidance note |
Height of cone = $\frac{a}{\tan \alpha} = 3a$ | M1 A1 ↓ M1 |
Hence $h = \frac{4}{3}a$ | |
$\tan \theta = \frac{a}{\frac{4}{3}a} = \frac{4}{3} \Rightarrow \theta = 53.1°$ | M1 A1 (5) |
1st M1 (generous) allow any trig ratio to get height of cone (e.g. using sin) | | Guidance note
3rd M1 For correct trig ratio on a suitable triangle to get $\theta$ or complement (even if they call the angle by another name – hence if they are aware or not that they are getting the required angle) | | Guidance note
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\begin{figure}[h]
\begin{center}
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\caption{Figure 1}
\includegraphics[alt={},max width=\textwidth]{25b3ece7-69ed-4ec4-a6c7-4cd83ec2cc5e-03_513_399_303_785}
\end{center}
\end{figure}
A uniform solid right circular cone has base radius $a$ and semi-vertical angle $\alpha$, where $\tan \alpha = \frac { 1 } { 3 }$. The cone is freely suspended by a string attached at a point $A$ on the rim of its base, and hangs in equilibrium with its axis of symmetry making an angle of $\theta ^ { \circ }$ with the upward vertical, as shown in Figure 1.
Find, to one decimal place, the value of $\theta$.\\
\hfill \mbox{\textit{Edexcel M3 2007 Q2 [5]}}