| Exam Board | Edexcel |
|---|---|
| Module | M3 (Mechanics 3) |
| Marks | 12 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Centre of Mass 1 |
| Type | Solid with removed cone from cone or cylinder |
| Difficulty | Challenging +1.2 This is a standard M3 centre of mass problem requiring systematic application of composite body techniques and equilibrium conditions. Part (a) involves routine volume and centre of mass calculations for cones using standard formulas, while part (b) applies equilibrium to find a geometric relationship. The multi-step nature and algebraic manipulation elevate it slightly above average difficulty, but the methods are well-practiced in M3 courses with no novel insight required. |
| Spec | 6.04d Integration: for centre of mass of laminas/solids |
| Answer | Marks |
|---|---|
| (a) Let mass of cone = \(M\), so mass removed = \(\frac{1}{8}M\), remainder = \(\frac{7}{8}M\) | M1 A1 |
| \(M \frac{h}{4} = M\left(\frac{h}{2} + \frac{h}{8}\right) + \frac{1}{8}M y\) | M1 A1 I1 M1 A1 |
| \(\frac{7}{8}y - \frac{h}{4} - \frac{5h}{64} = \frac{11h}{64}\) | |
| \(y = \frac{11h}{56}\) | |
| (b) \(\tan \alpha = \frac{1}{2}\left(\frac{h}{2} - \frac{11h}{56}\right) = \frac{28c}{17h}\) | M1 A1 M1 A1 I1 M1 |
| \(\frac{28c}{17h} = \frac{1}{2}\) | |
| \(h : r = 56 : 17\) | M1 A1 I1 M1 |
| Total: 12 marks |
**(a)** Let mass of cone = $M$, so mass removed = $\frac{1}{8}M$, remainder = $\frac{7}{8}M$ | M1 A1 |
$M \frac{h}{4} = M\left(\frac{h}{2} + \frac{h}{8}\right) + \frac{1}{8}M y$ | M1 A1 I1 M1 A1 |
$\frac{7}{8}y - \frac{h}{4} - \frac{5h}{64} = \frac{11h}{64}$ | |
$y = \frac{11h}{56}$ | |
**(b)** $\tan \alpha = \frac{1}{2}\left(\frac{h}{2} - \frac{11h}{56}\right) = \frac{28c}{17h}$ | M1 A1 M1 A1 I1 M1 |
$\frac{28c}{17h} = \frac{1}{2}$ | |
$h : r = 56 : 17$ | M1 A1 I1 M1 |
| | **Total: 12 marks** |A uniform solid right circular cone has height $h$ and base radius $r$. The top part of the cone is removed by cutting through the cone parallel to the base at a height $\frac{h}{2}$.
\includegraphics{figure_5}
\begin{enumerate}[label=(\alph*)]
\item Show that the centre of mass of the remaining solid is at a height $\frac{11h}{56}$ above the base, along its axis of symmetry. [7 marks]
\end{enumerate}
The remaining part of the solid is suspended from the point $D$ on the circumference of its smaller circular face, and the axis of symmetry then makes an angle $\alpha$ with the vertical, where $\tan \alpha = \frac{1}{2}$.
\begin{enumerate}[label=(\alph*)]
\setcounter{enumi}{1}
\item Find the value of the ratio $h : r$. [5 marks]
\end{enumerate}
\hfill \mbox{\textit{Edexcel M3 Q5 [12]}}