| Exam Board | Edexcel |
|---|---|
| Module | M3 (Mechanics 3) |
| Year | 2006 |
| Session | January |
| Marks | 9 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Centre of Mass 1 |
| Type | Composite solid with hemisphere and cylinder/cone |
| Difficulty | Standard +0.3 This is a standard M3 centre of mass question requiring application of standard formulas (hemisphere CM at 3r/8 from base) and moment equilibrium for toppling. The calculation is straightforward with given masses, and the toppling condition is a routine application of vertical line through CM passing through the pivot point. Slightly easier than average due to clear setup and standard techniques. |
| Spec | 6.04c Composite bodies: centre of mass6.04e Rigid body equilibrium: coplanar forces |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Marks | Guidance |
| \(5M\bar{x} = 3M \times \dfrac{h}{2} + 2M\left(h + \dfrac{3}{8}r\right)\) | M1 A2(1,0) | |
| \(5\bar{x} = \dfrac{3h}{2} + 2h + \dfrac{3}{4}r = \dfrac{7h}{2} + \dfrac{3}{4}r\) | ||
| \(\bar{x} = \dfrac{14h + 3r}{20}\) ✱ | M1 A1 | cso |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Marks | Guidance |
| \(\tan\alpha = \dfrac{20r}{14h + 3r} = \dfrac{4}{3}\) | M1 A1 | |
| Leading to \(h = \dfrac{6}{7}r\) | M1 A1 |
## Question 4:
**Part (a)**
| Answer/Working | Marks | Guidance |
|---|---|---|
| $5M\bar{x} = 3M \times \dfrac{h}{2} + 2M\left(h + \dfrac{3}{8}r\right)$ | M1 A2(1,0) | |
| $5\bar{x} = \dfrac{3h}{2} + 2h + \dfrac{3}{4}r = \dfrac{7h}{2} + \dfrac{3}{4}r$ | | |
| $\bar{x} = \dfrac{14h + 3r}{20}$ ✱ | M1 A1 | cso |
**(5 marks)**
**Part (b)**
| Answer/Working | Marks | Guidance |
|---|---|---|
| $\tan\alpha = \dfrac{20r}{14h + 3r} = \dfrac{4}{3}$ | M1 A1 | |
| Leading to $h = \dfrac{6}{7}r$ | M1 A1 | |
**(4 marks) — Total 9 marks**
---4.
\begin{figure}[h]
\begin{center}
\captionsetup{labelformat=empty}
\caption{Figure 2}
\includegraphics[alt={},max width=\textwidth]{67a9cf74-833f-4b4a-9fde-3c62dcc08e8c-3_531_387_1226_845}
\end{center}
\end{figure}
A body consists of a uniform solid circular cylinder $C$, together with a uniform solid hemisphere $H$ which is attached to $C$. The plane face of $H$ coincides with the upper plane face of $C$, as shown in Figure 2. The cylinder $C$ has base radius $r$, height $h$ and mass 3M. The mass of $H$ is $2 M$. The point $O$ is the centre of the base of $C$.
\begin{enumerate}[label=(\alph*)]
\item Show that the distance of the centre of mass of the body from $O$ is
$$\frac { 14 h + 3 r } { 20 } .$$
The body is placed with its plane face on a rough plane which is inclined at an angle $\alpha$ to the horizontal, where tan $\alpha = \frac { 4 } { 3 }$. The plane is sufficiently rough to prevent slipping. Given that the body is on the point of toppling,
\item find $h$ in terms of $r$.
\end{enumerate}
\hfill \mbox{\textit{Edexcel M3 2006 Q4 [9]}}