| Exam Board | Edexcel |
|---|---|
| Module | M3 (Mechanics 3) |
| Year | 2011 |
| Session | June |
| 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 volume/mass calculations for standard shapes (hemisphere and cylinder), taking moments about a point, and applying toppling condition (vertical through COM passes through edge). All techniques are routine for this module with no novel insight required, making it slightly easier than average. |
| Spec | 6.04c Composite bodies: centre of mass6.04e Rigid body equilibrium: coplanar forces |
| Answer | Marks | Guidance |
|---|---|---|
| Working/Answer | Marks | Notes |
| Mass ratio: \(\pi(3l)^2\times 5l\rho\) : \(\frac{2}{3}\pi(3l)^3\times 2\rho\) giving \(5:4\) (total \(9\)) | B1 | |
| Distances from \(O\): \(\frac{5}{2}l\) and \(-\frac{3}{8}\times 3l\) | B1 | |
| Moments equation: \(5\times\frac{5}{2}l - 4\times\frac{9}{8}l = 9\bar{x}\) | M1 A1 ft | |
| \(\bar{x} = \frac{8}{9}l\) | A1 | (5) |
| Answer | Marks | Guidance |
|---|---|---|
| Working/Answer | Marks | Notes |
| \(GX = 5l - \frac{8}{9}l = \frac{37}{9}l\) | B1 ft | |
| \(\tan\theta^\circ = \dfrac{3l}{\frac{37}{9}l} = \dfrac{27}{37}\) | M1 A1 ft | |
| \(\theta^\circ = 36.1°\) accept \(36°\), \(0.63\) or \(0.630\) rad or better | A1 | (4) Total: 9 |
# Question 3:
## Part (a)
| Working/Answer | Marks | Notes |
|---|---|---|
| Mass ratio: $\pi(3l)^2\times 5l\rho$ : $\frac{2}{3}\pi(3l)^3\times 2\rho$ giving $5:4$ (total $9$) | B1 | |
| Distances from $O$: $\frac{5}{2}l$ and $-\frac{3}{8}\times 3l$ | B1 | |
| Moments equation: $5\times\frac{5}{2}l - 4\times\frac{9}{8}l = 9\bar{x}$ | M1 A1 ft | |
| $\bar{x} = \frac{8}{9}l$ | A1 | **(5)** |
## Part (b)
| Working/Answer | Marks | Notes |
|---|---|---|
| $GX = 5l - \frac{8}{9}l = \frac{37}{9}l$ | B1 ft | |
| $\tan\theta^\circ = \dfrac{3l}{\frac{37}{9}l} = \dfrac{27}{37}$ | M1 A1 ft | |
| $\theta^\circ = 36.1°$ accept $36°$, $0.63$ or $0.630$ rad or better | A1 | **(4) Total: 9** |3.
\begin{figure}[h]
\begin{center}
\includegraphics[alt={},max width=\textwidth]{826ad8ff-6e5c-4224-88ba-e78b79d1bc21-04_542_469_219_735}
\captionsetup{labelformat=empty}
\caption{Figure 2}
\end{center}
\end{figure}
A solid consists of a uniform solid right cylinder of height $5 l$ and radius $3 l$ joined to a uniform solid hemisphere of radius $3 l$. The plane face of the hemisphere coincides with a circular end of the cylinder and has centre $O$, as shown in Figure 2.
The density of the hemisphere is twice the density of the cylinder.
\begin{enumerate}[label=(\alph*)]
\item Find the distance of the centre of mass of the solid from $O$.\\
(5)
\begin{figure}[h]
\begin{center}
\includegraphics[alt={},max width=\textwidth]{826ad8ff-6e5c-4224-88ba-e78b79d1bc21-04_618_807_1327_571}
\captionsetup{labelformat=empty}
\caption{Figure 3}
\end{center}
\end{figure}
The solid is now placed with its circular face on a plane inclined at an angle $\theta ^ { \circ }$ to the horizontal, as shown in Figure 3. The plane is sufficiently rough to prevent the solid slipping. The solid is on the point of toppling.
\item Find the value of $\theta$.
\end{enumerate}
\hfill \mbox{\textit{Edexcel M3 2011 Q3 [9]}}