| Exam Board | Edexcel |
|---|---|
| Module | M3 (Mechanics 3) |
| Year | 2008 |
| Session | June |
| Marks | 13 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Centre of Mass 1 |
| Type | Solid with removed cylinder or hemisphere from solid |
| Difficulty | Standard +0.3 This is a standard M3 centre of mass question with a familiar structure: composite solid with removal, then combined with cylinder, then toppling analysis. Uses standard formulas (hemisphere COM at 3r/8), straightforward algebraic manipulation, and routine toppling condition. The 'show that' format guides students to the answer, reducing problem-solving demand. Slightly easier than average due to its textbook nature and clear structure. |
| Spec | 6.04c Composite bodies: centre of mass6.04d Integration: for centre of mass of laminas/solids6.04e Rigid body equilibrium: coplanar forces |
| Answer | Marks |
|---|---|
| Mass \(a^3 \cdot \frac{2}{3}\pi \times\): 216, 8, 208; 27, 1, 26 | M1A1 |
| C of M from O: \(\frac{3}{8} \times 6a\), \(\frac{3}{8} \times 2a\), \(\bar{x}\); Use of \(\frac{3}{8}r\) | M1 |
| Moment: \(216 \times \frac{6a \times 3}{8} = 8 \times \frac{2a \times 3}{8} + 208\bar{x}\) | M1 |
| \(\bar{x} = \frac{480a}{208} = \frac{30a}{13}\) | A1cso (5) |
| Answer | Marks |
|---|---|
| Mass \(\pi a^3 \times\): \(\frac{416}{3}\), \(+24\), \(=\frac{488}{3}\) | B1 |
| C of M: \(\frac{30}{13}a\), \(+9a\), \(=\bar{y}\) | B1 |
| Moments: \(320a + 216a = \frac{488}{3}\bar{y}\) | M1 |
| \(\bar{y} = \frac{201}{61}a\) | A1cso (4) |
| Answer | Marks |
|---|---|
| \(\tan\theta = \frac{2a}{12a - \frac{201}{61}a}\); \(\tan\theta = \frac{2a}{...}\); \(12a - \frac{201}{61}a\) | M1 M1 |
| \(\theta = 12.93...\) | A1 |
| so critical angle \(= 12.93...\) \(\therefore\) if \(\theta = 12°\) it will NOT topple | A1∨ (4) |
**Part (a):**
Mass $a^3 \cdot \frac{2}{3}\pi \times$: 216, 8, 208; 27, 1, 26 | M1A1 |
C of M from O: $\frac{3}{8} \times 6a$, $\frac{3}{8} \times 2a$, $\bar{x}$; Use of $\frac{3}{8}r$ | M1 |
Moment: $216 \times \frac{6a \times 3}{8} = 8 \times \frac{2a \times 3}{8} + 208\bar{x}$ | M1 |
$\bar{x} = \frac{480a}{208} = \frac{30a}{13}$ | A1cso (5) |
**Part (b):**
Mass $\pi a^3 \times$: $\frac{416}{3}$, $+24$, $=\frac{488}{3}$ | B1 |
C of M: $\frac{30}{13}a$, $+9a$, $=\bar{y}$ | B1 |
Moments: $320a + 216a = \frac{488}{3}\bar{y}$ | M1 |
$\bar{y} = \frac{201}{61}a$ | A1cso (4) |
**Part (c):**
$\tan\theta = \frac{2a}{12a - \frac{201}{61}a}$; $\tan\theta = \frac{2a}{...}$; $12a - \frac{201}{61}a$ | M1 M1 |
$\theta = 12.93...$ | A1 |
so critical angle $= 12.93...$ $\therefore$ if $\theta = 12°$ it will NOT topple | A1∨ (4) |
**Total: 13 marks**
---4.
\begin{figure}[h]
\begin{center}
\includegraphics[alt={},max width=\textwidth]{f07b8a65-ccb5-423f-96cc-b303bd05ad1f-07_454_614_239_662}
\captionsetup{labelformat=empty}
\caption{Figure 3}
\end{center}
\end{figure}
A uniform solid hemisphere, of radius $6 a$ and centre $O$, has a solid hemisphere of radius $2 a$, and centre $O$, removed to form a bowl $B$ as shown in Figure 3.
\begin{enumerate}[label=(\alph*)]
\item Show that the centre of mass of $B$ is $\frac { 30 } { 13 } a$ from $O$.
\begin{figure}[h]
\begin{center}
\includegraphics[alt={},max width=\textwidth]{f07b8a65-ccb5-423f-96cc-b303bd05ad1f-07_735_614_1126_662}
\captionsetup{labelformat=empty}
\caption{Figure 4}
\end{center}
\end{figure}
The bowl $B$ is fixed to a plane face of a uniform solid cylinder made from the same material as $B$. The cylinder has radius $2 a$ and height $6 a$ and the combined solid $S$ has an axis of symmetry which passes through $O$, as shown in Figure 4.
\item Show that the centre of mass of $S$ is $\frac { 201 } { 61 } a$ from $O$.
The plane surface of the cylindrical base of $S$ is placed on a rough plane inclined at $12 ^ { \circ }$ to the horizontal. The plane is sufficiently rough to prevent slipping.
\item Determine whether or not $S$ will topple.
\section*{\begin{center}
\includegraphics[max width=\textwidth, alt={}]{f07b8a65-ccb5-423f-96cc-b303bd05ad1f-08_56_366_251_178}
\end{center}}
\end{enumerate}
\hfill \mbox{\textit{Edexcel M3 2008 Q4 [13]}}