| Exam Board | Edexcel |
|---|---|
| Module | M3 (Mechanics 3) |
| Year | 2005 |
| Session | June |
| Marks | 9 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Centre of Mass 1 |
| Type | Conical or hemispherical shell composite |
| Difficulty | Standard +0.3 This is a standard M3 centre of mass question requiring knowledge of standard results (hemisphere COM at 3a/8 from centre, disc at centre), then a straightforward equilibrium calculation with moments. The 'show that' part guides students through the first step, and the second part is routine application of taking moments about the contact point. Slightly above average due to the 3D geometry visualization required and two-part structure, but follows textbook methods throughout. |
| Spec | 6.04b Find centre of mass: using symmetry6.04c Composite bodies: centre of mass6.04d Integration: for centre of mass of laminas/solids6.04e Rigid body equilibrium: coplanar forces |
| Answer | Marks | Guidance |
|---|---|---|
| Working/Answer | Marks | Guidance |
| Mass ratio Bowl : Lid : \(C\) = \(2:1:3\) | B1 | anything in ratio \(2:1:3\) |
| \(\bar{y}\) values: \(\frac{1}{2}a\), \(0\), \(\bar{y}\) | B1 | |
| \(M(O)\): \(2 \times \frac{1}{2}a = 3\bar{y}\) | M1 | |
| \(\bar{y} = \frac{1}{3}a\) * | A1 | cso |
| Total | (4) |
| Answer | Marks | Guidance |
|---|---|---|
| Working/Answer | Marks | Guidance |
| \(M(A)\): \(Mg \times \frac{1}{3}a\sin\theta = \frac{1}{2}Mg \times a\cos\theta\) | M1 A1=A1 | |
| \(\tan\theta = \frac{3}{2}\) | M1 | |
| \(\theta \approx 56°\) | A1 | cao |
| Total | (5) [9] |
| Answer | Marks | Guidance |
|---|---|---|
| Working/Answer | Marks | Guidance |
| Finding both coordinates of \(G'\): \(\frac{2}{3}a = 3\bar{y} \Rightarrow \bar{y} = \frac{2}{9}a\) | M1 A1 | |
| \(a = 3\bar{x} \Rightarrow \bar{x} = \frac{1}{3}a\) | A1 | |
| \(\tan\theta = \frac{\frac{1}{3}a}{\frac{2}{9}a} = \frac{3}{2}\) | M1 | |
| \(\theta \approx 56°\) | A1 | cao |
| Total | (5) |
| Answer | Marks | Guidance |
|---|---|---|
| Working/Answer | Marks | Guidance |
| \(GG':G'P = \frac{1}{2}M : M = 1:2\); \(OG = \frac{1}{3}a\), \(OP = a\) | ||
| By similar triangles: \(ON = \frac{1}{3}OP = \frac{1}{3}a\); \(NG' = \frac{2}{3}OG = \frac{2}{9}a\) | M1 A1 A1 | |
| \(\tan\theta = \frac{ON}{NG'} = \frac{\frac{1}{3}a}{\frac{2}{9}a} = \frac{3}{2}\) | M1 | |
| \(\theta \approx 56°\) | A1 | cao |
| Total | (5) |
# Question 2:
## Part (a):
| Working/Answer | Marks | Guidance |
|---|---|---|
| Mass ratio Bowl : Lid : $C$ = $2:1:3$ | B1 | anything in ratio $2:1:3$ |
| $\bar{y}$ values: $\frac{1}{2}a$, $0$, $\bar{y}$ | B1 | |
| $M(O)$: $2 \times \frac{1}{2}a = 3\bar{y}$ | M1 | |
| $\bar{y} = \frac{1}{3}a$ * | A1 | cso |
| **Total** | **(4)** | |
## Part (b):
| Working/Answer | Marks | Guidance |
|---|---|---|
| $M(A)$: $Mg \times \frac{1}{3}a\sin\theta = \frac{1}{2}Mg \times a\cos\theta$ | M1 A1=A1 | |
| $\tan\theta = \frac{3}{2}$ | M1 | |
| $\theta \approx 56°$ | A1 | cao |
| **Total** | **(5) [9]** | |
### First Alternative (b):
| Working/Answer | Marks | Guidance |
|---|---|---|
| Finding both coordinates of $G'$: $\frac{2}{3}a = 3\bar{y} \Rightarrow \bar{y} = \frac{2}{9}a$ | M1 A1 | |
| $a = 3\bar{x} \Rightarrow \bar{x} = \frac{1}{3}a$ | A1 | |
| $\tan\theta = \frac{\frac{1}{3}a}{\frac{2}{9}a} = \frac{3}{2}$ | M1 | |
| $\theta \approx 56°$ | A1 | cao |
| **Total** | **(5)** | |
### Second Alternative (b):
| Working/Answer | Marks | Guidance |
|---|---|---|
| $GG':G'P = \frac{1}{2}M : M = 1:2$; $OG = \frac{1}{3}a$, $OP = a$ | | |
| By similar triangles: $ON = \frac{1}{3}OP = \frac{1}{3}a$; $NG' = \frac{2}{3}OG = \frac{2}{9}a$ | M1 A1 A1 | |
| $\tan\theta = \frac{ON}{NG'} = \frac{\frac{1}{3}a}{\frac{2}{9}a} = \frac{3}{2}$ | M1 | |
| $\theta \approx 56°$ | A1 | cao |
| **Total** | **(5)** | |
---2. A closed container $C$ consists of a thin uniform hollow hemispherical bowl of radius $a$, together with a lid. The lid is a thin uniform circular disc, also of radius $a$. The centre $O$ of the disc coincides with the centre of the hemispherical bowl. The bowl and its lid are made of the same material.
\begin{enumerate}[label=(\alph*)]
\item Show that the centre of mass of $C$ is at a distance $\frac { 1 } { 3 } a$ from $O$.
The container $C$ has mass $M$. A particle of mass $\frac { 1 } { 2 } M$ is attached to the container at a point $P$ on the circumference of the lid. The container is then placed with a point of its curved surface in contact with a horizontal plane. The container rests in equilibrium with $P , O$ and the point of contact in the same vertical plane.
\item Find, to the nearest degree, the angle made by the line $P O$ with the horizontal.
\end{enumerate}
\hfill \mbox{\textit{Edexcel M3 2005 Q2 [9]}}