| Exam Board | Edexcel |
|---|---|
| Module | M3 (Mechanics 3) |
| Year | 2004 |
| Session | June |
| Marks | 13 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Centre of Mass 1 |
| Type | Composite solid with cone and cylinder |
| Difficulty | Standard +0.3 This is a standard M3 centre of mass question requiring application of standard formulas for cone and cylinder centres of mass, then using composite body techniques. Part (a) is routine calculation, part (b) uses standard equilibrium geometry, and part (c) applies the toppling condition. All techniques are textbook exercises with no novel insight required, making it slightly easier than average. |
| 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 ratios: Cylinder \((36\pi r^3)\) = 3, Cone \((12\pi r^3)\) = 1, Toy \((48\pi r^3)\) = 4 | B1 | |
| Distances from \(O\): Cylinder = \(2r\), Cone = \((-r)\), Toy = \(\bar{x}\) | B1 | |
| \((3 \times 2r) - r = 4\bar{x}\) | M1 A1 | M1 for clear attempt at \(\Sigma mx = \bar{x}\,\Sigma m\) — correct no. of terms |
| \(\dfrac{5r}{4} = \bar{x}\) | A1 | (5 marks); If distances not measured from \(O\), B1B1M1A1 available |
| Answer | Marks | Guidance |
|---|---|---|
| Working/Answer | Marks | Guidance |
| \(AG\) vertical, seen or implied | M1 | |
| \(\tan\theta = \dfrac{3r}{4r - \bar{x}}\) | M1 A1 | second M1 for use of tan |
| \(\theta = 47.5°\) (1 d.p.) | A1 | (4 marks) |
| Answer | Marks | Guidance |
|---|---|---|
| Working/Answer | Marks | Guidance |
| Similar \(\Delta\)'s: \(\dfrac{OX}{3r} = \dfrac{3r}{4r}\) \((= \tan\alpha)\) | M1 | |
| \(\Rightarrow OX = \dfrac{9r}{4}\) | A1 | |
| \(\bar{x} < OX\) | M1 | independent mark, for the general idea |
| \(\Rightarrow\) won't topple | A1 c.s.o | (4 marks); (13 marks) |
## Question 5:
### Part (a):
| Working/Answer | Marks | Guidance |
|---|---|---|
| Mass ratios: Cylinder $(36\pi r^3)$ = 3, Cone $(12\pi r^3)$ = 1, Toy $(48\pi r^3)$ = 4 | B1 | |
| Distances from $O$: Cylinder = $2r$, Cone = $(-r)$, Toy = $\bar{x}$ | B1 | |
| $(3 \times 2r) - r = 4\bar{x}$ | M1 A1 | M1 for clear attempt at $\Sigma mx = \bar{x}\,\Sigma m$ — correct no. of terms |
| $\dfrac{5r}{4} = \bar{x}$ | A1 | (5 marks); If distances not measured from $O$, B1B1M1A1 available |
### Part (b):
| Working/Answer | Marks | Guidance |
|---|---|---|
| $AG$ vertical, seen or implied | M1 | |
| $\tan\theta = \dfrac{3r}{4r - \bar{x}}$ | M1 A1 | second M1 for use of tan |
| $\theta = 47.5°$ (1 d.p.) | A1 | (4 marks) |
### Part (c):
| Working/Answer | Marks | Guidance |
|---|---|---|
| Similar $\Delta$'s: $\dfrac{OX}{3r} = \dfrac{3r}{4r}$ $(= \tan\alpha)$ | M1 | |
| $\Rightarrow OX = \dfrac{9r}{4}$ | A1 | |
| $\bar{x} < OX$ | M1 | independent mark, for the general idea |
| $\Rightarrow$ won't topple | A1 c.s.o | (4 marks); **(13 marks)** |
---5.
\begin{figure}[h]
\begin{center}
\captionsetup{labelformat=empty}
\caption{Figure 2}
\includegraphics[alt={},max width=\textwidth]{b9e9b91c-7e6d-4b84-9f0e-180b626887c2-3_522_659_1043_701}
\end{center}
\end{figure}
A toy is formed by joining a uniform solid right circular cone, of base radius $3 r$ and height $4 r$, to a uniform solid cylinder, also of radius $3 r$ and height $4 r$. The cone and the cylinder are made from the same material, and the plane face of the cone coincides with a plane face of the cylinder, as shown in Fig. 2. The centre of this plane face is $O$.
\begin{enumerate}[label=(\alph*)]
\item Find the distance of the centre of mass of the toy from $O$.
The point $A$ lies on the edge of the plane face of the cylinder which forms the base of the toy. The toy is suspended from $A$ and hangs in equilibrium.
\item Find, in degrees to one decimal place, the angle between the axis of symmetry of the toy and the vertical.
The toy is placed with the curved surface of the cone on horizontal ground.
\item Determine whether the toy will topple.\\
(4)
\end{enumerate}
\hfill \mbox{\textit{Edexcel M3 2004 Q5 [13]}}