| Exam Board | Edexcel |
|---|---|
| Module | M3 (Mechanics 3) |
| Year | 2011 |
| Session | January |
| Marks | 8 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Centre of Mass 1 |
| Type | Centre of mass with variable parameter |
| Difficulty | Standard +0.3 This is a standard M3 centre of mass question requiring application of standard formulas for hemisphere and cone centres of mass, then using the principle that the centre of mass lies on the vertical through the suspension point. Part (a) is straightforward algebraic manipulation with known formulas, and part (b) involves basic geometry with the 30° angle. Both parts follow predictable patterns for this topic 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 |
|---|---|
| Mass ratio: \(4m\), \(km\) | B1 |
| Distance from \(O\): \(\frac{3}{8}r\), \(-\frac{1}{2}r\) | B1 |
| Moments about \(O\): \(\frac{3}{8}r \times 4m = \frac{1}{2}r \times km\) | M1 |
| \(k = 3\) | A1 |
| Answer | Marks |
|---|---|
| \(\tan 30° = \frac{OG}{r}\) | B1 |
| \(OG = \frac{\lambda}{(7 + \lambda)} \times 2r\) | M1 |
| \(\frac{1}{\sqrt{3}} = \frac{\lambda}{(7 + \lambda)} \times 2r \times \frac{1}{r}\) | A1 |
| \(7 + \lambda = 2\sqrt{3}\lambda\) | |
| \(\lambda = \frac{7}{(2\sqrt{3} - 1)}\) (o.e.) or \(2.84\) | A1 |
## Part (a):
Mass ratio: $4m$, $km$ | B1 |
Distance from $O$: $\frac{3}{8}r$, $-\frac{1}{2}r$ | B1 |
Moments about $O$: $\frac{3}{8}r \times 4m = \frac{1}{2}r \times km$ | M1 |
$k = 3$ | A1 |
## Part (b):
$\tan 30° = \frac{OG}{r}$ | B1 |
$OG = \frac{\lambda}{(7 + \lambda)} \times 2r$ | M1 |
$\frac{1}{\sqrt{3}} = \frac{\lambda}{(7 + \lambda)} \times 2r \times \frac{1}{r}$ | A1 |
$7 + \lambda = 2\sqrt{3}\lambda$ | |
$\lambda = \frac{7}{(2\sqrt{3} - 1)}$ (o.e.) or $2.84$ | A1 |\includegraphics{figure_1}
A toy is formed by joining a uniform solid hemisphere, of radius $r$ and mass $4m$, to a uniform right circular solid cone of mass $km$. The cone has vertex $A$, base radius $r$ and height $2r$. The plane face of the cone coincides with the plane face of the hemisphere. The centre of the plane face of the hemisphere is $O$ and $OB$ is a radius of its plane face as shown in Figure 1. The centre of mass of the toy is at $O$.
\begin{enumerate}[label=(\alph*)]
\item Find the value of $k$.
[4]
\end{enumerate}
A metal stud of mass $2m$ is attached to the toy at $A$. The toy is now suspended by a light string attached to $B$ and hangs freely at rest. The angle between $OB$ and the vertical is $30°$.
\begin{enumerate}[label=(\alph*)]
\setcounter{enumi}{1}
\item Find the value of $\lambda$.
[4]
\end{enumerate}
\hfill \mbox{\textit{Edexcel M3 2011 Q2 [8]}}