| Exam Board | Edexcel |
|---|---|
| Module | M3 (Mechanics 3) |
| Year | 2006 |
| Session | June |
| Marks | 10 |
| 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 involving composite bodies. Part (a) requires routine application of the formula for removing a hemisphere from another (using negative mass), and part (b) involves combining the bowl with liquid using the standard centre of mass formula. Both parts follow textbook methods with straightforward algebra, making it slightly easier than average for M3 level. |
| Spec | 6.04c Composite bodies: centre of mass6.04d Integration: for centre of mass of laminas/solids |
A bowl consists of a uniform solid metal hemisphere, of radius $a$ and centre $O$, from which is removed the solid hemisphere of radius $\frac{1}{4}a$ with the same centre $O$.
\begin{enumerate}[label=(\alph*)]
\item Show that the distance of the centre of mass of the bowl from $O$ is $\frac{45}{112}a$.
[5]
\end{enumerate}
The bowl is fixed with its plane face uppermost and horizontal. It is now filled with liquid. The mass of the bowl is $M$ and the mass of the liquid is $kM$, where $k$ is a constant. Given that the distance of the centre of mass of the bowl and liquid together from $O$ is $\frac{17}{48}a$,
\begin{enumerate}[label=(\alph*)]
\setcounter{enumi}{1}
\item Find the value of $k$.
[5]
\end{enumerate}
\hfill \mbox{\textit{Edexcel M3 2006 Q2 [10]}}