| Exam Board | Edexcel |
|---|---|
| Module | M3 (Mechanics 3) |
| Year | 2012 |
| Session | June |
| Marks | 10 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Centre of Mass 1 |
| Type | Solid with removed cone from cone or cylinder |
| Difficulty | Standard +0.3 This is a standard M3 centre of mass question requiring routine application of composite body formulas and equilibrium conditions. Part (a) uses the standard cone COM formula with straightforward subtraction of volumes and moments. Part (b) applies basic moment equilibrium about V with given geometry. While multi-step, it follows predictable M3 patterns without requiring novel insight or complex problem-solving. |
| Spec | 6.04c Composite bodies: centre of mass |
\includegraphics{figure_2}
Figure 2 shows the cross-section $AVBC$ of the solid $S$ formed when a uniform right circular cone of base radius $a$ and height $a$, is removed from a uniform right circular cone of base radius $a$ and height $2a$. Both cones have the same axis $VCO$, where $O$ is the centre of the base of each cone.
\begin{enumerate}[label=(\alph*)]
\item Show that the distance of the centre of mass of $S$ from the vertex $V$ is $\frac{5}{4}a$. [5]
\end{enumerate}
The mass of $S$ is $M$. A particle of mass $kM$ is attached to $S$ at $B$. The system is suspended by a string attached to the vertex $V$, and hangs freely in equilibrium. Given that $VA$ is at an angle $45°$ to the vertical through $V$,
\begin{enumerate}[label=(\alph*)]
\setcounter{enumi}{1}
\item find the value of $k$. [5]
\end{enumerate}
\hfill \mbox{\textit{Edexcel M3 2012 Q4 [10]}}