| Exam Board | Edexcel |
|---|---|
| Module | M3 (Mechanics 3) |
| Year | 2005 |
| Session | January |
| Marks | 9 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Centre of Mass 1 |
| Type | Composite solid with hemisphere and cylinder/cone |
| Difficulty | Standard +0.8 This is a multi-step centre of mass problem requiring knowledge of standard results for hemisphere and cone centres of mass, algebraic manipulation to prove a given result, and then applying equilibrium conditions with toppling analysis. Part (b) requires setting up a moments equation about the pivot point OA and solving an inequality, which goes beyond routine calculation. The constraint 2m < M and the specific geometry add complexity, making this harder than average but still within standard M3 scope. |
| Spec | 6.04c Composite bodies: centre of mass6.04e Rigid body equilibrium: coplanar forces |
\includegraphics{figure_1}
A child's toy consists of a uniform solid hemisphere, of mass $M$ and base radius $r$, joined to a uniform solid right circular cone of mass $m$, where $2m < M$. The cone has vertex $O$, base radius $r$ and height $3r$. Its plane face, with diameter $AB$, coincides with the plane face of the hemisphere, as shown in Figure 1.
\begin{enumerate}[label=(\alph*)]
\item Show that the distance of the centre of mass of the toy from $AB$ is
$$\frac{3(M - 2m)}{8(M + m)}r.$$ [5]
\end{enumerate}
The toy is placed with $OA$ on a horizontal surface. The toy is released from rest and does not remain in equilibrium.
\begin{enumerate}[label=(\alph*)]
\setcounter{enumi}{1}
\item Show that $M > 26m$. [4]
\end{enumerate}
\hfill \mbox{\textit{Edexcel M3 2005 Q2 [9]}}