| Exam Board | Edexcel |
|---|---|
| Module | M3 (Mechanics 3) |
| Year | 2002 |
| Session | January |
| Marks | 10 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Centre of Mass 1 |
| Type | Toppling on inclined plane |
| Difficulty | Standard +0.3 This is a standard M3 centre of mass question requiring volume integration using the formula x̄ = ∫x dV/∫dV for a solid of revolution, followed by a routine toppling condition (tan α ≤ base radius/height to centre of mass). The integration is straightforward with y² = rx, and part (b) is a direct application of the toppling criterion. Slightly easier than average due to the guided 'show that' in part (a) and standard techniques throughout. |
| Spec | 4.08d Volumes of revolution: about x and y axes6.04d Integration: for centre of mass of laminas/solids6.04e Rigid body equilibrium: coplanar forces |
\includegraphics{figure_2}
Figure 2 shows the region $R$ bounded by the curve with equation $y^2 = rx$, where $r$ is a positive constant, the $x$-axis and the line $x = r$. A uniform solid of revolution $S$ is formed by rotating $R$ through one complete revolution about the $x$-axis.
\begin{enumerate}[label=(\alph*)]
\item Show that the distance of the centre of mass of $S$ from $O$ is $\frac{4}{5}r$. [6]
\end{enumerate}
The solid is placed with its plane face on a plane which is inclined at an angle $\alpha$ to the horizontal. The plane is sufficiently rough to prevent $S$ from sliding. Given that $S$ does not topple,
\begin{enumerate}[label=(\alph*)]
\setcounter{enumi}{1}
\item find, to the nearest degree, the maximum value of $\alpha$. [4]
\end{enumerate}
\hfill \mbox{\textit{Edexcel M3 2002 Q4 [10]}}