| Exam Board | SPS |
|---|---|
| Module | SPS FM Mechanics (SPS FM Mechanics) |
| Year | 2021 |
| Session | January |
| Marks | 11 |
| Topic | Centre of Mass 2 |
| Type | Deriving standard centre of mass formulae by integration |
| Difficulty | Standard +0.8 This is a multi-part Further Maths mechanics question requiring integration to find centre of mass of a 3D solid of revolution, followed by toppling analysis with friction constraints. While the integration setup is standard (disc method), the 3D geometry, proof requirement, and friction inequality analysis elevate it above typical A-level questions. The toppling condition and friction range require careful force/moment analysis but follow established methods. |
| Spec | 6.04d Integration: for centre of mass of laminas/solids6.04e Rigid body equilibrium: coplanar forces |
The triangular region shown below is rotated through $360°$ around the $x$-axis, to form a solid cone.
\includegraphics{figure_1}
The coordinates of the vertices of the triangle are $(0, 0)$, $(8, 0)$ and $(0, 4)$.
All units are in centimetres.
\begin{enumerate}[label=(\alph*)]
\item State an assumption that you should make about the cone in order to find the position of its centre of mass.
[1 mark]
\item Using integration, prove that the centre of mass of the cone is $2$cm from its plane face.
[5 marks]
\item The cone is placed with its plane face on a rough board. One end of the board is lifted so that the angle between the board and the horizontal is gradually increased. Eventually the cone topples without sliding.
\begin{enumerate}[label=(\roman*)]
\item Find the angle between the board and the horizontal when the cone topples, giving your answer to the nearest degree.
[2 marks]
\item Find the range of possible values for the coefficient of friction between the cone and the board.
[3 marks]
\end{enumerate}
\end{enumerate}
\hfill \mbox{\textit{SPS SPS FM Mechanics 2021 Q2 [11]}}