| Exam Board | Edexcel |
|---|---|
| Module | M3 (Mechanics 3) |
| Year | 2002 |
| Session | January |
| Marks | 15 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Circular Motion 2 |
| Type | Particle on inner surface of sphere/bowl |
| Difficulty | Standard +0.8 This is a substantial M3 circular motion problem requiring energy conservation, circular motion dynamics, and projectile motion. Part (b) requires solving for the leaving angle using the condition that normal reaction becomes zero, involving trigonometric manipulation. Part (c) adds projectile motion analysis. While systematic, it demands careful coordination of multiple mechanics principles and algebraic manipulation across 15 marks, placing it moderately above average difficulty. |
| Spec | 3.02i Projectile motion: constant acceleration model6.02i Conservation of energy: mechanical energy principle6.05d Variable speed circles: energy methods |
\includegraphics{figure_3}
Figure 3 shows a fixed hollow sphere of internal radius $a$ and centre $O$. A particle $P$ of mass $m$ is projected horizontally from the lowest point $A$ of a sphere with speed $\sqrt{\left(\frac{5}{4}ag\right)}$. It moves in a vertical circle, centre $O$, on the smooth inner surface of the sphere. The particle passes through the point $B$, which is in the same horizontal plane as $O$. It leaves the surface of the sphere at the point $C$, where $OC$ makes an angle $\theta$ with the upward vertical.
\begin{enumerate}[label=(\alph*)]
\item Find, in terms of $m$ and $g$, the normal reaction between $P$ and the surface of the sphere at $B$. [4]
\item Show that $\theta = 60°$. [7]
\end{enumerate}
After leaving the surface of the sphere, $P$ meets it again at the point $A$.
\begin{enumerate}[label=(\alph*)]
\setcounter{enumi}{2}
\item Find, in terms of $a$ and $g$, the time $P$ takes to travel from $C$ to $A$. [4]
\end{enumerate}
\hfill \mbox{\textit{Edexcel M3 2002 Q7 [15]}}