| Exam Board | Edexcel |
|---|---|
| Module | M3 (Mechanics 3) |
| Marks | 11 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Circular Motion 2 |
| Type | Particle on outer surface of sphere |
| Difficulty | Standard +0.8 This is a sophisticated M3 mechanics problem requiring energy conservation, circular motion dynamics, and analysis of when contact is lost. While the individual techniques (energy methods, resolving forces radially) are standard M3 content, the problem requires careful setup of equations in a non-trivial geometry, algebraic manipulation to reach given results, and conceptual understanding of the physical constraint for leaving the surface. The multi-part structure with 'show that' proofs and the discussion element elevate it above routine exercises, but it remains within the expected scope of M3 without requiring exceptional insight. |
| Spec | 6.02i Conservation of energy: mechanical energy principle6.05f Vertical circle: motion including free fall |
| Answer | Marks |
|---|---|
| (a) (i) \(\frac{1}{2}mv^2 = \frac{1}{2}mu^2 + mgr(1 - \cos \theta)\) | M1 A1 A1 |
| \(v^2 = u^2 + 2gr(1 - \cos \theta)\) | |
| (ii) \(mg \cos \theta - X = \frac{mv^2}{r}\) | M1 A1 |
| \(X = mg \cos \theta - \frac{mu^2}{r} - 2mg(1 - \cos \theta)\) | M1 A1 |
| \(X = mg[3\cos \theta - 2 - \frac{u^2}{gr}]\) | A1 |
| (b) Leaves sphere when \(X = 0\), i.e. when \(3\cos \theta = 2 + \frac{u^2}{gr}\), etc. | M1 A1 A1 |
| (c) If \(u^2 \geq gr\), P leaves the surface as soon as it is projected | B2 |
(a) (i) $\frac{1}{2}mv^2 = \frac{1}{2}mu^2 + mgr(1 - \cos \theta)$ | M1 A1 A1 |
$v^2 = u^2 + 2gr(1 - \cos \theta)$ | |
(ii) $mg \cos \theta - X = \frac{mv^2}{r}$ | M1 A1 |
$X = mg \cos \theta - \frac{mu^2}{r} - 2mg(1 - \cos \theta)$ | M1 A1 |
$X = mg[3\cos \theta - 2 - \frac{u^2}{gr}]$ | A1 |
(b) Leaves sphere when $X = 0$, i.e. when $3\cos \theta = 2 + \frac{u^2}{gr}$, etc. | M1 A1 A1 |
(c) If $u^2 \geq gr$, P leaves the surface as soon as it is projected | B2 |
**Total: 11 marks**
---A particle $P$ is projected horizontally with speed $u$ ms$^{-1}$ from the highest point of a smooth sphere of radius $r$ m and centre $O$. It moves on the surface in a vertical plane, and at a particular instant the radius $OP$ makes an angle $\theta$ with the upward vertical, as shown. At this instant $P$ has speed $v$ ms$^{-1}$ and the magnitude of the reaction between $P$ and the sphere is $X$ N.
\includegraphics{figure_2}
\begin{enumerate}[label=(\alph*)]
\item Assuming that $u^2 < gr$, show that
\begin{enumerate}[label=(\roman*)]
\item $v^2 = u^2 + 2gr(1 - \cos \theta)$, [2 marks]
\item $X = mg\left(3\cos \theta - 2 - \frac{u^2}{gr}\right)$. [4 marks]
\end{enumerate}
\item Show that $P$ leaves the surface of the sphere when $\cos \theta = \frac{u^2 + 2gr}{3gr}$. [3 marks]
\item Discuss what happens if $u^2 \geq gr$. [2 marks]
\end{enumerate}
\hfill \mbox{\textit{Edexcel M3 Q5 [11]}}