| Exam Board | Edexcel |
|---|---|
| Module | M3 (Mechanics 3) |
| Marks | 13 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Circular Motion 1 |
| Type | Smooth ring on rotating string |
| Difficulty | Challenging +1.2 This is a standard M3 conical pendulum problem requiring resolution of forces and geometric constraints. Part (a) involves straightforward force balance on two particles with one at rest. Part (b) extends to both particles rotating, requiring simultaneous equations but following predictable M3 methodology. The algebra is moderate and the setup is a textbook scenario, making it above average difficulty but not requiring novel insight. |
| Spec | 3.03m Equilibrium: sum of resolved forces = 06.05c Horizontal circles: conical pendulum, banked tracks |
| Answer | Marks |
|---|---|
| (a) \(P: T = 0.06g\) | B1 M1 A1 A1 |
| \(Q: 0.06g \cos \theta = 0.04g\) | |
| \(\cos \theta = \frac{2}{3}\) | |
| Hence \(\frac{2}{3}(l-d) = d\) | M1 A1 |
| \(d = \frac{2}{5}l\) | |
| (b) \(P: T \sin \beta = 0.06g \sin \beta \omega^2\) | M1 A1 |
| \(T = 0.06\omega^2\) | |
| \(Q: T \sin \alpha = 0.04(l-e) \sin \alpha \omega^2\) | M1 A1 |
| \(T = 0.04(l-e)\omega^2\) | |
| Equate: \(3e = 2(l-e)\) | M1 A1 A1 |
| \(e = \frac{2}{5}l\), \(OQ = \frac{3}{5}l\) |
**(a)** $P: T = 0.06g$ | B1 M1 A1 A1 |
$Q: 0.06g \cos \theta = 0.04g$ | |
$\cos \theta = \frac{2}{3}$ | |
Hence $\frac{2}{3}(l-d) = d$ | M1 A1 |
$d = \frac{2}{5}l$ | |
**(b)** $P: T \sin \beta = 0.06g \sin \beta \omega^2$ | M1 A1 |
$T = 0.06\omega^2$ | |
$Q: T \sin \alpha = 0.04(l-e) \sin \alpha \omega^2$ | M1 A1 |
$T = 0.04(l-e)\omega^2$ | |
Equate: $3e = 2(l-e)$ | M1 A1 A1 |
$e = \frac{2}{5}l$, $OQ = \frac{3}{5}l$ | |
**Total: 13 marks**A light inelastic string of length $l$ m passes through a small smooth ring which is fixed at a point $O$ and is free to rotate about a vertical axis through $O$. Particles $P$ and $Q$, of masses 0.06 kg and 0.04 kg respectively, are attached to the ends of the string.
\begin{enumerate}[label=(\alph*)]
\item $Q$ describes a horizontal circle with centre $P$, while $P$ hangs at rest at a depth $d$ m below $O$. Show that $d = \frac{2l}{5}$. [6 marks]
\item $P$ and $Q$ now both move in horizontal circles with the same angular velocity $\omega$ rad s$^{-1}$ about a vertical axis through $O$. Show that $OQ = \frac{3l}{5}$ m. [7 marks]
\end{enumerate}
\includegraphics{figure_5}
\hfill \mbox{\textit{Edexcel M3 Q5 [13]}}