| Exam Board | Edexcel |
|---|---|
| Module | M3 (Mechanics 3) |
| Marks | 12 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Circular Motion 1 |
| Type | String through hole – lower particle also moves in horizontal circle (conical pendulum below) |
| Difficulty | Standard +0.8 This M3 circular motion problem requires understanding of conical pendulum motion, limiting friction, and two distinct scenarios with different constraint conditions. Part (a) involves resolving forces in 3D with friction at limiting equilibrium (non-trivial setup), part (b) applies circular motion principles, and part (c) reverses the configuration requiring careful force analysis with friction now opposing outward motion. The multi-stage reasoning and need to correctly identify force directions in different scenarios elevates this above standard circular motion exercises. |
| Spec | 6.05b Circular motion: v=r*omega and a=v^2/r6.05c Horizontal circles: conical pendulum, banked tracks |
| Answer | Marks | Guidance |
|---|---|---|
| (a) \(T = F = \mu R\), so \(T = \frac{1}{4}g\) | B1 M1 A1 | |
| \(T \cos \theta = 0.2g\) | A1 | |
| \(\cos \theta = 0.8\), \(\theta = 36.9°\) | ||
| (b) \(T \sin \theta = 0.2v^2/(0.4 \sin \theta)\) | M1 A1 | |
| \(v^2 = 0.5g \sin^2 \theta = 1.764\) | ||
| \(v = \sqrt{1.764} = 1.33 \text{ ms}^{-1}\) | A1 | |
| (c) Now \(T = 0.2g\) | B1 M1 A1 | |
| \(0.2g + 0.2g = 0.5\frac{v^2}{r}\) | ||
| \(0.4g = 0.5(0.84)^2 v r\) | M1 A1 | |
| \(r = 0.08\) | Total: 12 marks |
**(a)** $T = F = \mu R$, so $T = \frac{1}{4}g$ | B1 M1 A1 |
$T \cos \theta = 0.2g$ | A1 |
$\cos \theta = 0.8$, $\theta = 36.9°$ | |
**(b)** $T \sin \theta = 0.2v^2/(0.4 \sin \theta)$ | M1 A1 |
$v^2 = 0.5g \sin^2 \theta = 1.764$ | |
$v = \sqrt{1.764} = 1.33 \text{ ms}^{-1}$ | A1 |
**(c)** Now $T = 0.2g$ | B1 M1 A1 |
$0.2g + 0.2g = 0.5\frac{v^2}{r}$ | |
$0.4g = 0.5(0.84)^2 v r$ | M1 A1 |
$r = 0.08$ | | **Total: 12 marks**A particle $P$, of mass 0·5 kg, rests on the surface of a rough horizontal table. The coefficient of friction between $P$ and the table is 0·5. $P$ is connected to a particle $Q$, of mass 0·2 kg, by a light inextensible string passing through a small smooth hole at a point $O$ on the table, such that the distance $OQ$ is 0·4 m. $Q$ moves in a horizontal circle while $P$ remains in limiting equilibrium.
\includegraphics{figure_5}
\begin{enumerate}[label=(\alph*)]
\item Calculate the angle $\theta$ which $OQ$ makes with the vertical. [4 marks]
\item Show that the speed of $Q$ is 1·33 ms$^{-1}$. [3 marks]
\end{enumerate}
The motion is altered so that $Q$ hangs at rest below $O$ and $P$ moves in a horizontal circle on the table with speed 0·84 ms$^{-1}$, at a constant distance $r$ m from $O$ but tending to slip away from $O$.
\begin{enumerate}[label=(\alph*)]
\setcounter{enumi}{2}
\item Find the value of $r$. [5 marks]
\end{enumerate}
\hfill \mbox{\textit{Edexcel M3 Q5 [12]}}