| Exam Board | Edexcel |
|---|---|
| Module | M3 (Mechanics 3) |
| Year | 2014 |
| Session | June |
| Marks | 13 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Circular Motion 1 |
| Type | Particle on cone surface – with string attached to vertex or fixed point |
| Difficulty | Standard +0.8 This is a challenging M3 circular motion problem requiring careful 3D geometry with the cone constraint, resolution of forces in multiple directions, and an inequality proof involving the normal reaction condition. The 7+6 mark allocation and need to handle the cone contact condition (R≥0) to derive a time inequality elevates this above standard circular motion exercises. |
| Spec | 6.05c Horizontal circles: conical pendulum, banked tracks |
| Answer | Marks | Guidance |
|---|---|---|
| \(T \sin 60° + R \sin 60° = mg\) | M1 A1 | |
| \(T \cos 60° - R \cos 60° = ml \cos 60°\omega^2\) | M1 A1 A1 | |
| \(T = \frac{1}{3}m(l\omega^2 + \frac{2}{\sqrt{3}}g)\) | DM1 A1 | Solve for \(T\) |
| (7) |
| Answer | Marks | Guidance |
|---|---|---|
| \(R = \frac{1}{3}m(\frac{2}{\sqrt{3}}g - l\omega^2)\) | M1 A1 | |
| \(\frac{1}{3}m(\frac{2}{\sqrt{3}}g - l\omega^2) > 0\) | DM1 | |
| \(\omega < \sqrt{\frac{2g}{l\sqrt{3}}}\) | A1 | |
| \(t > 2\pi\sqrt{\frac{l\sqrt{3}}{2g}}\) ** | DM1 A1 | Obtaining an inequality for \(t\) |
| (6) | ||
| 13 |
## 2(a)
$T \sin 60° + R \sin 60° = mg$ | M1 A1 |
$T \cos 60° - R \cos 60° = ml \cos 60°\omega^2$ | M1 A1 A1 |
$T = \frac{1}{3}m(l\omega^2 + \frac{2}{\sqrt{3}}g)$ | DM1 A1 | Solve for $T$
| (7) |
## 2(b)
$R = \frac{1}{3}m(\frac{2}{\sqrt{3}}g - l\omega^2)$ | M1 A1 |
$\frac{1}{3}m(\frac{2}{\sqrt{3}}g - l\omega^2) > 0$ | DM1 |
$\omega < \sqrt{\frac{2g}{l\sqrt{3}}}$ | A1 |
$t > 2\pi\sqrt{\frac{l\sqrt{3}}{2g}}$ ** | DM1 A1 | Obtaining an inequality for $t$
| (6) |
| 13 |
---\includegraphics{figure_1}
A cone of semi-vertical angle $60°$ is fixed with its axis vertical and vertex upwards. A particle of mass $m$ is attached to one end of a light inextensible string of length $l$. The other end of the string is attached to a fixed point vertically above the vertex of the cone. The particle moves in a horizontal circle on the smooth outer surface of the cone with constant angular speed $\omega$, with the string making a constant angle $60°$ with the horizontal, as shown in Figure 1.
\begin{enumerate}[label=(\alph*)]
\item Find the tension in the string, in terms of $m$, $l$, $\omega$ and $g$. [7]
\end{enumerate}
The particle remains on the surface of the cone.
\begin{enumerate}[label=(\alph*)]
\setcounter{enumi}{1}
\item Show that the time for the particle to make one complete revolution is greater than
$$2\pi\sqrt{\frac{l\sqrt{3}}{2g}}$$ [6]
\end{enumerate}
\hfill \mbox{\textit{Edexcel M3 2014 Q2 [13]}}