| Exam Board | Edexcel |
|---|---|
| Module | M3 (Mechanics 3) |
| Year | 2010 |
| Session | June |
| Marks | 7 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Circular Motion 1 |
| Type | Conical pendulum – horizontal circle in free space (no surface) |
| Difficulty | Moderate -0.8 This is a standard conical pendulum problem requiring basic resolution of forces and application of circular motion formula (T cos θ = mg, T sin θ = mv²/r). The geometry is given (5-12-13 triangle), making it straightforward with no problem-solving insight needed—just routine application of two equations. |
| Spec | 6.05c Horizontal circles: conical pendulum, banked tracks |
| Answer | Marks | Guidance |
|---|---|---|
| Working | Marks | Notes |
| \(\cos\alpha = \dfrac{12}{13}\) | B1 | |
| \(R(\uparrow)\quad T\cos\alpha = mg\) | M1 | Resolving vertically |
| \(T \times \dfrac{12}{13} = mg\) | ||
| \(T = \dfrac{13}{12}mg\) | A1 | (3) oe |
| Answer | Marks | Guidance |
|---|---|---|
| Working | Marks | Notes |
| Equation of motion: \(T\sin\alpha = m\dfrac{v^2}{5l}\) | M1 A1 | |
| \(\dfrac{13mg}{12} \times \dfrac{5}{13} = m\dfrac{v^2}{5l}\) | M1 dep | |
| \(v^2 = \dfrac{25gl}{12}\) | ||
| \(v = \dfrac{5}{2}\sqrt{\dfrac{gl}{3}}\) | A1 | (4) Accept \(5\sqrt{\dfrac{gl}{12}}\) or \(\sqrt{\dfrac{25gl}{12}}\) or any equivalent |
# Question 1:
## Part (a)
| Working | Marks | Notes |
|---------|-------|-------|
| $\cos\alpha = \dfrac{12}{13}$ | B1 | |
| $R(\uparrow)\quad T\cos\alpha = mg$ | M1 | Resolving vertically |
| $T \times \dfrac{12}{13} = mg$ | | |
| $T = \dfrac{13}{12}mg$ | A1 | (3) oe |
## Part (b)
| Working | Marks | Notes |
|---------|-------|-------|
| Equation of motion: $T\sin\alpha = m\dfrac{v^2}{5l}$ | M1 A1 | |
| $\dfrac{13mg}{12} \times \dfrac{5}{13} = m\dfrac{v^2}{5l}$ | M1 dep | |
| $v^2 = \dfrac{25gl}{12}$ | | |
| $v = \dfrac{5}{2}\sqrt{\dfrac{gl}{3}}$ | A1 | (4) Accept $5\sqrt{\dfrac{gl}{12}}$ or $\sqrt{\dfrac{25gl}{12}}$ or any equivalent |
**Total: [7]**
---1.
\begin{figure}[h]
\begin{center}
\includegraphics[alt={},max width=\textwidth]{049ea68c-d15f-41f8-860e-0816d36a2748-02_458_516_281_712}
\captionsetup{labelformat=empty}
\caption{Figure 1}
\end{center}
\end{figure}
A garden game is played with a small ball $B$ of mass $m$ attached to one end of a light inextensible string of length 13l. The other end of the string is fixed to a point $A$ on a vertical pole as shown in Figure 1. The ball is hit and moves with constant speed in a horizontal circle of radius $5 l$ and centre $C$, where $C$ is vertically below $A$. Modelling the ball as a particle, find
\begin{enumerate}[label=(\alph*)]
\item the tension in the string,
\item the speed of the ball.
\end{enumerate}
\hfill \mbox{\textit{Edexcel M3 2010 Q1 [7]}}