| Exam Board | Edexcel |
|---|---|
| Module | FM2 (Further Mechanics 2) |
| Year | 2022 |
| Session | June |
| Marks | 8 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Circular Motion 1 |
| Type | Smooth ring on rotating string |
| Difficulty | Standard +0.8 This is a Further Mechanics 2 conical pendulum problem requiring resolution of forces in two directions, use of circular motion equations, and geometric reasoning with the given tan θ. While it involves multiple steps and careful force analysis, it follows a standard FM2 approach without requiring novel insight. The given tan θ simplifies the trigonometry significantly. Slightly above average difficulty due to the two-string geometry and FM2 content. |
| Spec | 6.05b Circular motion: v=r*omega and a=v^2/r6.05c Horizontal circles: conical pendulum, banked tracks |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Marks | Guidance |
| Resolve vertically | M1 | Need all terms. Condone \(\sin/\cos\) confusion |
| \(T\cos\theta = mg\) | A1 | Correct unsimplified equation |
| \(T = \frac{mg}{\cos\theta} = \frac{6.8mg}{6} = \frac{17mg}{15}\) | A1 | Correct answer only. \(1.1mg\) or better \((1.13\ldots mg)\). Do not ignore subsequent working if they try to combine with tension in \(AR\) |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Marks | Guidance |
| Equation of motion | M1 | Equation for circular motion. Need all terms and dimensionally correct. Condone \(\sin/\cos\) confusion and sign errors. Any correct form for acceleration |
| \(mr\omega^2 = T + T\sin\theta \quad \left(m\times 3.2a\omega^2 = \text{their } T\!\left(1 + \frac{8}{17}\right)\right)\) | A1 | Unsimplified equation with at most one error |
| A1 | Correct unsimplified equation | |
| Solves for \(\omega\) or \(\omega^2\) | M1 | |
| \(\frac{r\omega^2}{g} = \frac{1+\sin\theta}{\cos\theta} = \frac{6.8+3.2}{6},\quad \omega^2 = \frac{10g}{6\times 3.2a} \quad \omega = \sqrt{\frac{25g}{48a}} = \frac{5}{4}\sqrt{\frac{g}{3a}}\) | A1 |
## Question 4:
### Part (a):
| Answer/Working | Marks | Guidance |
|---|---|---|
| Resolve vertically | M1 | Need all terms. Condone $\sin/\cos$ confusion |
| $T\cos\theta = mg$ | A1 | Correct unsimplified equation |
| $T = \frac{mg}{\cos\theta} = \frac{6.8mg}{6} = \frac{17mg}{15}$ | A1 | Correct answer only. $1.1mg$ or better $(1.13\ldots mg)$. Do not ignore subsequent working if they try to combine with tension in $AR$ |
### Part (b):
| Answer/Working | Marks | Guidance |
|---|---|---|
| Equation of motion | M1 | Equation for circular motion. Need all terms and dimensionally correct. Condone $\sin/\cos$ confusion and sign errors. Any correct form for acceleration |
| $mr\omega^2 = T + T\sin\theta \quad \left(m\times 3.2a\omega^2 = \text{their } T\!\left(1 + \frac{8}{17}\right)\right)$ | A1 | Unsimplified equation with at most one error |
| | A1 | Correct unsimplified equation |
| Solves for $\omega$ or $\omega^2$ | M1 | |
| $\frac{r\omega^2}{g} = \frac{1+\sin\theta}{\cos\theta} = \frac{6.8+3.2}{6},\quad \omega^2 = \frac{10g}{6\times 3.2a} \quad \omega = \sqrt{\frac{25g}{48a}} = \frac{5}{4}\sqrt{\frac{g}{3a}}$ | A1 | |4.
\begin{figure}[h]
\begin{center}
\includegraphics[alt={},max width=\textwidth]{1f39620e-c10f-4344-89f1-626fff36d187-12_640_645_258_699}
\captionsetup{labelformat=empty}
\caption{Figure 2}
\end{center}
\end{figure}
A small smooth ring $R$ of mass $m$ is threaded onto a light inextensible string. One end of the string is attached to a fixed point $A$ and the other end of the string is attached to the fixed point $B$ such that $B$ is vertically above $A$ and $A B = 6 a$
The ring moves with constant angular speed $\omega$ in a horizontal circle with centre $A$. The string is taut and $B R$ makes a constant angle $\theta$ with the downward vertical, as shown in Figure 2.
The ring is modelled as a particle.\\
Given that $\tan \theta = \frac { 8 } { 15 }$
\begin{enumerate}[label=(\alph*)]
\item find, in terms of $m$ and $g$, the magnitude of the tension in the string,
\item find $\omega$ in terms of $a$ and $g$
\end{enumerate}
\hfill \mbox{\textit{Edexcel FM2 2022 Q4 [8]}}