| Exam Board | Edexcel |
|---|---|
| Module | FM2 AS (Further Mechanics 2 AS) |
| Year | 2020 |
| Session | June |
| Marks | 13 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Circular Motion 1 |
| Type | Smooth ring on rotating string |
| Difficulty | Standard +0.3 This is a standard Further Mechanics circular motion problem with a smooth bead on a string. Part (a) is straightforward geometry using Pythagoras, part (b) requires resolving forces and applying F=mrω², part (c) involves eliminating tension to find a constraint, and parts (d)-(e) are qualitative. The setup is typical for FM2 with clear geometric configuration and standard techniques, making it slightly easier than average for Further Maths. |
| Spec | 6.05b Circular motion: v=r*omega and a=v^2/r6.05c Horizontal circles: conical pendulum, banked tracks |
| Answer | Marks | Guidance |
|---|---|---|
| Working | Mark | Notes |
| \((a\sqrt{3})^2 + (3a - AP)^2 = AP^2\) | M1 | Use of Pythagoras: \(3a^2 + 9a^2 - 6a \times AP + AP^2 = AP^2 \Rightarrow 6a \times AP = 12a^2\) |
| \(AP = 2a\) * | A1* | Given answer |
| Answer | Marks | Guidance |
|---|---|---|
| Working | Mark | Notes |
| Equation of motion horizontally | M1 | Use of horizontal equation with correct no. of terms |
| \(T + T \times \frac{1}{2} = \frac{mU^2}{a}\) | A1 | Equation with at most one error |
| A1 | Correct equation | |
| \(T = \frac{2mU^2}{3a}\) | A1 | Correct answer |
| Answer | Marks | Guidance |
|---|---|---|
| Working | Mark | Notes |
| Resolving vertically | M1 | Use of vertical resolution with correct no. of terms |
| \(R + T \times \frac{\sqrt{3}}{2} = mg\) | A1 | Correct equation |
| On the table \(\Rightarrow R > 0\) | M1 | Use of \(R > 0\) |
| \(mg - \frac{2mU^2\sqrt{3}}{3a \times 2} > 0\) | A1 | Correct inequality |
| \(U^2 < ag\sqrt{3}\) * | A1* | Correctly obtained given answer |
| Answer | Marks | Guidance |
|---|---|---|
| Working | Mark | Notes |
| Bead would lift off the table | B1 | Clear comment |
| Answer | Marks | Guidance |
|---|---|---|
| Working | Mark | Notes |
| Tension would vary along the string | B1 | Clear explanation |
## Question 2:
### Part (a)
| Working | Mark | Notes |
|---------|------|-------|
| $(a\sqrt{3})^2 + (3a - AP)^2 = AP^2$ | M1 | Use of Pythagoras: $3a^2 + 9a^2 - 6a \times AP + AP^2 = AP^2 \Rightarrow 6a \times AP = 12a^2$ |
| $AP = 2a$ * | A1* | Given answer |
### Part (b)
| Working | Mark | Notes |
|---------|------|-------|
| Equation of motion horizontally | M1 | Use of horizontal equation with correct no. of terms |
| $T + T \times \frac{1}{2} = \frac{mU^2}{a}$ | A1 | Equation with at most one error |
| | A1 | Correct equation |
| $T = \frac{2mU^2}{3a}$ | A1 | Correct answer |
### Part (c)
| Working | Mark | Notes |
|---------|------|-------|
| Resolving vertically | M1 | Use of vertical resolution with correct no. of terms |
| $R + T \times \frac{\sqrt{3}}{2} = mg$ | A1 | Correct equation |
| On the table $\Rightarrow R > 0$ | M1 | Use of $R > 0$ |
| $mg - \frac{2mU^2\sqrt{3}}{3a \times 2} > 0$ | A1 | Correct inequality |
| $U^2 < ag\sqrt{3}$ * | A1* | Correctly obtained given answer |
### Part (d)
| Working | Mark | Notes |
|---------|------|-------|
| Bead would lift off the table | B1 | Clear comment |
### Part (e)
| Working | Mark | Notes |
|---------|------|-------|
| Tension would vary along the string | B1 | Clear explanation |
---2.
\begin{figure}[h]
\begin{center}
\includegraphics[alt={},max width=\textwidth]{0567d068-e23c-446e-9e11-f0c292972093-06_531_837_258_632}
\captionsetup{labelformat=empty}
\caption{Figure 2}
\end{center}
\end{figure}
One end of a string of length $3 a$ is attached to a point $A$ and the other end is attached to a point $B$ on a smooth horizontal table. The point $B$ is vertically below $A$ with $A B = a \sqrt { 3 }$ A small smooth bead, $P$, of mass $m$ is threaded on to the string. The bead $P$ moves on the table in a horizontal circle, with centre $B$, with constant speed $U$. Both portions, $A P$ and $B P$, of the string are taut, as shown in Figure 2.
The string is modelled as being light and inextensible and the bead is modelled as a particle.
\begin{enumerate}[label=(\alph*)]
\item Show that $A P = 2 a$
\item Find, in terms of $m , U$ and $a$, the tension in the string.
\item Show that $U ^ { 2 } < a g \sqrt { 3 }$
\item Describe what would happen if $U ^ { 2 } > a g \sqrt { 3 }$
\item State briefly how the tension in the string would be affected if the string were not modelled as being light.
\end{enumerate}
\hfill \mbox{\textit{Edexcel FM2 AS 2020 Q2 [13]}}