| Exam Board | Edexcel |
|---|---|
| Module | FM2 AS (Further Mechanics 2 AS) |
| Year | 2022 |
| Session | June |
| Marks | 11 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Circular Motion 1 |
| Type | Banked track – with friction (find maximum/minimum speed or friction coefficient) |
| Difficulty | Standard +0.8 This is a standard Further Mechanics banked track problem requiring resolution of forces in two directions, application of friction laws, and circular motion equations. Part (a) involves algebraic manipulation to reach a given result, part (b) is simpler (frictionless case), and part (c) requires inequality comparison. While multi-step and requiring careful force resolution, it follows a well-established template for FM2 circular motion questions without requiring novel insight. |
| Spec | 3.03e Resolve forces: two dimensions3.03r Friction: concept and vector form3.03t Coefficient of friction: F <= mu*R model6.05c Horizontal circles: conical pendulum, banked tracks |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| Resolving vertically: \(R\cos\alpha - F\sin\alpha = mg\) | M1 A1 | Correct no. of terms, dim correct, condone sin/cos confusion and sign errors; Correct equation |
| Equation of motion horizontally: \(R\sin\alpha + F\cos\alpha = \frac{mV^2}{r}\) | M1 A1 | Correct no. of terms, dim correct, condone sin/cos confusion and sign errors; Correct equation |
| Use of \(F = \mu R\) | M1 | Independent but must be used in an equation |
| Solve for \(V\) | M1 | Substitute for trig and solve for \(V\). Dependent on preceding M marks |
| \(V = \sqrt{\dfrac{(3+4\mu)rg}{4-3\mu}}\) | A1* | Correct given answer correctly obtained |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| Use of \(\mu = 0\) | M1 | If \(\mu \neq 0\), need to see first 6 marks from (a) without friction |
| \(U = \sqrt{\dfrac{3rg}{4}}\) | A1 | cao |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| Since \(3 + 4\mu > 3\) and \(4 - 3\mu < 4\) | M1 | Any convincing argument. SC: Allow M1A0 if they work in reverse to show if \(U < V\) then \(\mu > 0\) and make appropriate comment |
| \(\dfrac{3}{4} < \dfrac{3+4\mu}{4-3\mu}\) and hence \(U < V\) | A1* | Given answer correctly obtained |
## Question 3:
**Part 3(a):**
| Answer/Working | Mark | Guidance |
|---|---|---|
| Resolving vertically: $R\cos\alpha - F\sin\alpha = mg$ | M1 A1 | Correct no. of terms, dim correct, condone sin/cos confusion and sign errors; Correct equation |
| Equation of motion horizontally: $R\sin\alpha + F\cos\alpha = \frac{mV^2}{r}$ | M1 A1 | Correct no. of terms, dim correct, condone sin/cos confusion and sign errors; Correct equation |
| Use of $F = \mu R$ | M1 | Independent but must be used in an equation |
| Solve for $V$ | M1 | Substitute for trig and solve for $V$. Dependent on preceding M marks |
| $V = \sqrt{\dfrac{(3+4\mu)rg}{4-3\mu}}$ | A1* | Correct given answer correctly obtained |
**Part 3(b):**
| Answer/Working | Mark | Guidance |
|---|---|---|
| Use of $\mu = 0$ | M1 | If $\mu \neq 0$, need to see first 6 marks from (a) without friction |
| $U = \sqrt{\dfrac{3rg}{4}}$ | A1 | cao |
**Part 3(c):**
| Answer/Working | Mark | Guidance |
|---|---|---|
| Since $3 + 4\mu > 3$ and $4 - 3\mu < 4$ | M1 | Any convincing argument. SC: Allow M1A0 if they work in reverse to show if $U < V$ then $\mu > 0$ and make appropriate comment |
| $\dfrac{3}{4} < \dfrac{3+4\mu}{4-3\mu}$ and hence $U < V$ | A1* | Given answer correctly obtained |\begin{enumerate}
\item A cyclist is travelling around a circular track which is banked at an angle $\alpha$ to the horizontal, where $\tan \alpha = \frac { 3 } { 4 }$
\end{enumerate}
The cyclist moves with constant speed in a horizontal circle of radius $r$.\\
In an initial model,
\begin{itemize}
\item the cyclist and her cycle are modelled as a particle
\item the track is modelled as being rough so that there is sideways friction between the tyres of the cycle and the track, with coefficient of friction $\mu$, where $\mu < \frac { 4 } { 3 }$\\
Using this model, the maximum speed that the cyclist can travel around the track in a horizontal circle of radius $r$, without slipping sideways, is $V$.\\
(a) Show that $V = \sqrt { \frac { ( 3 + 4 \mu ) r g } { 4 - 3 \mu } }$
\end{itemize}
In a new simplified model,
\begin{itemize}
\item the cyclist and her cycle are modelled as a particle
\item the motion is now modelled so that there is no sideways friction between the tyres of the cycle and the track
\end{itemize}
Using this new model, the speed that the cyclist can travel around the track in a horizontal circle of radius $r$, without slipping sideways, is $U$.\\
(b) Find $U$ in terms of $r$ and $g$.\\
(c) Show that $U < V$.
\hfill \mbox{\textit{Edexcel FM2 AS 2022 Q3 [11]}}