| Exam Board | Edexcel |
|---|---|
| Module | FM2 AS (Further Mechanics 2 AS) |
| Year | 2023 |
| Session | June |
| Marks | 9 |
| 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 Further Maths banked track problem requiring resolution of forces in two directions, application of friction laws, and circular motion equations. While the setup is standard for FM2, students must correctly handle the minimum speed condition (friction acts up the slope) and manipulate simultaneous equations involving trigonometric terms, which is more demanding than typical AS-level mechanics. |
| 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 |
| Equation of motion horizontally | M1 | First equation: Dimensionally correct. Condone sign errors and sine/cosine confusion |
| \(\frac{55\times4.5^{2}}{50}=R\sin15°-F\cos15°\) | A1, A1 | Unsimplified equation with at most one error; Correct unsimplified equation |
| Resolve vertically | M1 | Second equation: Dimensionally correct. Condone sign errors and sine/cosine confusion |
| \(55g = R\cos15°+F\sin15°\) | A1ft, A1ft | Unsimplified with at most one error; Correct unsimplified. Follow their direction for \(F\) parallel to slope |
| Uses \(F = \mu R\) | B1 | Seen anywhere in the solution |
| Solve for \(\mu\) | dM1 | Complete method to obtain \(\mu\). Dependent on both previous M marks |
| \(\mu = 0.22\) or \(\mu = 0.224\) | A1 | Value correct 2 or 3 sf (follows 9.8) |
## Question 3:
| Answer/Working | Mark | Guidance |
|---|---|---|
| Equation of motion horizontally | M1 | First equation: Dimensionally correct. Condone sign errors and sine/cosine confusion |
| $\frac{55\times4.5^{2}}{50}=R\sin15°-F\cos15°$ | A1, A1 | Unsimplified equation with at most one error; Correct unsimplified equation |
| Resolve vertically | M1 | Second equation: Dimensionally correct. Condone sign errors and sine/cosine confusion |
| $55g = R\cos15°+F\sin15°$ | A1ft, A1ft | Unsimplified with at most one error; Correct unsimplified. Follow their direction for $F$ parallel to slope |
| Uses $F = \mu R$ | B1 | Seen anywhere in the solution |
| Solve for $\mu$ | dM1 | Complete method to obtain $\mu$. Dependent on both previous M marks |
| $\mu = 0.22$ or $\mu = 0.224$ | A1 | Value correct 2 or 3 sf (follows 9.8) |
---\begin{enumerate}
\item A girl is cycling round a circular track.
\end{enumerate}
The girl and her bicycle have a combined mass of 55 kg .\\
The coefficient of friction between the track surface and the tyres of the bicycle is $\mu$.\\
The track is banked at an angle of $15 ^ { \circ }$ to the horizontal.\\
The girl and her bicycle are modelled as a particle moving in a horizontal circle of radius 50 m\\
The minimum speed at which the girl can cycle round this circle without slipping is $4.5 \mathrm {~m} \mathrm {~s} ^ { - 1 }$
Using the model, find the value of $\mu$.
\hfill \mbox{\textit{Edexcel FM2 AS 2023 Q3 [9]}}