| Exam Board | OCR |
|---|---|
| Module | Further Mechanics AS (Further Mechanics AS) |
| Year | 2020 |
| Session | November |
| Marks | 9 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Circular Motion 1 |
| Type | Banked track – no friction (find speed or radius) |
| Difficulty | Standard +0.3 This is a standard banked track problem requiring resolution of forces (weight and normal reaction) and application of F=ma for circular motion. Part (a) is a routine 'show that' derivation, part (b) is direct substitution, and part (c) tests understanding of modeling assumptions (friction). Slightly above average due to the 3D force resolution and modeling discussion, but follows a well-established template for Further Mechanics. |
| Spec | 3.03r Friction: concept and vector form6.05c Horizontal circles: conical pendulum, banked tracks |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| \(\uparrow C\cos 30° = mg\) | M1* | Where \(C\) is the (normal) contact force between car and track - Allow sin/cos confusion, allow \(\theta\) instead of \(30°\) |
| \(-C\sin 30° = \dfrac{mv^2}{r}\) | M1* | NII with centripetal acceleration - Allow sin/cos confusion |
| \(\Rightarrow \dfrac{C\sin 30°}{C\cos 30°} = \dfrac{\frac{mv^2}{r}}{mg}\) | M1ft | Dividing so that \(C\) and \(m\) cancel. May see \(\tan\theta\) or \(\tan 30\) instead of sin/cos - Or rearrange one equation for \(C\) and substitute into the other |
| \(\Rightarrow \tan 30° = \dfrac{1}{\sqrt{3}} = \dfrac{v^2}{rg} \Rightarrow 3v^2 = rg\) | A1 | AG - \(\theta\) must be clearly stated and correctly used to gain this mark |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| \(\sqrt{3}v^2 = 24 \times 9.8\) | M1 | Using the formula from (a) |
| \(v = \text{awrt } 11.7\) | A1 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| The model implies that only a single value for the speed is possible for a given radius, so any change in speed should cause the car to move in a different circle | B1 | Or equivalent - Do not allow discussion of assumptions here. Or any equivalent comment about a possible consequence according to the model of a change in speed or the radius |
| The track should be modelled as resisting sideways motion | B1 | Accept 'model track as rough' or 'include friction' etc without explicit reference to 'sideways' - Must be relevant to the question. Do not accept references that ignore friction |
## Question 7:
### Part (a):
| Answer | Marks | Guidance |
|--------|-------|----------|
| $\uparrow C\cos 30° = mg$ | M1* | Where $C$ is the (normal) contact force between car and track - Allow sin/cos confusion, allow $\theta$ instead of $30°$ |
| $-C\sin 30° = \dfrac{mv^2}{r}$ | M1* | NII with centripetal acceleration - Allow sin/cos confusion |
| $\Rightarrow \dfrac{C\sin 30°}{C\cos 30°} = \dfrac{\frac{mv^2}{r}}{mg}$ | M1ft | Dividing so that $C$ and $m$ cancel. May see $\tan\theta$ or $\tan 30$ instead of sin/cos - Or rearrange one equation for $C$ and substitute into the other |
| $\Rightarrow \tan 30° = \dfrac{1}{\sqrt{3}} = \dfrac{v^2}{rg} \Rightarrow 3v^2 = rg$ | A1 | AG - $\theta$ must be clearly stated and correctly used to gain this mark |
### Part (b):
| Answer | Marks | Guidance |
|--------|-------|----------|
| $\sqrt{3}v^2 = 24 \times 9.8$ | M1 | Using the formula from (a) |
| $v = \text{awrt } 11.7$ | A1 | |
### Part (c):
| Answer | Marks | Guidance |
|--------|-------|----------|
| The model implies that only a single value for the speed is possible for a given radius, so any change in speed should cause the car to move in a different circle | B1 | Or equivalent - Do not allow discussion of assumptions here. Or any equivalent comment about a possible consequence according to the model of a change in speed or the radius |
| The track should be modelled as resisting sideways motion | B1 | Accept 'model track as rough' or 'include friction' etc without explicit reference to 'sideways' - Must be relevant to the question. Do not accept references that ignore friction |7 It is required to model the motion of a car of mass $m \mathrm {~kg}$ travelling at a constant speed $v \mathrm {~ms} ^ { - 1 }$ around a circular portion of banked track. The track is banked at $30 ^ { \circ }$ (see diagram).\\
\includegraphics[max width=\textwidth, alt={}, center]{0501e5a4-2137-4e7d-98ff-2ee81941cbf3-5_414_624_356_242}
In a model, the following modelling assumptions are made.
\begin{itemize}
\item The track is smooth.
\item The car is a particle.
\item The car follows a horizontal circular path with radius $r \mathrm {~m}$.
\begin{enumerate}[label=(\alph*)]
\item Show that, according to the model, $\sqrt { 3 } \mathrm { v } ^ { 2 } = \mathrm { gr }$.
\end{itemize}
For a particular portion of banked track, $r = 24$.
\item Find the value of $v$ as predicted by the model.
A car is being driven on this portion of the track at the constant speed calculated in part (b). The driver finds that in fact he can drive a little slower or a little faster than this while still moving in the same horizontal circle.
\item Explain
\begin{itemize}
\item how this contrasts with what the model predicts,
\item how to improve the model to account for this.
\end{itemize}
\end{enumerate}
\hfill \mbox{\textit{OCR Further Mechanics AS 2020 Q7 [9]}}