| Exam Board | OCR MEI |
|---|---|
| Module | Further Mechanics Major (Further Mechanics Major) |
| Year | 2022 |
| Session | June |
| Marks | 10 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Circular Motion 2 |
| Type | Particle on inner surface of sphere/bowl |
| Difficulty | Standard +0.8 This is a classic circular motion energy problem requiring multiple steps: applying conservation of energy from A to general point P, then using circular motion dynamics (centripetal force equation), and finally finding the critical condition for maintaining contact. While the techniques are standard for Further Mechanics (energy conservation + F=mv²/r), it requires careful coordinate setup, algebraic manipulation to reach the given expression, and understanding that minimum speed occurs at the top. The 7-mark part (a) indicates substantial working, but this is a well-practiced problem type in FM syllabi, making it moderately challenging but not exceptional. |
| Spec | 6.02i Conservation of energy: mechanical energy principle6.05d Variable speed circles: energy methods |
| Answer | Marks | Guidance |
|---|---|---|
| 10 | (a) | 1 |
| Answer | Marks |
|---|---|
| r | M1 |
| Answer | Marks |
|---|---|
| [7] | 3.3 |
| Answer | Marks |
|---|---|
| 2.2a | Use of conservation of energy between A and C (or |
| Answer | Marks |
|---|---|
| (b) | 2h |
| Answer | Marks |
|---|---|
| 2 2 | M1 |
| Answer | Marks |
|---|---|
| [2] | 3.1b |
| 3.2a | Set θ=πand R = 0, > 0 or …0 |
| Answer | Marks | Guidance |
|---|---|---|
| (c) | Include air resistance in the model | B1 |
| [1] | 3.5c | Any correct refinement e.g. model the track as rough |
Question 10:
10 | (a) | 1
mgh= mu 2
C
2
1
At angle θ,PE =mgr(1−cosθ)and KE = mv2
2
1 1
mu 2 =mgr(1−cosθ)+ mv2
C
2 2
v2 =2gh−2gr(1−cosθ)
mv2
R−mgcosθ=
r
m
R−mgcosθ= ( 2gh−2gr(1−cosθ))
r
2mgh
R=mgcosθ+ −2mg+2mgcosθ
r
2h
⇒R=mg3cosθ−2+
r | M1
B1
M1
A1
M1*
M1dep*
A1
[7] | 3.3
1.1
3.3
1.1
3.3
3.4
2.2a | Use of conservation of energy between A and C (or
B) loss in PE = gain in KE (where u is the speed at
C
C) – this mark may be applied by later working
Correct PE and KE expressions for P at angle θ
Use of conservation of energy between C (or A) and
when P is at angle θ
Correct expression for the speed or speed-squared
when P is at an angle θ
N2L radially with correct number of terms and weight
resolved
Substitute expression for v2(with correct number of
terms)
AG – sufficient working must be shown
(b) | 2h
mg3cosπ−2+ …0
r
5 5
h… r∴least value of h is r
2 2 | M1
A1
[2] | 3.1b
3.2a | Set θ=πand R = 0, > 0 or …0
5
Allow h> 5r or h… r
2 2
(c) | Include air resistance in the model | B1
[1] | 3.5c | Any correct refinement e.g. model the track as rough\includegraphics{figure_10}
A small toy car runs along a track in a vertical plane.
The track consists of three sections: a curved section AB, a horizontal section BC which rests on the floor, and a circular section that starts at C with centre O and radius $r$ m.
The section BC is tangential to the curved section at B and tangential to the circular section at C, as shown in the diagram.
The car, of mass $m$ kg, is placed on the track at A, at a height $h$ m above the floor, and released from rest. The car runs along the track from A to C and enters the circular section at C. It can be assumed that the track does not obstruct the car moving on to the circular section at C.
The track is modelled as being smooth, and the car is modelled as a particle P.
\begin{enumerate}[label=(\alph*)]
\item Show that, while P remains in contact with the circular section of the track, the magnitude of the normal contact force between P and the circular section is
$$mg\left(3\cos\theta - 2 + \frac{2h}{r}\right)\text{N},$$
where $\theta$ is the angle between OC and OP. [7]
\item Hence determine, in terms of $r$, the least possible value of $h$ so that P can complete a vertical circle. [2]
\item Apart from not modelling the car as a particle, state one refinement that would make the model more realistic. [1]
\end{enumerate}
\hfill \mbox{\textit{OCR MEI Further Mechanics Major 2022 Q10 [10]}}