| Exam Board | OCR MEI |
|---|---|
| Module | Further Mechanics Major (Further Mechanics Major) |
| Session | Specimen |
| Marks | 6 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Circular Motion 1 |
| Type | Vertical circle – surface contact (sphere/track, leaving surface) |
| Difficulty | Standard +0.3 This is a standard particle-on-sphere problem requiring energy conservation and circular motion dynamics—both routine Further Maths techniques. The question guides students through each step explicitly, and the final part involves straightforward substitution. While it requires multiple concepts, the approach is well-established and no novel insight is needed. |
| Spec | 3.03i Normal reaction force6.02i Conservation of energy: mechanical energy principle6.05d Variable speed circles: energy methods |
| Answer | Marks | Guidance |
|---|---|---|
| 4 | (i) | Conservation of energy: |
| Answer | Marks |
|---|---|
| v2 (cid:32)u2 (cid:14)2ga(1(cid:16)cos(cid:84)) AG | M1 |
| Answer | Marks |
|---|---|
| [2] | 3.4 |
| 1.1 | Using conservation of energy or |
| Answer | Marks | Guidance |
|---|---|---|
| 4 | (ii) | mv2 |
| Answer | Marks |
|---|---|
| a | B1 |
| Answer | Marks |
|---|---|
| [2] | 3.4 |
| 1.1 | N |
| Answer | Marks | Guidance |
|---|---|---|
| M | N2L is Newton's second law | |
| 4 | (iii) | 3 |
| Answer | Marks |
|---|---|
| 2 | M1 |
| Answer | Marks |
|---|---|
| [2] | I |
| Answer | Marks |
|---|---|
| 1.1 | Both used |
Question 4:
4 | (i) | Conservation of energy:
1 1
mv2 (cid:32) mu2 (cid:14)mga(cid:11)1(cid:16)cos(cid:84)(cid:12)
2 2
v2 (cid:32)u2 (cid:14)2ga(1(cid:16)cos(cid:84)) AG | M1
A1
[2] | 3.4
1.1 | Using conservation of energy or
work-energy equation
Correct use of c.o.e. leading to
given answer
4 | (ii) | mv2
N2L: R(cid:32)mgcos(cid:84)(cid:16)
a
m
(cid:32)mgcos(cid:84)(cid:16) (cid:11) u2 (cid:14)2ga(cid:11)1(cid:16)cos(cid:84)(cid:12)(cid:12)
a
mu2
R(cid:32)mg(cid:11)3cos(cid:84)(cid:16)2(cid:12)(cid:16) . AG
a | B1
A1
[2] | 3.4
1.1 | N
CElearly shown
M | N2L is Newton's second law
4 | (iii) | 3
Use R = 0, cos(cid:84)(cid:32)
4
1
u(cid:32) ga
2 | M1
A1
[2] | I
C
3.1b
1.1 | Both usedA fixed smooth sphere has centre O and radius $a$. A particle P of mass $m$ is placed at the highest point of the sphere and given an initial horizontal speed $u$.
For the first part of its motion, P remains in contact with the sphere and has speed $v$ when OP makes an angle $\theta$ with the upward vertical. This is shown in Fig. 4.
\includegraphics{figure_4}
\begin{enumerate}[label=(\roman*)]
\item By considering the energy of P, show that $v^2 = u^2 + 2ga(1 - \cos\theta)$. [2]
\item Show that the magnitude of the normal contact force between the sphere and particle P is
$$mg(3\cos\theta - 2) - \frac{mv^2}{a}.$$ [2]
\end{enumerate}
The particle loses contact with the sphere when $\cos\theta = \frac{3}{4}$.
\begin{enumerate}[label=(\roman*)]
\setcounter{enumi}{2}
\item Find an expression for $u$ in terms of $a$ and $g$. [2]
\end{enumerate}
\hfill \mbox{\textit{OCR MEI Further Mechanics Major Q4 [6]}}