| Exam Board | OCR MEI |
|---|---|
| Module | Further Mechanics Major (Further Mechanics Major) |
| Year | 2023 |
| Session | June |
| Marks | 9 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Work done and energy |
| Type | Connected particles with pulley |
| Difficulty | Standard +0.3 This is a standard Further Maths mechanics problem involving connected particles, friction, and the work-energy principle. Part (a) requires straightforward calculation of friction work using W = μRd. Part (b) applies work-energy principle with given final velocity—methodical but routine for FM students. The coefficient of friction simplifies nicely (√3/6), and the multi-step nature is typical for this level. Slightly above average difficulty due to the FM context and work-energy application, but follows standard problem-solving patterns without requiring novel insight. |
| Spec | 3.03r Friction: concept and vector form3.03t Coefficient of friction: F <= mu*R model3.03u Static equilibrium: on rough surfaces6.01a Dimensions: M, L, T notation6.02a Work done: concept and definition6.02b Calculate work: constant force, resolved component6.02i Conservation of energy: mechanical energy principle |
| Answer | Marks | Guidance |
|---|---|---|
| 7 | (a) | R = 2 g c o s 3 0 |
| Answer | Marks | Guidance |
|---|---|---|
| 2 3 | M1dep* | 3.4 |
| Work done against friction is 12 g d | A1 | 1.1 |
| Answer | Marks | Guidance |
|---|---|---|
| 7 | (b) | PE (lost by) B: 4gd |
| PE (gained by) A: 2 g ( d s i n 3 0 ) ( = g d ) | B1 | 1.1 |
| Answer | Marks | Guidance |
|---|---|---|
| 2 2 | B1 | 1.1 |
| Answer | Marks | Guidance |
|---|---|---|
| 12 ( 2 ) (1 .7 5 ) 2 + 12 ( 4 ) (1 .7 5 ) 2 = ( 4 g d − g d ) − 12 g d | M1 | 3.4 |
| Answer | Marks | Guidance |
|---|---|---|
| d = 0.375 | A1 | 2.2a |
| Answer | Marks | Guidance |
|---|---|---|
| 7 | (c) | • Consider the dimensions of the pulley or block(s) |
| Answer | Marks | Guidance |
|---|---|---|
| • Elastic rope | B1 | 3.5c |
Question 7:
7 | (a) | R = 2 g c o s 3 0 | M1* | 3.3 | Resolving perpendicular to the plane for A – correct
number of terms – allow sin/cos confusion (M0 if 2
used for the weight)
1
F = ( 2 g c o s 3 0 ) ( = 12 g )
2 3 | M1dep* | 3.4 | Use of F =R with correct and their R
Work done against friction is 12 g d | A1 | 1.1 | Must be in terms of g and d
[3]
7 | (b) | PE (lost by) B: 4gd | B1 | 1.1 | Allow 4 g d or 3 9 .2 d
PE (gained by) A: 2 g ( d s i n 3 0 ) ( = g d ) | B1 | 1.1 | Allow g d or 9.8d
KE (gained by) A and B: 1(2)(1.75)2+1(4)(1.75)2
2 2 | B1 | 1.1 | KE (gained) for either A or B (so B1 for either
3.0625 or 6.125 or 9.1875)
12 ( 2 ) (1 .7 5 ) 2 + 12 ( 4 ) (1 .7 5 ) 2 = ( 4 g d − g d ) − 12 g d | M1 | 3.4 | Work energy principle – condone sign errors and slips
but must be the correct number of terms (so must be
considering KE for A and B and three terms in d) and
dimensionally correct
d = 0.375 | A1 | 2.2a
[5]
7 | (c) | • Consider the dimensions of the pulley or block(s)
• Consider the weight/mass of the rope
• More accurate value of g
• Friction at the pulley
• Elastic rope | B1 | 3.5c | Must be suggesting an improvement so B0 for ‘do not
modelled blocks as particles’ (oe)
B0 for ‘include air resistance/resistance/wind’ or
‘friction/resistance should be proportional to speed or
speed squared’
[1]One end of a rope is attached to a block A of mass 2 kg. The other end of the rope is attached to a second block B of mass 4 kg. Block A is held at rest on a fixed rough ramp inclined at $30°$ to the horizontal. The rope is taut and passes over a small smooth pulley P which is fixed at the top of the ramp. The part of the rope from A to P is parallel to a line of greatest slope of the ramp. Block B hangs vertically below P, at a distance $d$ m above the ground, as shown in the diagram.
\includegraphics{figure_7}
Block A is more than $d$ m from P. The blocks are released from rest and A moves up the ramp.
The coefficient of friction between A and the ramp is $\frac{1}{2\sqrt{3}}$.
The blocks are modelled as particles, the rope is modelled as light and inextensible, and air resistance can be ignored.
\begin{enumerate}[label=(\alph*)]
\item Determine, in terms of $g$ and $d$, the work done against friction as A moves $d$ m up the ramp. [3]
\item Given that the speed of B immediately before it hits the ground is $1.75 \text{ m s}^{-1}$, use the work–energy principle to determine the value of $d$. [5]
\item Suggest one improvement, apart from including air resistance, that could be made to the model to make it more realistic. [1]
\end{enumerate}
\hfill \mbox{\textit{OCR MEI Further Mechanics Major 2023 Q7 [9]}}