OCR MEI Further Mechanics Minor (Further Mechanics Minor) Specimen

1 In this question, \(\mathbf { i }\) and \(\mathbf { j }\) are perpendicular unit vectors in a horizontal plane. A particle \(P\) has mass 10 kg and a speed of \(20 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) in the direction of \(4 \mathbf { i } + 3 \mathbf { j }\). A force of \(( - 4 \mathbf { i } + 15 \mathbf { j } ) \mathrm { N }\) acts on P for 8 seconds.

1. Calculate the impulse of the force over the 8 seconds.
2. Hence find the speed of P at the end of the 8 seconds.

2 A car of mass 1200 kg is travelling in a straight line along a horizontal road. At a time when the power of the driving force is 25 kW , the car has a speed of \(12.5 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) and is accelerating at \(1.5 \mathrm {~m} \mathrm {~s} ^ { - 2 }\). Calculate the magnitude of the resistance to the motion of the car.

3

1. Find the dimensions of
   • density and
   • pressure (force per unit area).
   The frequency, \(f\), of the note emitted by an air horn is modelled as \(f = k s ^ { \alpha } p ^ { \beta } d ^ { \gamma }\), where
   • \(s\) is the length of the horn,
   • \(\quad p\) is the air pressure,
   • \(d\) is the air density,
   • \(k\) is a dimensionless constant.
   • Determine the values of \(\alpha , \beta\) and \(\gamma\).
   A particular air horn emits a note at a frequency of 512 Hz and the air pressure and air density are recorded. At another time it is found that the air pressure has fallen by \(2 \%\) and the air density has risen by \(1 \%\). The length of the horn is unchanged.
2. Calculate the new frequency predicted by the model.

4 Fig. 4 shows a non-uniform rigid plank AB of weight 900 N and length 2.5 m . The centre of mass of the plank is at G which is 2 m from A . The end A rests on rough horizontal ground and does not slip. The plank is held in equilibrium at \(20 ^ { \circ }\) above the horizontal by a force of \(T \mathrm {~N}\) applied at B at an angle of \(55 ^ { \circ }\) above the horizontal as shown in Fig. 4. \begin{figure}[h] \end{figure}

1. Show that \(T = 700\) (correct to 3 significant figures).
2. Determine the possible values of the coefficient of friction between the plank and the ground.

5 A young man of mass 60 kg swings on a trapeze. A simple model of this situation is as follows. The trapeze is a light seat suspended from a fixed point by a light inextensible rope. The man's centre of mass, G , moves on an arc of a circle of radius 9 m with centre O , as shown in Fig. 5. The point C is 9 m vertically below O . B is a point on the arc where angle COB is \(45 ^ { \circ }\). \begin{figure}[h] \end{figure}

1. Calculate the gravitational potential energy lost by the man if he swings from B to C . In this model it is also assumed that there is no resistance to the man's motion and he starts at rest from B.
2. Using an energy method, find the man's speed at C . A new model is proposed which also takes into account resistance to the man's motion.
3. State whether you would expect any such model to give a larger, smaller or the same value for the man's speed at C . Give a reason for your answer. A particular model takes account of the resistance by assuming that there is a force of constant magnitude 15 N always acting in the direction opposing the man's motion. This new model also takes account of the man 'pushing off' along the arc from B to C with a speed of \(1.5 \mathrm {~m} \mathrm {~s} ^ { - 1 }\).
4. Using an energy method, find the man's speed at C .

6 My cat Jeoffry has a mass of 4 kg and is sitting on rough ground near a sledge of mass 8 kg . The sledge is on a large area of smooth horizontal ice. Initially, the sledge is at rest and Jeoffry jumps and lands on it with a horizontal velocity of \(2.25 \mathrm {~ms} ^ { - 1 }\) parallel to the runners of the sledge. On landing, Jeoffry grips the sledge with his claws so that he does not move relative to the sledge in the subsequent motion.

1. Show that the sledge with Jeoffry on it moves off with a speed of \(0.75 \mathrm {~ms} ^ { - 1 }\). With the sledge and Jeoffry moving at \(0.75 \mathrm {~m} \mathrm {~s} ^ { - 1 }\), the sledge collides directly with a stationary stone of mass 3 kg . The stone may move freely over the ice. The coefficient of restitution in the collision is \(\frac { 4 } { 15 }\).
2. Calculate the velocity of the sledge and Jeoffry immediately after the collision. In a new situation, Jeoffry is initially sitting at rest on the sledge when it is stationary on the ice. He then walks from the back to the front of the sledge.
3. Giving a brief reason for your answer, describe what happens to the sledge during his walk. Jeoffry is again sitting on the sledge when it is stationary on the ice. He jumps off and, after he has lost contact with the sledge, has a horizontal speed relative to the sledge of \(3 \mathrm {~ms} ^ { - 1 }\).
4. Determine the speed of the sledge after Jeoffry loses contact with it. Fig. 7 shows a container for flowers which is a vertical cylindrical shell with a closed horizontal base. Its radius and its height are both \(\frac { 1 } { 2 } \mathrm {~m}\). Both the curved surface and the base are made of the same thin uniform material. The mass of the container is \(M \mathrm {~kg}\). \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{54711a46-83ce-4fb9-b6d3-53b264725c74-6_323_709_447_767} \captionsetup{labelformat=empty} \caption{Fig. 7}
   \end{figure}
5. Find, as a fraction, the height above the base of the centre of mass of the container. The container would hold \(\frac { 3 } { 2 } M \mathrm {~kg}\) of soil when full to the top. Some soil is put into the empty container and levelled with its top surface \(y \mathrm {~m}\) above the base. The centre of mass of the container with this much soil is zm above the base.
6. Show that \(z = \frac { 1 + 9 y ^ { 2 } } { 6 ( 1 + 3 y ) }\).
7. It is given that \(\frac { \mathrm { d } z } { \mathrm {~d} y } = 0\) when \(y = 0.14\) (to 2 significant figures) and that \(\frac { \mathrm { d } ^ { 2 } z } { \mathrm {~d} y ^ { 2 } } > 0\) at this value of \(y\). When putting in the soil, how might you use this information if the container is to be placed on slopes without it tipping over? \section*{END OF QUESTION PAPER} OCR is committed to seeking permission to reproduce all third-party content that it uses in the assessment materials. OCR has attempted to identify and contact all copyright holders whose work is used in this paper. To avoid the issue of disclosure of answer-related information to candidates, all copyright acknowledgements are reproduced in the OCR Copyright Acknowledgements booklet. This is produced for each series of examinations and is freely available to download from our public website (\href{http://www.ocr.org.uk}{www.ocr.org.uk}) after the live examination series. If OCR has unwittingly failed to correctly acknowledge or clear any third-party content in this assessment material, OCR will be happy to correct its mistake at the earliest possible opportunity. For queries or further information please contact the Copyright Team, First Floor, 9 Hills Road, Cambridge CB2 1GE.
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