Questions Further Mechanics Minor (42 questions)

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OCR MEI Further Mechanics Minor 2020 November Q2
2 The speed of propagation, \(c\), of a soundwave travelling in air is given by the formula \(c = k p ^ { \alpha } d ^ { \beta }\),
where
  • \(p\) is the air pressure,
  • \(d\) is the air density,
  • \(k\) is a dimensionless constant.
    1. Use dimensional analysis to determine the values of \(\alpha\) and \(\beta\).
During a series of experiments the speed of propagation of soundwaves travelling in air is initially recorded as \(340 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). At a later time it is found that the air pressure has increased by \(1 \%\) and the air density has fallen by \(0.5 \%\).
  • Determine, for the later time, the speed of propagation of the soundwaves.
    •
    OCR MEI Further Mechanics Minor 2020 November Q3
    3 In this question \(\mathbf { i }\) and \(\mathbf { j }\) are perpendicular unit vectors and \(c\) is a positive real number.
    The resultant of two forces \(c \mathbf { i N }\) and \(- \mathbf { i } + 2 \sqrt { c } \mathbf { j N }\) is denoted by \(R \mathrm {~N}\).
    1. Show that the magnitude of \(R\) is \(c + 1\). A car of mass 900 kg travels along a straight horizontal road with constant resistance to motion of magnitude \(( c + 1 ) \mathrm { N }\). The car passes through point A on the road with speed \(6 \mathrm {~ms} ^ { - 1 }\), and 8 seconds later passes through a point B on the same road. The power developed by the car while travelling from A to B is zero. Furthermore, while travelling between A and B, the car's direction of motion is unchanged.
    2. Determine the range of possible values of \(c\). The car later passes through a point C on the road. While travelling between B and C the power developed by the car is modelled as constant and equal to 18 kW . The car passes through C with speed \(5 \mathrm {~ms} ^ { - 1 }\) and acceleration \(3.5 \mathrm {~ms} ^ { - 2 }\).
    3. Determine the value of \(c\).
    4. Suggest how one of the modelling assumptions made in this question could be improved.
    OCR MEI Further Mechanics Minor 2020 November Q4
    4 A block of mass 20 kg is placed on a rough plane inclined at an angle \(30 ^ { \circ }\) to the horizontal. The block is pulled up the plane by a constant force acting parallel to a line of greatest slope.
    The block passes through points A and B on the plane with speeds \(9 \mathrm {~ms} ^ { - 1 }\) and \(4 \mathrm {~ms} ^ { - 1 }\) respectively with B higher up the plane than A . The distance between A and B is \(x \mathrm {~m}\) and the coefficient of friction between the block and the plane is \(\frac { \sqrt { 3 } } { 49 }\). Use an energy method to determine the range of possible values of \(x\).
    OCR MEI Further Mechanics Minor 2020 November Q5
    2 marks
    5 A uniform rod AB , of mass \(3 m\) and length \(2 a\), rests with the end A on a rough horizontal surface. A small object of mass \(m\) is attached to the rod at B . The rod is maintained in equilibrium at an angle of \(60 ^ { \circ }\) to the horizontal by a force acting at an angle of \(\theta\) to the vertical at a point C , where the distance \(\mathrm { AC } = \frac { 6 } { 5 } a\). The force acting at C is in the same vertical plane as the rod (see Fig. 5). \begin{figure}[h]
    \includegraphics[alt={},max width=\textwidth]{6418c1b7-092a-4747-bc88-1b57815c6ad9-4_800_648_932_255} \captionsetup{labelformat=empty} \caption{Fig. 5}
    \end{figure}
    1. On the copy of Fig. 5 in the Printed Answer Booklet, mark all the forces acting on the rod. [2]
    2. Show that the magnitude of the force acting at C can be expressed as \(\frac { 25 m g } { 6 ( \cos \theta + \sqrt { 3 } \sin \theta ) }\).
    3. Given that the rod is in limiting equilibrium and the coefficient of friction between the rod and the surface is \(\frac { 3 } { 4 }\), determine the value of \(\theta\).
    OCR MEI Further Mechanics Minor 2020 November Q6
    6 Stones A and B have masses \(m \mathrm {~kg}\) and \(3 m \mathrm {~kg}\) respectively. They lie at rest on a large area of smooth horizontal ice and may move freely over the ice. Stone A is given a horizontal impulse of magnitude \(m u \mathrm {~N} s\) towards B so that the stones collide directly. After the collision the direction of motion of A is reversed. The coefficient of restitution between A and B is denoted by \(e\).
    1. Find the range of possible values of \(e\). After the collision, B subsequently collides with a vertical smooth wall perpendicular to its path and rebounds. The coefficient of restitution between \(B\) and the wall is the same as the coefficient of restitution between A and B .
    2. Show that A and B will collide again unless the collision between B and the wall is perfectly elastic.
    3. Explain why modelling the collision between B and the wall as perfectly elastic is possibly unrealistic.
    4. Given that the kinetic energy lost in the first collision between A and B is \(\frac { 5 } { 24 } m u ^ { 2 }\), determine the value of \(e\).
    5. Given that B was 2 metres from the wall when the stones first collided, determine the distance of the stones from the wall when they next collide.
    OCR MEI Further Mechanics Minor 2021 November Q1
    1
    1. State the dimensions of force. The force \(F\) required to keep a car moving at constant speed on a circular track is given by the formula $$\mathrm { F } = \frac { \mathrm { mv } ^ { 2 } } { \mathrm { r } }$$ where
      • \(m\) is the constant mass of the car,
      • \(v\) is the speed of the car,
      • \(r\) is the radius of the circular track.
      • Verify that the formula is dimensionally consistent.
      • Determine the percentage increase in force required to keep a car moving on a circular track if the speed of the car were to increase by \(10 \%\) and if the track radius were to decrease by \(10 \%\).
      It is proposed that a new unit of force, the trackforce (Tr), should be adopted in motor-racing. 1 Tr is defined as the amount of force required to accelerate a mass of 1 ton at a rate of 1 mile per hour per second. It is given that 1 ton \(= 1016 \mathrm {~kg}\) and 1 mile \(= 1609 \mathrm {~m}\).
    2. Determine the number of newtons that are equivalent to 1 Tr .
    OCR MEI Further Mechanics Minor 2021 November Q2
    2 The diagram shows a uniform beam AB that rests with its end A on rough horizontal ground and its end B against a smooth vertical wall. The beam makes an angle of \(\theta ^ { \circ }\) with the ground.
    \includegraphics[max width=\textwidth, alt={}, center]{b3e369f4-13f7-457b-9a43-04ed2e2a2bba-3_812_588_347_246} The weight of the beam is \(W N\). The beam is in limiting equilibrium and the coefficient of friction between the beam and the ground is \(\mu\). It is given that the magnitude of the contact force at A is 70 N and the magnitude of the contact force at B is 20 N . Determine, in any order,
    • the value of \(W\),
    • the value of \(\mu\),
    • the value of \(\theta\).
    OCR MEI Further Mechanics Minor 2021 November Q3
    3 The diagram shows an electric winch raising two crates A and B , with masses 40 kg and 25 kg , respectively. The cable connecting the winch to A , and the cable connecting A to B may both be modelled as light and inextensible. Furthermore, it can be assumed that there are no resistances to motion.
    \includegraphics[max width=\textwidth, alt={}, center]{b3e369f4-13f7-457b-9a43-04ed2e2a2bba-4_499_300_447_246} Throughout the entire motion, the power \(P \mathrm {~W}\) developed by the winch is constant.
    Crates A and B are both being raised at a constant speed \(\nu \mathrm { m } \mathrm { s } ^ { - 1 }\) when the cable connecting A and B breaks. After the cable between A and B breaks, crate A continues to be raised by the winch. Crate A now accelerates until it reaches a new constant speed of \(( v + 3 ) \mathrm { m } \mathrm { s } ^ { - 1 }\). Determine
    • the value of \(v\),
    • the value of \(P\).
    OCR MEI Further Mechanics Minor 2021 November Q4
    4 A child throws a ball of mass \(m \mathrm {~kg}\) vertically upwards with a speed of \(7.2 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). The ball leaves the child's hand at a height of 1.6 m above horizontal ground.
    1. Ignoring any possible air resistance, use an energy method to determine the maximum height reached by the ball above the ground. In fact, the ball only reaches a height of 4.1 m above the ground. For the rest of this question you should assume that the air resistance may be modelled as a constant force acting in the opposite direction to the ball's motion.
    2. Show that the ball does 0.568 mJ of work against air resistance per metre travelled.
    3. Calculate the speed of the ball just before it hits the ground. The ball bounces off the ground and first comes instantaneously to rest 2.8 m above the ground.
    4. Determine the coefficient of restitution between the ball and the ground. In the first impact between the ball and the ground, the magnitude of the impulse exerted on the ball by the ground is 12 Ns .
    5. Determine the value of \(m\).
    OCR MEI Further Mechanics Minor 2021 November Q5
    5 Fig. 5.1 shows a solid L-shaped ornament, of uniform density. The ornament is 3 cm thick. The \(x , y\) and \(z\) axes are shown, along with the dimensions of the ornament. The measurements are in centimetres. \begin{figure}[h]
    \includegraphics[alt={},max width=\textwidth]{b3e369f4-13f7-457b-9a43-04ed2e2a2bba-6_556_887_406_244} \captionsetup{labelformat=empty} \caption{Fig. 5.1}
    \end{figure}
    1. Determine, with reference to the axes shown, the coordinates of the ornament's centre of mass. Fig. 5.2 shows the ornament placed so that the shaded face (indicated in Fig. 5.1) is in contact with a plane inclined at \(\theta ^ { \circ }\) to the horizontal, with the 4 cm edge parallel to a line of greatest slope. The surface of the plane is sufficiently rough so that the ornament will not slip down the plane. \begin{figure}[h]
      \includegraphics[alt={},max width=\textwidth]{b3e369f4-13f7-457b-9a43-04ed2e2a2bba-6_646_844_1452_242} \captionsetup{labelformat=empty} \caption{Fig. 5.2}
      \end{figure}
    2. Determine the minimum and maximum possible values of \(\theta\) for which the ornament does not topple. The ornament is now placed with its shaded face in contact with a rough horizontal surface. A force of magnitude \(P\) N, acting parallel to the planes of the L -shaped faces, is applied to one of the edges of the ornament, as shown in Fig. 5.3. The force is inclined at an angle of \(30 ^ { \circ }\) to the horizontal. The coefficient of friction between the ornament and the surface is \(\mu\). \begin{figure}[h]
      \includegraphics[alt={},max width=\textwidth]{b3e369f4-13f7-457b-9a43-04ed2e2a2bba-7_524_680_452_246} \captionsetup{labelformat=empty} \caption{Fig. 5.3}
      \end{figure} The value of \(P\) is gradually increased until the ornament is on the point of toppling but does not slide.
    3. Determine the minimum value of \(\mu\).
    4. Explain how your answer to part (c) would change if the angle between \(P\) and the horizontal was less than \(30 ^ { \circ }\).
    OCR MEI Further Mechanics Minor 2021 November Q6
    6 A block rests on a horizontal surface. The coefficient of friction between the block and the surface is \(\mu\).
    1. Show that if the block is given an initial speed of \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\), it will move a distance of \(\frac { \mathrm { v } ^ { 2 } } { 2 \mu \mathrm {~g} }\) before coming to rest. Block B rests on the same horizontal surface as a sphere S . On the other side of S is a vertical wall, as shown below. The mass of \(B\) is 8 times the mass of \(S\).
      \includegraphics[max width=\textwidth, alt={}, center]{b3e369f4-13f7-457b-9a43-04ed2e2a2bba-8_211_1013_662_244} S is projected directly towards B with speed \(u \mathrm {~m} \mathrm {~s} ^ { - 1 }\) and hits B . It is given that
      • the coefficient of restitution between S and B is 0.8 ,
      • collisions between S and the wall are perfectly elastic,
      • the wall is perpendicular to the direction of motion of S and B .
      Furthermore, you should model the contact between B and the surface as rough and model the contact between S and the surface as smooth.
    2. Determine, in terms of \(u\), expressions for
      • the speed of S
      • the speed of B
        immediately after the first collision between S and B . In each case stating the corresponding direction of motion.
      It is given that B has sufficient time to come to rest before each subsequent collision with S .
      Let \(\mathrm { X } _ { \mathrm { n } }\) be the distance B moves after the \(n\)th impact between S and B .
    3. Explain why \(\mathrm { x } _ { \mathrm { n } + 1 } = \frac { 9 } { 25 } \mathrm { x } _ { \mathrm { n } }\).
    4. Given that \(u = 11.2\) and the coefficient of friction between B and the surface is \(\frac { 1 } { 7 }\), show that B will travel a total distance that cannot exceed 2.8 m . \section*{END OF QUESTION PAPER} \section*{OCR
      Oxford Cambridge and RSA}
    OCR MEI Further Mechanics Minor Specimen Q1
    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.
    OCR MEI Further Mechanics Minor Specimen Q2
    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.
    OCR MEI Further Mechanics Minor Specimen Q3
    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.
    OCR MEI Further Mechanics Minor Specimen Q4
    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]
    \includegraphics[alt={},max width=\textwidth]{54711a46-83ce-4fb9-b6d3-53b264725c74-3_426_672_539_605} \captionsetup{labelformat=empty} \caption{Fig. 4}
    \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.
    OCR MEI Further Mechanics Minor Specimen Q5
    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]
    \includegraphics[alt={},max width=\textwidth]{54711a46-83ce-4fb9-b6d3-53b264725c74-4_383_371_552_852} \captionsetup{labelformat=empty} \caption{Fig. 5}
    \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 .
    OCR MEI Further Mechanics Minor Specimen Q6
    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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