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OCR MEI Further Mechanics Minor Specimen Q3
8 marks Standard +0.3
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
9 marks Standard +0.8
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
11 marks Standard +0.3
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
14 marks Moderate -0.8
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.
    OCR is part of the Cambridge Assessment Group; Cambridge Assessment is the brand name of University of Cambridge Local Examinations Syndicate (UCLES), which is itself a department of the University of Cambridge. }\section*{}
OCR MEI Further Mechanics Major 2019 June Q1
5 marks Standard +0.3
1 Three forces represented by the vectors \(- 4 \mathbf { i } , \mathbf { i } + 2 \mathbf { j }\) and \(k \mathbf { i } - 2 \mathbf { j }\) act at the points with coordinates \(( 0,0 ) , ( 3,0 )\) and \(( 0,4 )\) respectively.
  1. Given that the three forces form a couple, find the value of \(k\).
  2. Find the magnitude and direction of the couple.
OCR MEI Further Mechanics Major 2019 June Q2
4 marks Moderate -0.5
2 The Reynolds number, \(R\), is an important dimensionless quantity in fluid dynamics; it can be used to predict flow patterns when a fluid is in motion relative to a surface.
The Reynolds number is defined as \(\quad R = \frac { \rho u l } { \mu }\),
where \(\rho\) is the density of the fluid, \(u\) is the velocity of the fluid relative to the surface, \(l\) is the distance travelled by the fluid and \(\mu\) is the viscosity of the fluid. Find the dimensions of \(\mu\).
OCR MEI Further Mechanics Major 2019 June Q3
5 marks Standard +0.3
3 A ball of mass 2 kg is moving with velocity \(( 3 \mathbf { i } - 2 \mathbf { j } ) \mathrm { ms } ^ { - 1 }\) when it is struck by a bat. The impulse on the ball is \(( - 8 \mathbf { i } + 10 \mathbf { j } )\) Ns.
  1. Find the speed of the ball immediately after the impact.
  2. State one modelling assumption you have used in answering part (a). \begin{figure}[h]
    \includegraphics[alt={},max width=\textwidth]{a69c3e7a-dfc4-438b-84ba-e88c15d421ea-03_373_558_315_258} \captionsetup{labelformat=empty} \caption{Fig. 4}
    \end{figure} Fig. 4 shows a uniform lamina ABCDE such that ABDE is a rectangle and BCD is an isosceles triangle. \(\mathrm { AB } = 5 a , \mathrm { AE } = 4 a\) and \(\mathrm { BC } = \mathrm { CD }\). The point F is the midpoint of BD and \(\mathrm { FC } = a\).
  3. Find, in terms of \(a\), the exact distance of the centre of mass of the lamina from AE. The lamina is freely suspended from B and hangs in equilibrium.
  4. Find the angle between AB and the downward vertical.
OCR MEI Further Mechanics Major 2019 June Q5
7 marks Standard +0.3
5 A particle P of mass 4 kilograms moves in such a way that its position vector at time \(t\) seconds is r metres, where \(\mathbf { r } = 3 t \mathbf { i } + 2 \mathrm { e } ^ { - 3 t } \mathbf { j }\).
  1. Find the initial kinetic energy of P .
  2. Find the time when the acceleration of P is 2 metres per second squared. Section B (93 marks)
OCR MEI Further Mechanics Major 2019 June Q6
7 marks Standard +0.3
6 \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{a69c3e7a-dfc4-438b-84ba-e88c15d421ea-04_483_828_370_246} \captionsetup{labelformat=empty} \caption{Fig. 6}
\end{figure} The rim of a smooth hemispherical bowl is a circle of centre O and radius \(a\). The bowl is fixed with its rim horizontal and uppermost. A particle P of mass \(m\) is released from rest at a point A on the rim as shown in Fig. 6. When P reaches the lowest point of the bowl it collides directly with a stationary particle Q of mass \(\frac { 1 } { 2 } m\). After the collision Q just reaches the rim of the bowl. Find the coefficient of restitution between P and Q .
OCR MEI Further Mechanics Major 2019 June Q8
11 marks Standard +0.8
8 A car of mass 800 kg travels up a line of greatest slope of a straight road inclined at \(5 ^ { \circ }\) to the horizontal. The power developed by the car is constant and equal to 25 kW . The resistance to the motion of the car is constant and equal to 750 N . The car passes through a point A on the road with speed \(7 \mathrm {~ms} ^ { - 1 }\).
  1. Find
    • the acceleration of the car at A ,
    • the greatest steady speed at which the car can travel up the hill.
    The car later passes through a point B on the road where \(\mathrm { AB } = 131 \mathrm {~m}\). The time taken to travel from A to B is 10.4 s .
  2. Calculate the speed of the car at B. \begin{figure}[h]
    \includegraphics[alt={},max width=\textwidth]{a69c3e7a-dfc4-438b-84ba-e88c15d421ea-06_442_346_292_251} \captionsetup{labelformat=empty} \caption{Fig. 9}
    \end{figure} A particle P of mass \(m\) is joined to a fixed point O by a light inextensible string of length \(l . \mathrm { P }\) is released from rest with the string taut and making an acute angle \(\alpha\) with the downward vertical, as shown in Fig. 9. At a time \(t\) after P is released the string makes an angle \(\theta\) with the downward vertical and the tension in the string is \(T\). Angles \(\alpha\) and \(\theta\) are measured in radians.
  3. Show that $$\left( \frac { \mathrm { d } \theta } { \mathrm {~d} t } \right) ^ { 2 } = \frac { 2 g } { l } \cos \theta + k _ { 1 } ,$$ where \(k _ { 1 }\) is a constant to be determined in terms of \(g , l\) and \(\alpha\).
  4. Show that $$T = 3 m g \cos \theta + k _ { 2 } ,$$ where \(k _ { 2 }\) is a constant to be determined in terms of \(m , g\) and \(\alpha\). It is given that \(\alpha\) is small enough for \(\alpha ^ { 2 }\) to be negligible.
  5. Find, in terms of \(m\) and \(g\), the approximate tension in the string.
  6. Show that the motion of P is approximately simple harmonic.
OCR MEI Further Mechanics Major 2019 June Q10
8 marks Challenging +1.8
10 A particle P , of mass \(m\), moves on a rough horizontal table. P is attracted towards a fixed point O on the table by a force of magnitude \(\frac { k m g } { x ^ { 2 } }\), where \(x\) is the distance OP. The coefficient of friction between P and the table is \(\mu\).
P is initially projected in a direction directly away from O . The velocity of P is first zero at a point A which is a distance \(a\) from O .
  1. Show that the velocity \(v\) of P , when P is moving away from O , satisfies the differential equation $$\frac { \mathrm { d } } { \mathrm {~d} x } \left( v ^ { 2 } \right) + \frac { 2 k g } { x ^ { 2 } } + 2 \mu g = 0$$
  2. Verify that $$v ^ { 2 } = 2 g k \left( \frac { 1 } { x } - \frac { 1 } { a } \right) + 2 \mu g ( a - x )$$
  3. Find, in terms of \(k\) and \(a\), the range of values of \(\mu\) for which P remains at A .
OCR MEI Further Mechanics Major 2019 June Q11
14 marks Standard +0.3
11 Two uniform smooth spheres A and B have equal radii and are moving on a smooth horizontal surface. The mass of \(A\) is 0.2 kg and the mass of \(B\) is 0.6 kg . The spheres collide obliquely. When the spheres collide the line joining their centres is parallel to \(\mathbf { i }\). Immediately before the collision the velocity of A is \(\mathbf { u } _ { \mathrm { A } } \mathrm { ms } ^ { - 1 }\) and the velocity of B is \(\mathbf { u } _ { \mathrm { B } } \mathrm { ms } ^ { - 1 }\). The coefficient of restitution between A and B is 0.5. Immediately after the collision the velocity of A is \(( - 4 \mathbf { i } + 2 \mathbf { j } ) \mathrm { ms } ^ { - 1 }\) and the velocity of B is \(( 2 \mathbf { i } + 3 \mathbf { j } ) \mathrm { ms } ^ { - 1 }\).
  1. Find \(\mathbf { u } _ { \mathrm { A } }\) and \(\mathbf { u } _ { \mathrm { B } }\). After the collision B collides with a smooth vertical wall which is parallel to \(\mathbf { j }\).
    The loss in kinetic energy of B caused by the collision with the wall is 1.152 J .
  2. Find the coefficient of restitution between B and the wall.
  3. Find the angle through which the direction of motion of B is deflected as a result of the collision with the wall. \begin{figure}[h]
    \includegraphics[alt={},max width=\textwidth]{a69c3e7a-dfc4-438b-84ba-e88c15d421ea-08_730_476_264_251} \captionsetup{labelformat=empty} \caption{Fig. 12}
    \end{figure} The ends of a light inextensible string are fixed to two points A and B in the same vertical line, with A above B. The string passes through a small smooth ring of mass \(m\). The ring is fastened to the string at a point P . When the string is taut the angle APB is a right angle, the angle BAP is \(\theta\) and the perpendicular distance of P from AB is \(r\). The ring moves in a horizontal circle with constant angular velocity \(\omega\) and the string taut as shown in Fig. 12.
  4. By resolving horizontally and vertically, show that the tension in the part of the string BP is \(m \left( r \omega ^ { 2 } \cos \theta - g \sin \theta \right)\).
  5. Find a similar expression, in terms of \(r , \omega , m , g\) and \(\theta\), for the tension in the part of the string AP. It is given that \(\mathrm { AB } = 5 a\) and \(\mathrm { AP } = 4 a\).
  6. Show that \(16 a \omega ^ { 2 } > 5 g\). The ring is now free to move on the string but remains in the same position on the string as before. The string remains taut and the ring continues to move in a horizontal circle.
  7. Find the period of the motion of the ring, giving your answer in terms of \(a , g\) and \(\pi\). \begin{figure}[h]
    \includegraphics[alt={},max width=\textwidth]{a69c3e7a-dfc4-438b-84ba-e88c15d421ea-09_838_1132_280_248} \captionsetup{labelformat=empty} \caption{Fig. 13}
    \end{figure} A step-ladder has two sides AB and AC , each of length \(4 a\). Side AB has weight \(W\) and its centre of mass is at the half-way point; side AC is light. The step-ladder is smoothly hinged at A and the two parts of the step-ladder, AB and AC , are connected by a light taut rope DE , where D is on \(\mathrm { AB } , \mathrm { E }\) is on AC and \(\mathrm { AD } = \mathrm { AE } = a\). A man of weight \(4 W\) stands at a point F on AB , where \(\mathrm { BF } = x\).
    The system is in equilibrium with B and C on a smooth horizontal floor and the sides AB and AC are each at an angle \(\theta\) to the vertical, as shown in Fig. 13.
  8. By taking moments about A for side AB of the step-ladder and then for side AC of the step-ladder show that the tension in the rope is $$W \left( 1 + \frac { 2 x } { a } \right) \tan \theta .$$ The rope is elastic with natural length \(\frac { 1 } { 4 } a\) and modulus of elasticity \(W\).
  9. Show that the condition for equilibrium is that $$x = \frac { 1 } { 2 } a ( 8 \cos \theta - \cot \theta - 1 ) .$$ \section*{In this question you must show detailed reasoning.}
  10. Hence determine, in terms of \(a\), the maximum value of \(x\) for which equilibrium is possible.
OCR MEI Further Mechanics Major 2022 June Q1
5 marks Moderate -0.8
1 \includegraphics[max width=\textwidth, alt={}, center]{cbe25a5a-0ca7-4e1b-b5b1-141a49186944-02_645_609_459_246} Three forces of magnitudes \(4 \mathrm {~N} , 7 \mathrm {~N}\) and P N act at a point in the directions shown in the diagram. The forces are in equilibrium.
  1. Draw a closed figure to represent the three forces.
  2. Hence, or otherwise, find the following.
    1. The value of \(\theta\).
    2. The value of \(P\).
OCR MEI Further Mechanics Major 2022 June Q2
4 marks Standard +0.8
2 \includegraphics[max width=\textwidth, alt={}, center]{cbe25a5a-0ca7-4e1b-b5b1-141a49186944-03_359_931_251_255} A particle is projected with speed V from a point O on horizontal ground. The angle of projection is \(\theta\) above the horizontal. The particle passes, in succession, through two points A and B , each at a height h above the ground and a distance d apart, as shown in the diagram. You are given that \(d ^ { 2 } = \frac { v ^ { \alpha } \sin ^ { 2 } 2 \theta } { g ^ { \beta } } - \frac { 8 h v ^ { 2 } \cos ^ { 2 } \theta } { g }\). Use dimensional analysis to find \(\alpha\) and \(\beta\).
OCR MEI Further Mechanics Major 2022 June Q3
6 marks Moderate -0.3
3 A particle, of mass 2 kg , is placed at a point A on a rough horizontal surface. There is a straight vertical wall on the surface and the point on the wall nearest to \(A\) is \(B\). The distance \(A B\) is 5 m . The particle is projected with speed \(4.2 \mathrm {~ms} ^ { - 1 }\) along the surface from A towards B . The particle hits the wall directly and rebounds. The coefficient of friction between the particle and the surface is 0.1 .
  1. Determine the speed of the particle immediately before impact with the wall. The magnitude of the impulse that the wall exerts on the particle is 9.8 Ns .
  2. Find the speed of the particle immediately after impact with the wall.
OCR MEI Further Mechanics Major 2022 June Q4
7 marks Standard +0.3
4 \includegraphics[max width=\textwidth, alt={}, center]{cbe25a5a-0ca7-4e1b-b5b1-141a49186944-04_629_835_260_251} The diagram shows a particle P , of mass 0.1 kg , which is attached by a light inextensible string of length 0.5 m to a fixed point O . P moves with constant angular speed \(5 \mathrm { rad } \mathrm { s } ^ { - 1 }\) in a horizontal circle with centre vertically below O . The string is inclined at an angle \(\theta\) to the vertical.
  1. Determine the tension in the string.
  2. Find the value of \(\theta\).
  3. Find the kinetic energy of P.
OCR MEI Further Mechanics Major 2022 June Q5
7 marks Standard +0.3
5 At time \(t\) seconds, where \(t \geqslant 0\), a particle P of mass 2 kg is moving on a smooth horizontal surface. The particle moves under the action of a constant horizontal force of ( \(- 2 \mathbf { i } + 6 \mathbf { j }\) ) N and a variable horizontal force of \(( 2 \cos 2 t \mathbf { i } + 4 \sin t \mathbf { j } ) \mathrm { N }\). The acceleration of P at time t seconds is \(\mathrm { am } \mathrm { S } ^ { - 2 }\).
  1. Find a in terms of t. The particle P is at rest when \(\mathrm { t } = 0\).
  2. Determine the speed of P at the instant when \(\mathrm { t } = 2\). Answer all the questions.
    Section B (91 marks)
OCR MEI Further Mechanics Major 2022 June Q7
12 marks Standard +0.3
7 Two small uniform smooth spheres A and B , of masses 2 kg and 3 kg respectively, are moving in opposite directions along the same straight line towards each other on a smooth horizontal surface. Sphere \(A\) has speed \(2 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) and B has speed \(1 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) before they collide. The coefficient of restitution between A and B is e .
  1. Show that the velocity of B after the collision, in the original direction of motion of A , is \(\frac { 1 } { 5 } ( 1 + 6 e ) \mathrm { m } \mathrm { s } ^ { - 1 }\) and find a similar expression for the velocity of A after the collision.
  2. The following three parts are independent of each other, and each considers a different scenario regarding the collision between A and B .
    1. In the collision between A and B the spheres coalesce to form a combined body C . State the speed of C after the collision.
    2. In the collision between A and B the direction of motion of A is reversed. Find the range of possible values of e .
    3. The total loss in kinetic energy due to the collision is 3 J . Determine the value of e.
OCR MEI Further Mechanics Major 2022 June Q8
13 marks Standard +0.3
8 A particle P is projected from a fixed point O with initial velocity \(\mathbf { u i } + \mathrm { kuj }\), where k is a positive constant. The unit vectors \(\mathbf { i }\) and \(\mathbf { j }\) are in the horizontal and vertically upward directions respectively. \(P\) moves with constant gravitational acceleration of magnitude \(g\). At time \(t \geqslant 0\), particle \(P\) has position vector \(\mathbf { r }\) relative to \(O\).
  1. Starting from an expression for \(\ddot { \mathbf { r } }\), use integration to derive the formula $$\mathbf { r } = u \mathbf { t } + \left( k u t - \frac { 1 } { 2 } g t ^ { 2 } \right) \mathbf { j } .$$ The position vector \(\mathbf { r }\) of P at time \(\mathrm { t } \geqslant 0\) can be expressed as \(\mathbf { r } = x \mathbf { i } + y \mathbf { j }\), where the axes Ox and Oy are horizontally and vertically upwards through O respectively. The axis Ox lies on horizontal ground.
  2. Show that the path of P has cartesian equation $$g x ^ { 2 } - 2 k u ^ { 2 } x + 2 u ^ { 2 } y = 0 .$$
  3. Hence find, in terms of \(\mathrm { g } , \mathrm { k }\) and u , the maximum height of P above the ground during its motion. The maximum height P reaches above the ground is equal to the distance OA , where A is the point where P first hits the ground.
  4. Determine the value of k .
OCR MEI Further Mechanics Major 2022 June Q9
11 marks Standard +0.8
9 [In this question you may use the facts that for a uniform solid right circular cone of height \(h\) and base radius \(r\) the volume is \(\frac { 1 } { 3 } \pi r ^ { 2 } h\) and the centre of mass is \(\frac { 1 } { 4 } h\) above the base on the line from the centre of the base to the vertex.] \includegraphics[max width=\textwidth, alt={}, center]{cbe25a5a-0ca7-4e1b-b5b1-141a49186944-07_677_963_395_248} The diagram shows the shaded region S bounded by the curve \(\mathrm { y } = \mathrm { e } ^ { \frac { 1 } { 2 } \mathrm { x } }\) for \(0 \leqslant x \leqslant 2\), the x -axis, the y -axis, and the line \(\mathrm { y } = \frac { 1 } { 4 } \mathrm { e } ( 6 - \mathrm { x } )\). The line \(\mathrm { y } = \frac { 1 } { 4 } \mathrm { e } ( 6 - \mathrm { x } )\) meets the curve \(\mathrm { y } = \mathrm { e } ^ { \frac { 1 } { 2 } \mathrm { x } }\) at the point A with coordinates (2,e).
The region S is rotated through \(2 \pi\) radians about the x -axis to form a uniform solid of revolution T .
  1. Show that the x -coordinate of the centre of mass of T is \(\frac { 3 \left( 5 \mathrm { e } ^ { 2 } + 1 \right) } { 7 \mathrm { e } ^ { 2 } - 3 }\). Solid T is freely suspended from A and hangs in equilibrium.
  2. Determine the angle between AO , where O is the origin, and the vertical. \includegraphics[max width=\textwidth, alt={}, center]{cbe25a5a-0ca7-4e1b-b5b1-141a49186944-08_725_1541_251_248} 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 mkg , is placed on the track at A , at a height hm 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 .
  3. 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 \(\mathrm { mg } \left( 3 \cos \theta - 2 + \frac { 2 \mathrm {~h} } { \mathrm { r } } \right) \mathrm { N }\),
    where \(\theta\) is the angle between OC and OP .
  4. Hence determine, in terms of r , the least possible value of h so that P can complete a vertical circle.
  5. Apart from not modelling the car as a particle, state one refinement that would make the model more realistic. \includegraphics[max width=\textwidth, alt={}, center]{cbe25a5a-0ca7-4e1b-b5b1-141a49186944-09_668_695_258_251} The diagram shows two small identical uniform smooth spheres, A and B, just before A collides with B. Sphere B is at rest on a horizontal table with its centre vertically above the edge of the table. Sphere A is projected vertically upwards so that, just before it collides with B, the speed of A is \(\mathrm { U } \mathrm { m } \mathrm { s } ^ { - 1 }\) and it is in contact with the vertical side of the table. The point of contact of A with the vertical side of the table and the centres of the spheres are in the same vertical plane.
  6. Show that on impact the line of centres makes an angle of \(30 ^ { \circ }\) with the vertical. The coefficient of restitution between A and B is \(\frac { 1 } { 2 }\). After the impact B moves freely under gravity.
  7. Determine, in terms of U and g , the time taken for B to first return to the table. \includegraphics[max width=\textwidth, alt={}, center]{cbe25a5a-0ca7-4e1b-b5b1-141a49186944-10_867_1045_255_255} The diagram shows a uniform square lamina ABCD , of weight W and side-length \(a\). The lamina is in equilibrium in a vertical plane that also contains the point O . The vertex A rests on a smooth plane inclined at an angle of \(30 ^ { \circ }\) to the horizontal. The vertex B rests on a smooth plane inclined at an angle of \(60 ^ { \circ }\) to the horizontal. OA is a line of greatest slope of the plane inclined at \(30 ^ { \circ }\) to the horizontal and OB is a line of greatest slope of the plane inclined at \(60 ^ { \circ }\) to the horizontal. The side AB is inclined at an angle \(\theta\) to the horizontal and the lamina is kept in equilibrium in this position by a clockwise couple of magnitude \(\frac { 1 } { 8 } \mathrm { aW }\).
  8. By resolving horizontally and vertically, determine, in terms of W, the magnitude of the normal contact force between the plane and the lamina at B .
  9. By taking moments about A , show that \(\theta\) satisfies the equation $$2 ( \sqrt { 3 } + 2 ) \sin \theta - 2 \cos \theta = 1 .$$
  10. Verify that \(\theta = 22.4 ^ { \circ }\), correct to 1 decimal place.
OCR MEI Further Mechanics Major 2023 June Q1
4 marks Standard +0.3
1 A car of mass 800 kg moves in a straight line along a horizontal road.
There is a constant resistance to the motion of the car of magnitude 600 N .
When the car is travelling at a speed of \(15 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) the power developed by the car is 27 kW .
Determine the acceleration of the car when it is travelling at \(15 \mathrm {~ms} ^ { - 1 }\).
OCR MEI Further Mechanics Major 2023 June Q2
4 marks Standard +0.3
2 \includegraphics[max width=\textwidth, alt={}, center]{41b1f65b-8806-4183-81a1-0276691e203c-02_232_609_840_239} Two small uniform smooth spheres A and B have masses 0.5 kg and 2 kg respectively. The two spheres are travelling in the same direction in the same straight line on a smooth horizontal surface. Sphere \(A\) is moving towards \(B\) with speed \(6 \mathrm {~ms} ^ { - 1 }\) and \(B\) is moving away from \(A\) with speed \(2 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) (see diagram). Spheres A and B collide. After this collision A moves with speed \(0.2 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). Determine the possible speeds with which B moves after the collision.
OCR MEI Further Mechanics Major 2023 June Q3
5 marks Standard +0.3
3 \includegraphics[max width=\textwidth, alt={}, center]{41b1f65b-8806-4183-81a1-0276691e203c-03_565_757_251_242} The diagram shows a particle P , of mass 0.2 kg , which is attached by a light inextensible string of length 0.75 m to a fixed point O . Particle P moves with constant angular speed \(\omega\) rads \(^ { - 1 }\) in a horizontal circle with centre vertically below O . The string is inclined at an angle \(\theta\) to the vertical. The greatest tension that the string can withstand without breaking is 15 N .
  1. Find the greatest possible value of \(\theta\), giving your answer to the nearest degree.
  2. Determine the greatest possible value of \(\omega\).
OCR MEI Further Mechanics Major 2023 June Q4
6 marks Standard +0.8
4 \includegraphics[max width=\textwidth, alt={}, center]{41b1f65b-8806-4183-81a1-0276691e203c-04_598_696_255_246} \includegraphics[max width=\textwidth, alt={}, center]{41b1f65b-8806-4183-81a1-0276691e203c-04_465_67_294_1023} A rigid lamina of negligible mass is in the form of a rhombus ABCD , where \(\mathrm { AC } = 6 \mathrm {~m}\) and \(\mathrm { BD } = 8 \mathrm {~m}\). Forces of magnitude \(2 \mathrm {~N} , 4 \mathrm {~N} , 3 \mathrm {~N}\) and 5 N act along its sides \(\mathrm { AB } , \mathrm { BC } , \mathrm { CD }\) and DA , respectively, as shown in the diagram. A further force \(\mathbf { F }\) N, acting at A, and a couple of magnitude \(G N m\) are also applied to the lamina so that it is in equilibrium.
  1. Determine the magnitude and direction of \(\mathbf { F }\).
  2. Determine the value of \(G\).
OCR MEI Further Mechanics Major 2023 June Q5
7 marks Standard +0.3
5 A particle P of mass \(m \mathrm {~kg}\) is projected with speed \(u \mathrm {~m} \mathrm {~s} ^ { - 1 }\) along a rough horizontal surface. During the motion of P , a constant frictional force of magnitude \(F \mathrm {~N}\) acts on P . When the velocity of P is \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\), it experiences a force of magnitude \(k v \mathrm {~N}\) due to air resistance, where \(k\) is a constant.
  1. Determine the dimensions of \(k\). At time \(T\) s after projection P comes to rest. A formula approximating the value of \(T\) is \(\mathrm { T } = \frac { \mathrm { mu } } { \mathrm { F } } - \frac { \mathrm { kmu } ^ { 2 } } { 2 \mathrm {~F} ^ { 2 } } + \frac { 1 } { 3 } \mathrm { k } ^ { 2 } \mathrm {~m} ^ { \alpha } \mathrm { u } ^ { \beta } \mathrm { F } ^ { \gamma }\).
  2. Use dimensional analysis to find \(\alpha , \beta\) and \(\gamma\).