Questions — OCR MEI Further Mechanics A AS (52 questions)

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OCR MEI Further Mechanics A AS 2018 June Q1
1 Forces of magnitude \(4 \mathrm {~N} , 3 \mathrm {~N} , 5 \mathrm {~N}\) and \(R \mathrm {~N}\) act on a particle in the directions shown in Fig. 1. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{fa99d9e6-e174-42dd-ac92-7b7d112c08be-2_697_780_443_639} \captionsetup{labelformat=empty} \caption{Fig. 1}
\end{figure} The particle is in equilibrium. Find each of the following.
  • The value of \(R\).
  • The value of \(\theta\).
OCR MEI Further Mechanics A AS 2018 June Q2
2 A car of mass 1350 kg travels along a straight horizontal road. Throughout this question the resistance force to the motion of the car is modelled as constant and equal to 920 N .
  1. Calculate the power, in kW , developed by the car when the car is travelling at a constant speed of \(25 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). The car is now used to tow a caravan of mass 1050 kg along the same road. When the car tows the caravan at a constant speed of \(20 \mathrm {~ms} ^ { - 1 }\) the power developed by the car is 45 kW .
  2. Find the additional resistance force due to the caravan. In the remaining parts of this question the power developed by the car is constant and equal to 68 kW and the resistance force due to the caravan is modelled as constant and equal to the value found in part (ii). When the car and caravan pass a point A on the same straight horizontal road the speed of the car and caravan is \(20 \mathrm {~ms} ^ { - 1 }\).
  3. Find the acceleration of the car and caravan at point A . The car and caravan later pass a point B on the same straight horizontal road with speed \(28 \mathrm {~ms} ^ { - 1 }\). The distance \(A B\) is \(1024 m\).
  4. Find the time taken for the car and caravan to travel from point A to point B .
  5. Suggest one way in which any of the modelling assumptions used in this question could have been improved.
OCR MEI Further Mechanics A AS 2018 June Q3
3 Jodie is doing an experiment involving a simple pendulum. The pendulum consists of a small object tied to one end of a piece of string. The other end of the string is attached to a fixed point O and the object is allowed to swing between two fixed points A and B and back again, as shown in Fig. 3. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{fa99d9e6-e174-42dd-ac92-7b7d112c08be-3_328_350_584_886} \captionsetup{labelformat=empty} \caption{Fig. 3}
\end{figure} Jodie thinks that \(P\), the time the pendulum takes to swing from A to B and back again, depends on the mass, \(m\), of the small object, the length, \(l\), of the piece of string, and the acceleration due to gravity \(g\). She proposes the formula \(P = k m ^ { \alpha } l ^ { \beta } g ^ { \gamma }\).
  1. What is the significance of \(k\) in Jodie's formula?
  2. Use dimensional analysis to determine the values of \(\alpha , \beta\) and \(\gamma\). Jodie finds that when the mass of the object is 1.5 kg and the length of the string is 80 cm the time taken for the pendulum to swing from A to B and back again is 1.8 seconds.
  3. Use Jodie's formula and your answers to part (ii) to find each of the following.
    (A) The value of \(k\)
    (B) The time taken for the pendulum to swing from A to B and back again when the mass of the object is 0.9 kg and the length of the string is 1.4 m
  4. Comment on the assumption made by Jodie that the formula for the time taken for the pendulum to swing from A to B and back again is dependent on \(m , l\) and \(g\).
OCR MEI Further Mechanics A AS 2018 June Q4
4 A uniform lamina ABDE is in the shape of an equilateral triangle ABC of side 12 cm from which an equilateral triangle of side 6 cm has been removed from corner \(C\). The lamina is situated on coordinate axes as shown in Fig. 4. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{fa99d9e6-e174-42dd-ac92-7b7d112c08be-4_501_753_406_646} \captionsetup{labelformat=empty} \caption{Fig. 4}
\end{figure}
  1. Explain why angle \(\mathrm { BDA } = 90 ^ { \circ }\).
  2. Find the coordinates of the centre of mass of the lamina ABDE . The lamina ABDE is now freely suspended from D and hangs in equilibrium.
  3. Calculate the angle DE makes with the downward vertical.
OCR MEI Further Mechanics A AS 2018 June Q5
5 A small ball is held at a height of 160 cm above a horizontal surface. The ball is released from rest and rebounds from the surface. After its first bounce on the surface the ball reaches a height of 122.5 cm .
  1. Find the height reached by the ball after its second bounce on the surface. After \(n\) bounces the height reached by the ball is less than 10 cm .
  2. Find the minimum possible value of \(n\).
  3. State what would happen if the same ball is released from rest from a height of 160 cm above a different horizontal surface and
    (A) the coefficient of restitution between the ball and the new surface is 0 ,
    (B) the coefficient of restitution between the ball and the new surface is 1 .
OCR MEI Further Mechanics A AS 2018 June Q6
6 A uniform rod AB has length \(2 a\) and weight \(W\). The rod is in equilibrium in a horizontal position. The end A rests on a smooth plane which is inclined at an angle of \(30 ^ { \circ }\) to the horizontal. The force exerted on AB by the plane is \(R\). The end B is attached to a light inextensible string inclined at an angle of \(\theta\) to AB as shown in Fig. 6. The rod and string are in the same vertical plane, which also contains the line of greatest slope of the plane on which A lies. The tension in the string is \(T\). \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{fa99d9e6-e174-42dd-ac92-7b7d112c08be-5_474_862_479_616} \captionsetup{labelformat=empty} \caption{Fig. 6}
\end{figure}
  1. Add the forces \(R\) and \(T\) to the copy of Fig. 6 in the Printed Answer Booklet.
  2. By taking moments about B , find an expression for \(R\) in terms of \(W\).
  3. By resolving horizontally, show that \(6 T \cos \theta = W \sqrt { 3 }\).
  4. By finding a second equation connecting \(T\) and \(\theta\), determine
    • the value of \(\theta\),
    • an expression for \(T\) in terms of \(W\).
OCR MEI Further Mechanics A AS 2019 June Q1
1 A child is pulling a toy block in a straight line along a horizontal floor.
The block is moving with a constant speed of \(2 \mathrm {~ms} ^ { - 1 }\) by means of a constant force of magnitude 20 N acting at an angle of \(\theta ^ { \circ }\) above the horizontal. The work done by the force in 10 s is 350 J . Calculate the value of \(\theta\).
OCR MEI Further Mechanics A AS 2019 June Q2
2 The surface tension of a liquid allows a metal needle to be at rest on the surface of the liquid.
The greatest mass \(m\) of a needle of length \(l\) which can be supported in this way by a liquid of surface tension \(S\) is given by the formula
\(m = \frac { 2 S l } { g }\)
where \(g\) is the acceleration due to gravity.
  1. Determine the dimensions of surface tension. Surface tension also allows liquids to rise up capillary tubes. Molly is experimenting with liquids in capillary tubes and she arrives at the formula \(h = \frac { 2 S } { \rho g r }\), where \(h\) is the height to which a liquid of surface tension \(S\) rises, \(\rho\) is the density of the liquid, and \(r\) is the radius of the capillary tube.
  2. Show that the equation for \(h\) is dimensionally consistent. In SI units, the surface tension of mercury is \(0.475 \mathrm {~kg} \mathrm {~s} ^ { - 2 }\) and its density is \(13500 \mathrm {~kg} \mathrm {~m} ^ { - 3 }\).
  3. Find the diameter of a capillary tube in which mercury will rise to a height of 10 cm . In another experiment, Molly finds that when liquid of surface tension \(S\) is poured onto a horizontal surface, puddles of depth \(d\) are formed. For this experiment she finds that
    \(d = k S ^ { \alpha } \rho ^ { \beta } g ^ { \gamma }\)
    where \(k\) is a dimensionless constant.
  4. Determine the values of \(\alpha , \beta\) and \(\gamma\).
OCR MEI Further Mechanics A AS 2019 June Q3
3 A box weighing 130 N is on a rough plane inclined at \(12 ^ { \circ }\) to the horizontal.
The box is held at rest on the plane by the action of a force of magnitude 70 N acting up the plane in a direction parallel to a line of greatest slope of the plane.
The box is on the point of slipping up the plane.
  1. Find the coefficient of friction between the box and the plane. The force of magnitude 70 N is removed.
  2. Determine whether or not the box remains in equilibrium.
OCR MEI Further Mechanics A AS 2019 June Q4
4 A shovel consists of a blade and handle, as shown in Fig. 4.1 and Fig. 4.2. The dimensions shown in the figures are in metres.
The blade is modelled as a uniform rectangular lamina ABCD lying in the Oxy plane, where O is the mid-point of AB . The handle is modelled as a thin uniform rod EF . The handle lies in the Oyz plane, and makes an angle \(\alpha\) with \(\mathrm { O } y\), where \(\sin \alpha = \frac { 7 } { 25 }\). The rod and lamina are rigidly attached at E, the mid-point of CD.
The blade of the shovel has mass 1.25 kg and the handle of the shovel has mass 0.5 kg . \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{6b27d322-417e-4cea-85cc-65d3728173c8-3_746_671_1217_246} \captionsetup{labelformat=empty} \caption{Fig. 4.1}
\end{figure} \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{6b27d322-417e-4cea-85cc-65d3728173c8-3_664_766_1226_1064} \captionsetup{labelformat=empty} \caption{Fig. 4.2}
\end{figure}
  1. Find,
    1. the \(y\)-coordinate of the centre of mass of the shovel,
    2. the \(z\)-coordinate of the centre of mass of the shovel. The shovel is freely suspended from O and hangs in equilibrium.
  2. Calculate the angle that OE makes with the vertical.
OCR MEI Further Mechanics A AS 2019 June Q5
5 A car of mass 4000 kg travels up a line of greatest slope of a straight road inclined at an angle of \(\theta\) to the horizontal, where \(\sin \theta = 0.1\).
The power developed by the car's engine is constant and the resistance to the motion of the car is constant and equal to 850 N . The car passes through a point A on the road with speed \(18 \mathrm {~ms} ^ { - 1 }\) and acceleration \(0.75 \mathrm {~ms} ^ { - 2 }\).
  1. Calculate the power developed by the car. The car later passes through a point B on the road with speed \(25 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). The car takes 17.8 s to travel from A to B .
  2. Find the distance AB .
OCR MEI Further Mechanics A AS 2019 June Q6
6 Three particles, A, B and C are in a straight line on a smooth horizontal surface.
The particles have masses \(5 \mathrm {~kg} , 3 \mathrm {~kg}\) and 1 kg respectively. Particles B and C are at rest. Particle A is projected towards B with a speed of \(u \mathrm {~ms} ^ { - 1 }\) and collides with B . The coefficient of restitution between A and B is \(\frac { 1 } { 3 }\). Particle B subsequently collides with C. The coefficient of restitution between B and C is \(\frac { 1 } { 3 }\).
  1. Determine whether any further collisions occur.
  2. Given that the loss of kinetic energy during the initial collision between A and B is 4.8 J , find the value of \(u\). \begin{figure}[h]
    \includegraphics[alt={},max width=\textwidth]{6b27d322-417e-4cea-85cc-65d3728173c8-5_607_501_294_301} \captionsetup{labelformat=empty} \caption{Fig. 7}
    \end{figure} Fig. 7 shows a uniform rod AB of length \(4 a\) and mass \(m\).
    The end A rests against a rough vertical wall. A light inextensible string is attached to the rod at B and to a point C on the wall vertically above A , where \(\mathrm { AC } = 4 a\). The plane ABC is perpendicular to the wall and the angle ABC is \(30 ^ { \circ }\). The system is in limiting equilibrium. Find the coefficient of friction between the wall and the rod. \section*{END OF QUESTION PAPER}
OCR MEI Further Mechanics A AS 2022 June Q1
1
  1. Fig. 1.1 and Fig. 1.2 show rigid rods with forces acting as marked. The diagrams are to scale, and in each figure the side length of a grid square is 1 metre. \begin{figure}[h]
    \includegraphics[alt={},max width=\textwidth]{d1ec7861-dc8b-450b-8e05-c70479ab0dc2-2_428_552_443_319} \captionsetup{labelformat=empty} \caption{Fig. 1.1}
    \end{figure} \begin{figure}[h]
    \includegraphics[alt={},max width=\textwidth]{d1ec7861-dc8b-450b-8e05-c70479ab0dc2-2_431_553_440_1005} \captionsetup{labelformat=empty} \caption{Fig. 1.2}
    \end{figure}
    • On the copy of Fig. 1.1 in the Printed Answer Booklet, add, to scale, a force so that the overall system represents an anti-clockwise couple of magnitude 24 Nm .
    • On the copy of Fig. 1.2 in the Printed Answer Booklet, add, to scale, a force so that the overall system represents a clockwise couple of magnitude 1 Nm .
    • Fig. 1.3 shows a rectangular lamina with two coplanar forces acting as marked. Each grid square has side length 1 m .
    \begin{figure}[h]
    \includegraphics[alt={},max width=\textwidth]{d1ec7861-dc8b-450b-8e05-c70479ab0dc2-2_561_761_1452_315} \captionsetup{labelformat=empty} \caption{Fig. 1.3}
    \end{figure} A third coplanar force, of magnitude \(T \mathrm {~N}\), acts at A so that the resultant force on the lamina is zero.
    (i) Calculate the value of \(T\).
    (ii) Determine the magnitude and direction of the couple represented by this system of three forces.
OCR MEI Further Mechanics A AS 2022 June Q2
2 Three forces, of magnitudes \(33 \mathrm {~N} , 45 \mathrm {~N}\) and \(P \mathrm {~N}\), act at a point in the directions shown in the diagram. The system is in equilibrium.
\includegraphics[max width=\textwidth, alt={}, center]{d1ec7861-dc8b-450b-8e05-c70479ab0dc2-3_501_703_342_239}
  1. Draw a triangle of forces for the system shown above. Your diagram should include the magnitudes of the forces ( \(33 \mathrm {~N} , 45 \mathrm {~N}\) and \(P \mathrm {~N}\) ) and angle \(\theta\).
  2. If \(P = 38\), find, in degrees, the value of \(\theta\).
  3. If \(\theta = 40 ^ { \circ }\), determine the possible values for \(P\).
OCR MEI Further Mechanics A AS 2022 June Q3
3 Fig. 3.1 shows a thin rectangular frame ABCD , with part of it filled by a triangular lamina ABD . \(\mathrm { AD } = 30 \mathrm {~cm}\) and \(\mathrm { AB } = x \mathrm {~cm}\). Together they form the composite structure S . The centre of mass of \(S\) lies at a point \(M , 16.5 \mathrm {~cm}\) from \(A D\) and 11.7 cm from \(A B\). \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{d1ec7861-dc8b-450b-8e05-c70479ab0dc2-4_572_953_450_242} \captionsetup{labelformat=empty} \caption{Fig. 3.1}
\end{figure} The frame and the triangular lamina are both uniform but made of different materials. The mass of the frame is 1.7 kg .
  1. Show that the triangular lamina has a mass of 3.3 kg .
  2. Determine the value of \(x\), correct to \(\mathbf { 3 }\) significant figures. One end of a light inextensible string is attached to S at D . The other end is attached to a fixed point on a vertical wall. For S to hang in equilibrium with AD vertical, a force of magnitude \(Q N\) is applied to S as shown in Fig. 3.2. The line of action of this force lies in the same plane as S . The string is taut and lies in the same plane as S at an angle \(\phi\) to the downward vertical. \begin{figure}[h]
    \includegraphics[alt={},max width=\textwidth]{d1ec7861-dc8b-450b-8e05-c70479ab0dc2-4_611_994_1756_242} \captionsetup{labelformat=empty} \caption{Fig. 3.2}
    \end{figure}
  3. By taking moments about D , show that \(Q = 50.5\), correct to 3 significant figures.
  4. Determine, in degrees, the value of \(\phi\).
OCR MEI Further Mechanics A AS 2022 June Q4
4 The diagram shows two points A and B on a snowy slope. A is a vertical distance of 25 m above B.
\includegraphics[max width=\textwidth, alt={}, center]{d1ec7861-dc8b-450b-8e05-c70479ab0dc2-5_220_1376_306_244} A rider and snowmobile, with a combined mass of 240 kg , start at the top of the slope, heading in the direction of \(B\). As the snowmobile passes \(A\), with a speed of \(3 \mathrm {~ms} ^ { - 1 }\), the rider switches off the engine so that the snowmobile coasts freely. When the snowmobile passes B, it has a speed of \(18 \mathrm {~ms} ^ { - 1 }\). The resistances to motion can be modelled as a single, constant force of magnitude 120 N .
  1. Calculate the distance the snowmobile travels from A to B. The rider now turns the snowmobile around and brings it back to B, so that it faces up the slope. Starting from rest, the snowmobile ascends the slope so that it passes A with a speed of \(7 \mathrm {~ms} ^ { - 1 }\). It takes 30 seconds for the snowmobile to travel from B to A. The resistances to motion can still be modelled as a single, constant force of magnitude 120 N .
  2. Show that the snowmobile develops an average power of 2856 W during this time. The snowmobile can develop a maximum power of 6000 W . At a later point in the journey, the rider and snowmobile reach a different slope inclined at \(12 ^ { \circ }\) to the horizontal. The resistances to motion can still be modelled as a single, constant force of magnitude 120 N .
  3. Determine the maximum speed with which the rider and snowmobile can ascend. The power developed by a vehicle is sometimes given in the non-SI unit mechanical horsepower \(( \mathrm { hp } ) .1 \mathrm { hp }\) is the power required to lift 550 pounds against gravity, starting and ending at rest, by 1 foot in 1 second.
  4. Given that 1 metre \(\approx 3.28\) feet and \(1 \mathrm {~kg} \approx 2.2\) pounds, determine the number of watts that are equivalent to 1 hp .
OCR MEI Further Mechanics A AS 2022 June Q5
5 Fig. 5.1 shows a small smooth sphere A at rest on a smooth horizontal surface. At both ends of the surface is a smooth vertical wall. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{d1ec7861-dc8b-450b-8e05-c70479ab0dc2-6_97_1307_351_242} \captionsetup{labelformat=empty} \caption{Fig. 5.1}
\end{figure} Sphere A is projected directly towards the left-hand wall at a speed of \(5 \mathrm {~ms} ^ { - 1 }\). Sphere A collides directly with the left-hand wall, rebounds, then collides directly with the right-hand wall. After this second collision A has a speed of \(3.2 \mathrm {~ms} ^ { - 1 }\).
  1. Explain how it can be deduced that the collision between A and the left-hand wall was not inelastic. The coefficient of restitution between A and each wall is \(e\).
  2. Calculate the value of \(e\). Sphere A is now brought to rest and a second identical sphere B is placed on the surface. The surface is 1 m long, and A and B are positioned so that they are both 0.5 m from each wall, as shown in Fig. 5.2. \begin{figure}[h]
    \includegraphics[alt={},max width=\textwidth]{d1ec7861-dc8b-450b-8e05-c70479ab0dc2-6_241_1307_1322_242} \captionsetup{labelformat=empty} \caption{Fig. 5.2}
    \end{figure} Sphere A is projected directly towards the left-hand wall at a speed of \(0.2 \mathrm {~ms} ^ { - 1 }\). At the same time, B is projected directly towards the right-hand wall at a speed of \(0.3 \mathrm {~ms} ^ { - 1 }\). You may assume that the duration of impact of a sphere and a wall is negligible.
  3. Calculate the distance of A and B from the left-hand wall when they meet again.
OCR MEI Further Mechanics A AS 2022 June Q6
6 A block B of mass \(m \mathrm {~kg}\) rests on a rough slope inclined at angle \(\alpha\) to the horizontal. The coefficient of friction between \(B\) and the slope is \(\frac { 5 } { 9 }\).
  1. When B is in limiting equilibrium, show that \(\tan \alpha = \frac { 5 } { 9 }\).
  2. If \(\alpha = 40 ^ { \circ }\), determine the acceleration of B down the slope. A horizontal force of magnitude \(P \mathrm {~N}\) is now applied to B , as shown in the diagram below. At first B is at rest.
    \includegraphics[max width=\textwidth, alt={}, center]{d1ec7861-dc8b-450b-8e05-c70479ab0dc2-7_381_410_689_242}
    \(P\) is gradually increased.
  3. Show that, for B to slide on the slope, $$\mathrm { P } \left( \cos \alpha - \frac { 5 } { 9 } \sin \alpha \right) > \mathrm { mg } \left( \frac { 5 } { 9 } \cos \alpha + \sin \alpha \right) .$$
  4. Determine, in degrees, the least value of \(\alpha\) for which B will not slide no matter how large \(P\) becomes.
OCR MEI Further Mechanics A AS 2022 June Q7
7 The diagram shows a cannon fixed to a trolley. The trolley runs on a smooth horizontal track.
\includegraphics[max width=\textwidth, alt={}, center]{d1ec7861-dc8b-450b-8e05-c70479ab0dc2-8_310_1086_296_520} A driver boards the trolley with two cannon balls. The combined mass of the trolley, driver, cannon and cannon balls is 320 kg . Each cannon ball has a mass of 5 kg . Initially the trolley is at rest. A force of 480 N acts on the trolley in the forward direction for 4 seconds.
    1. Calculate the magnitude of the impulse of the force on the trolley.
    2. Calculate the speed of the trolley after the force stops acting. The driver now fires a cannon ball horizontally in the backward direction. The cannon ball and cannon separate at a rate of \(90 \mathrm {~ms} ^ { - 1 }\).
  2. Show that, after the firing of the cannon ball, the trolley moves with a speed of \(7.41 \mathrm {~ms} ^ { - 1 }\), correct to \(\mathbf { 3 }\) significant figures. The driver now reverses the direction of the cannon and fires the second cannon ball horizontally in the forward direction. Again, the cannon ball and cannon separate at a rate of \(90 \mathrm {~ms} ^ { - 1 }\).
  3. Calculate the overall percentage change in the kinetic energy of the trolley (alone) from before the first cannon ball is fired to after the second is fired, giving your answer correct to \(\mathbf { 2 }\) decimal places. You should make clear whether the change in kinetic energy is a gain or a loss.
  4. Give a reason why one of the modelling assumptions that was required in answering parts (a), (b) and (c) may not have been appropriate. \section*{END OF QUESTION PAPER}
OCR MEI Further Mechanics A AS 2023 June Q1
1 Throughout all parts of this question, the resistance to the motion of a car has magnitude \(\mathrm { kv } ^ { 2 } \mathrm {~N}\), where \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\) is the speed of the car and \(k\) is a constant. At first, the car travels along a straight horizontal road with constant speed \(20 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). The power developed by the car at this speed is 5000 W .
  1. Show that \(k = \frac { 5 } { 8 }\).
  2. Find the power the car must develop in order to maintain a constant speed of \(28 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) when travelling along the same horizontal road. The car climbs a hill which is inclined at an angle of \(2 ^ { \circ }\) to the horizontal. The power developed by the car is 13000 W , and the car has a constant speed of \(20 \mathrm {~m} \mathrm {~s} ^ { - 1 }\).
  3. Determine the mass of the car.
OCR MEI Further Mechanics A AS 2023 June Q2
2 A ball P of mass \(m \mathrm {~kg}\) is held at a height of 12.8 m above a horizontal floor. P is released from rest and rebounds from the floor. After the first bounce, P reaches a maximum height of 5 m above the floor. Two models, A and B , are suggested for the motion of P .
Model A assumes that air resistance may be neglected.
  1. Determine, according to model A , the coefficient of restitution between P and the floor. Model B assumes that the collision between P and the floor is perfectly elastic, but that work is done against air resistance at a constant rate of \(E\) joules per metre.
  2. Show that, according to model \(\mathrm { B } , \mathrm { E } = \frac { 39 } { 89 } \mathrm { mg }\).
  3. Show that both models predict that P will attain the same maximum height after the second bounce.
OCR MEI Further Mechanics A AS 2023 June Q3
3 The time period \(T\) of a satellite in circular orbit around a planet satisfies the equation
\(G M T ^ { 2 } = 4 \pi ^ { 2 } R ^ { 3 }\),
where
  • \(G\) is the universal gravitational constant,
  • \(M\) is the mass of the planet,
  • \(\quad R\) is the radius of the orbital circle.
    1. Find the dimensions of \(G\).
A student suggests the following formula to model the approach speed between two orbiting bodies.
\(v = k G { } ^ { \alpha } { } ^ { \beta } { } _ { r } \gamma _ { m _ { 1 } } m _ { 2 } \left( m _ { 1 } + m _ { 2 } \right)\),
where
  • \(\quad v\) is the approach speed of the two bodies,
  • \(k\) is a dimensionless constant,
  • \(\quad c\) is the speed of light,
  • \(\quad r\) is the distance between the two bodies,
  • \(\quad m _ { 1 }\) and \(m _ { 2 }\) are the masses of the bodies.
  • Use dimensional analysis to determine the values of \(\alpha , \beta\) and \(\gamma\).
  • Calculate, according to the student's model, how many times greater the approach speed is between a pair of stars which are 6.13 light-years apart and the same pair of stars if they were 8.64 light-years apart. (A light-year is a unit of distance.)
OCR MEI Further Mechanics A AS 2023 June Q4
4 The diagram shows three beads, A, B and C, of masses \(0.3 \mathrm {~kg} , 0.5 \mathrm {~kg}\) and 0.7 kg respectively, threaded onto a smooth wire circuit consisting of two straight and two semi-circular sections. The circuit occupies a vertical plane, with the two straight sections horizontal and the upper section 0.45 m directly above the lower section.
\includegraphics[max width=\textwidth, alt={}, center]{a87d62b8-406d-44cd-9ffa-384005329566-5_361_961_450_248} Initially, the beads are at rest. A and B are each given an impulse so that they move towards each other, A with a speed of \(8 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) and B with a speed of \(1.6 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). In the subsequent collision between A and \(\mathrm { B } , \mathrm { A }\) is brought to rest.
  1. Show that the coefficient of restitution between A and B is \(\frac { 1 } { 3 }\). Bead B next collides with C.
  2. Show that the speed of B before this collision is \(4.37 \mathrm {~m} \mathrm {~s} ^ { - 1 }\), correct to 3 significant figures. In this collision between B and C , B is brought to rest.
  3. Determine whether C next collides with A or with B .
  4. Explain why, if B has a greater mass than C , B could not be brought to rest in their collision.
OCR MEI Further Mechanics A AS 2023 June Q5
5 Fig. 5.1 shows the uniform cross-section of a solid S which is formed from a cylinder by boring two cylindrical tunnels the entire way through the cylinder. The radius of S is 50 cm , and the two tunnels have radii 10 cm and 30 cm . The material making up \(S\) has uniform density.
Coordinates refer to the axes shown in Fig. 5.1 and the units are centimetres. \begin{figure}[h]
\captionsetup{labelformat=empty} \caption{Fig. 5.1} \includegraphics[alt={},max width=\textwidth]{a87d62b8-406d-44cd-9ffa-384005329566-6_684_666_708_278}
\end{figure} The centre of mass of \(S\) is ( \(\mathrm { x } , \mathrm { y }\) ).
  1. Show that \(\bar { x } = 12\) and find the value of \(\bar { y }\). Solid \(S\) is placed onto two rails, \(A\) and \(B\), whose point of contacts with \(S\) are at ( \(- 30 , - 40\) ) and \(( 30 , - 40 )\) as shown in Fig. 5.2. Two points, \(\mathrm { P } ( 0,50 )\) and \(\mathrm { Q } ( 0 , - 50 )\), are marked on Fig. 5.2. \begin{figure}[h]
    \captionsetup{labelformat=empty} \caption{Fig. 5.2} \includegraphics[alt={},max width=\textwidth]{a87d62b8-406d-44cd-9ffa-384005329566-6_654_640_1875_251}
    \end{figure} At first, you should assume that the contact between S and the two rails is smooth.
  2. Determine the angle PQ makes with the vertical, after S settles into equilibrium. For the remainder of the question, you should assume that the contact between S and A is rough, that the contact between \(S\) and \(B\) is smooth, and that \(S\) does not move when placed on the rails. Fig. 5.3 shows only the forces exerted on S by the rails. The normal contact forces exerted by A and B on S have magnitude \(R _ { \mathrm { A } } \mathrm { N }\) and \(R _ { \mathrm { B } } \mathrm { N }\) respectively. The frictional force exerted by A on S has magnitude \(F\) N. \begin{figure}[h]
    \captionsetup{labelformat=empty} \caption{Fig. 5.3} \includegraphics[alt={},max width=\textwidth]{a87d62b8-406d-44cd-9ffa-384005329566-7_652_641_593_248}
    \end{figure} The weight of S is \(W \mathrm {~N}\).
  3. By taking moments about the origin, express \(F\) in the form \(\lambda W\), where \(\lambda\) is a constant to be determined.
  4. Given that S is in limiting equilibrium, find the coefficient of friction between A and S .
OCR MEI Further Mechanics A AS 2023 June Q6
6 A uniform beam of length 6 m and mass 10 kg rests horizontally on two supports A and B , which are 3.8 m apart. A particle \(P\) of mass 4 kg is attached 1.95 m from one end of the beam (see Fig. 6.1). \begin{figure}[h]
\captionsetup{labelformat=empty} \caption{Fig. 6.1} \includegraphics[alt={},max width=\textwidth]{a87d62b8-406d-44cd-9ffa-384005329566-8_257_1079_447_246}
\end{figure} When A is \(x \mathrm {~m}\) from the end of the beam, the supports exert forces of equal magnitude on the beam.
  1. Determine the value of \(x\). P is now removed. The same beam is placed on the supports so that B is 0.7 m from the end of the beam. The supports remain 3.8 m apart (see Fig. 6.2). \begin{figure}[h]
    \captionsetup{labelformat=empty} \caption{Fig. 6.2} \includegraphics[alt={},max width=\textwidth]{a87d62b8-406d-44cd-9ffa-384005329566-8_296_1082_1162_246}
    \end{figure} The contact between A and the beam is smooth. The contact between B and the beam is rough, with coefficient of friction 0.4. A small force of magnitude \(T \mathrm {~N}\) is applied to one end of the beam. The force acts in the same vertical plane as the beam and the angle the force makes with the beam is \(60 ^ { \circ }\). As \(T\) is increased, forces \(\mathrm { T } _ { \mathrm { L } }\) and \(\mathrm { T } _ { \mathrm { S } }\) are defined in the following way.
    • \(\quad \mathrm { T } _ { \mathrm { L } }\) is the value of \(T\) at which the beam would start lifting, assuming that is not already sliding.
    • \(\quad \mathrm { T } _ { \mathrm { S } }\) is the value of \(T\) at which the beam would start sliding, assuming that it has not already lifted.
    • Show that \(\mathrm { T } _ { \mathrm { L } } = 49.1\), correct to 3 significant figures.
    • Determine whether the beam will first slide or lift.
    \section*{END OF QUESTION PAPER}