Questions — OCR MEI (4456 questions)

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OCR MEI Further Mechanics A AS 2019 June Q3
7 marks Moderate -0.3
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
9 marks Standard +0.3
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
10 marks Standard +0.3
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
11 marks Standard +0.3
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
7 marks Moderate -0.3
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.
    1. Calculate the value of \(T\).
    2. Determine the magnitude and direction of the couple represented by this system of three forces.
OCR MEI Further Mechanics A AS 2022 June Q2
7 marks Standard +0.3
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
10 marks Standard +0.3
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
10 marks Standard +0.3
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
6 marks Standard +0.8
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
10 marks Standard +0.8
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
10 marks Moderate -0.3
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
7 marks Moderate -0.3
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
10 marks Standard +0.8
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
9 marks Moderate -0.3
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
OCR MEI Further Mechanics A AS 2023 June Q4
10 marks Standard +0.3
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
13 marks Standard +0.3
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
11 marks Standard +0.3
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.
    \section*{END OF QUESTION PAPER}
OCR MEI Further Mechanics A AS 2024 June Q1
4 marks Moderate -0.8
1 Two horizontal forces of magnitudes 7 N and 15 N act at a point O .
The 15 N force acts an angle of \(\theta ^ { \circ }\) above the positive \(x\)-axis.
The 7 N force acts at an angle of \(70 ^ { \circ }\) below the negative \(x\)-axis (see diagram). \includegraphics[max width=\textwidth, alt={}, center]{a96a0ebe-8f4f-4d79-9d11-9d348ef72314-2_606_773_402_239} The resultant of the two forces acts only in the positive \(x\)-direction.
  1. Calculate the value of \(\theta\).
  2. Calculate the magnitude of the resultant of the two forces.
OCR MEI Further Mechanics A AS 2024 June Q2
11 marks Moderate -0.8
2
  1. Find the dimensions of energy. The moment of inertia, \(I\), of a rigid body rotating about a fixed axis is measured in \(\mathrm { kg } \mathrm { m } ^ { 2 }\).
  2. State the dimensions of \(I\). The kinetic energy, \(E\), of a rigid body rotating about a fixed axis is given by the formula \(\mathrm { E } = \frac { 1 } { 2 } \mathrm { I } \omega ^ { 2 }\),
    where \(\omega\) is the angular velocity (angle per unit time) of the rigid body.
  3. Show that the formula for \(E\) is dimensionally consistent. When a rigid body is pivoted from one of its end points and allowed to swing freely, it forms a pendulum. The period, \(t\), of the pendulum is the time taken for it to complete one oscillation. A student conjectures the formula \(\mathrm { t } = \left. \mathrm { k } ( \mathrm { mg } ) ^ { \alpha } \mathrm { r } ^ { \beta } \right| ^ { \gamma }\),
    where
    The moment of inertia of a thin uniform rigid rod of mass 1.5 kg and length 0.8 m , rotating about one of its endpoints, is \(0.32 \mathrm {~kg} \mathrm {~m} ^ { 2 }\). The student suspends such a rod from one of its endpoints and allows it to swing freely. The student measures the period of this pendulum and finds that it is 1.47 seconds.
  4. Using the formula conjectured by the student, determine the value of \(k\).
OCR MEI Further Mechanics A AS 2024 June Q3
13 marks Standard +0.3
3 The diagram shows a uniform beam AB , of weight 80 N and length 7 m , resting in equilibrium in a vertical plane. The end A is in contact with a rough vertical wall, and the angle between the beam and the upward vertical is \(60 ^ { \circ }\). The beam is supported by a smooth peg at a point C , where \(\mathrm { AC } = 2 \mathrm {~m}\). \includegraphics[max width=\textwidth, alt={}, center]{a96a0ebe-8f4f-4d79-9d11-9d348ef72314-4_474_709_445_244}
  1. Complete the diagram in the Printed Answer Booklet to show all the forces acting on the beam.
    1. Show that the magnitude of the frictional force exerted on the beam by the wall is 25 N .
    2. Hence determine the magnitude of the total contact force exerted on the beam by the wall.
  3. Determine the direction of the total contact force exerted on the beam by the wall. The coefficient of friction between the beam and the wall is \(\mu\).
  4. Find the range of possible values for \(\mu\).
  5. Explain how your answer to part (b)(ii) would change if the peg were situated closer to A but the angle between the beam and the upward vertical remained at \(60 ^ { \circ }\).
OCR MEI Further Mechanics A AS 2024 June Q4
13 marks Standard +0.3
4 Three spheres A, B, and C, of equal radius are in the same straight line on a smooth horizontal surface. The masses of \(\mathrm { A } , \mathrm { B }\) and C are \(2 \mathrm {~kg} , 4 \mathrm {~kg}\) and 1 kg respectively. Initially the three spheres are at rest.
Spheres \(A\) and \(C\) are each given impulses so that \(A\) moves towards \(B\) with speed \(5 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) and C moves towards B with speed \(u \mathrm {~m} \mathrm {~s} ^ { - 1 }\) as shown in the diagram below. \includegraphics[max width=\textwidth, alt={}, center]{a96a0ebe-8f4f-4d79-9d11-9d348ef72314-5_325_1591_603_239} The coefficient of restitution between \(A\) and \(B\) is \(\frac { 4 } { 5 }\).
It is given that the first collision occurs between A and B .
  1. State how you can tell from the information given above that kinetic energy is lost when A collides with B .
  2. Show that the combined kinetic energy of A and B decreases by \(24 \%\) during their collision. Sphere B next collides with C. The coefficient of restitution between B and C is \(\frac { 2 } { 3 }\).
  3. Given that a third collision occurs, determine the range of possible values for \(u\).
  4. State one limitation of the model used in this question.
OCR MEI Further Mechanics A AS 2024 June Q5
9 marks Standard +0.3
5 In the diagram below, points \(\mathrm { A } , \mathrm { B }\) and C lie in the same vertical plane. The slope AB is inclined at an angle of \(30 ^ { \circ }\) to the horizontal and \(\mathrm { AB } = 5 \mathrm {~m}\). The point B is a vertical distance of 6.5 m above horizontal ground. The point C lies on the horizontal ground. \includegraphics[max width=\textwidth, alt={}, center]{a96a0ebe-8f4f-4d79-9d11-9d348ef72314-6_601_1285_395_244} Starting at A , a particle P , of mass \(m \mathrm {~kg}\), moves along the slope towards B , under the action of a constant force \(\mathbf { F }\). The force \(\mathbf { F }\) has a magnitude of 50 N and acts at an angle of \(\theta ^ { \circ }\) to AB in the same vertical plane as A and B . When P reaches \(\mathrm { B } , \mathbf { F }\) is removed, and P moves under gravity landing at C . It is given that
  • the speed of P at A is \(3 \mathrm {~m} \mathrm {~s} ^ { - 1 }\),
  • the speed of P at B is \(6 \mathrm {~ms} ^ { - 1 }\),
  • the speed of P at C is \(12 \mathrm {~m} \mathrm {~s} ^ { - 1 }\),
  • 58 J of work is done against non-gravitational resistances as P moves from A to B ,
  • 42 J of work is done against non-gravitational resistances as P moves from B to C .
    1. By considering the motion from B to C, show that \(m = 4.33\) correct to 3 significant figures.
    2. By considering the motion from A to B , determine the value of \(\theta\).
    3. Calculate the power of \(\mathbf { F }\) at the instant that P reaches B .
OCR MEI Further Mechanics A AS 2024 June Q6
10 marks Standard +0.8
6 A uniform lamina OABC is in the shape of a trapezium where O is the origin of the coordinate system in which the points \(A , B\) and \(C\) have coordinates \(( 12,0 ) , ( 12 + p , q )\) and \(( 0 , q )\) respectively. \includegraphics[max width=\textwidth, alt={}, center]{a96a0ebe-8f4f-4d79-9d11-9d348ef72314-7_536_917_349_239}
  1. Determine, in terms of \(p\) and \(q\), the coordinates of the centre of mass of OABC . The point D has coordinates \(( 7.6 , q )\). When OABC is suspended from D , the lamina hangs in equilibrium with BC horizontal.
  2. Determine the value of \(p\). When OABC is suspended from C, the lamina hangs in equilibrium with BC at an angle of \(35 ^ { \circ }\) to the downward vertical.
  3. Determine the value of \(q\), giving your answer correct to \(\mathbf { 3 }\) significant figures.
OCR MEI Further Mechanics A AS 2020 November Q1
4 marks Standard +0.3
1 Brent is riding his bicycle along a straight horizontal road.
While riding along this road Brent can attain a maximum speed of \(6.25 \mathrm {~ms} ^ { - 1 }\) and the wind resistance acting on Brent and his bicycle is constant and equal to 19.2 N . Brent and his bicycle have a combined mass of 72 kg . Brent later begins to ride up a hill which is inclined at an angle of \(3 ^ { \circ }\) to the horizontal.
Given that the wind resistance and the maximum power developed by the bicycle is unchanged, determine Brent's maximum speed up the hill.
OCR MEI Further Mechanics A AS 2020 November Q2
8 marks Standard +0.3
2 George is investigating the time it takes for a ball to reach a certain height when projected vertically upwards. George believes that the time, \(t\), for the ball to reach a certain height, \(h\), depends on
  • the ball's mass \(m\),
  • the projection speed \(u\), and
  • the height \(h\).
George suggests the following formula to model this situation \(t = k m ^ { \alpha } u ^ { \beta } h ^ { \gamma }\),
where \(k\) is a dimensionless constant.
  1. Use dimensional analysis to show that \(t = \frac { k h } { u }\).
  2. Hence explain why George's formula is unrealistic. Mandy argues that any model of this situation must consider the acceleration due to gravity, \(g\). She suggests the alternative formula \(t = \frac { u - \sqrt { u ^ { 2 } + g h } } { g }\).
  3. Show that Mandy's formula is dimensionally consistent.
  4. Explain why Mandy's formula is incorrect.