Questions — OCR MEI Further Mechanics Minor (42 questions)

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OCR MEI Further Mechanics Minor 2019 June Q1
1 Dilip and Anna are doing an experiment to find the power at which they each work when running up a staircase at school. The top of the staircase is a vertical distance of 16 m above the bottom of the staircase. Dilip, who has mass 75 kg , does the experiment first. Anna times him, and finds that he takes 5.6 seconds to run up the staircase.
  1. Find the average power generated by Dilip as he runs up the staircase. Anna, who has mass \(M \mathrm {~kg}\), then does the same experiment and runs up the staircase in 5.0 seconds. She works out that the average power she has generated is less than the corresponding value for Dilip.
  2. Find an inequality satisfied by \(M\). Gareth, who also has mass 75 kg , says that members of his sports club do an exercise similar to this, but they run up a 16 m high sand dune. Gareth can run up the sand dune in 8.4 seconds, but he claims that he generates more power than Dilip.
  3. Give a reason why Gareth's claim could be true.
OCR MEI Further Mechanics Minor 2019 June Q2
4 marks
2
  1. Write down the dimensions of pressure. The SI unit of pressure is the pascal (Pa). 15 Pa is equivalent to \(Q\) newtons per square centimetre.
  2. Find the value of \(Q\). Simon thinks the speed, \(v\), of sound in a gas is given by the formula
    \(v = k P ^ { x } d ^ { y } V ^ { z }\),
    where \(P\) is the pressure of the gas,
    \(d\) is the density of the gas,
    \(V\) is the volume of the gas,
    \(k\) is a dimensionless constant.
  3. Use dimensional analysis to
    • find the values of \(x\) and \(y\) and
    • show that \(z = 0\).
      [0pt] [4]
      At normal atmospheric pressure the density of air at sea level is \(1.29 \mathrm {~kg} \mathrm {~m} ^ { - 3 }\). Under the same conditions the density of helium is \(0.166 \mathrm {~kg} \mathrm {~m} ^ { - 3 }\).
    • Given that the speed of sound in air under these conditions is \(340 \mathrm {~m} \mathrm {~s} ^ { - 1 }\), use Simon's formula to find the speed of sound in helium under the same conditions.
OCR MEI Further Mechanics Minor 2019 June Q3
3 Two identical uniform rectangular laminas, P and Q , each having length \(k a\) and width \(a\) are fixed together, in the same plane, to form a lamina R.
With reference to coordinate axes, the corners of P are at ( 0,0 ), ( \(k a , 0\) ), ( \(k a , a\) ) and ( \(0 , a\) ) and the corners of Q are at \(( k a , 0 ) , ( k a + a , 0 ) , ( k a + a , k a )\) and \(( k a , k a )\), as shown in Fig. 3. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{3b808042-95b8-4862-8355-3979c1981089-3_704_1102_459_244} \captionsetup{labelformat=empty} \caption{Fig. 3}
\end{figure} Determine the range of values of \(k\) for which the centre of mass of R lies outside the boundary of R.
OCR MEI Further Mechanics Minor 2019 June Q4
4 Two model railway trucks, A of mass 0.1 kg and B of mass 0.2 kg , are constrained to move on a smooth straight level track.
Initially B is stationary and A is moving towards B with speed \(u \mathrm {~m} \mathrm {~s} ^ { - 1 }\) before they collide. The coefficient of restitution between A and B is \(e\).
  1. Find the speed of A and the speed of B after the collision, giving your answers in terms of \(e\) and \(u\).
  2. Show that the loss of kinetic energy in the collision is \(\frac { 1 } { 30 } u ^ { 2 } \left( 1 - e ^ { 2 } \right)\).
  3. For the case in which the loss of kinetic energy is least
    • state the value of \(e\)
    • state the loss in kinetic energy
    • describe the subsequent motion of the trucks.
    • For the case in which the loss of kinetic energy is greatest
    • state the value of \(e\)
    • state the loss in kinetic energy
    • describe the subsequent motion of the trucks.
OCR MEI Further Mechanics Minor 2019 June Q5
5 Jack and Jemima are pulling a boat along a straight level canal.
The resistance to the motion of the boat is modelled as constant and equal to 1200 N .
Jack and Jemima walk in the same direction on paths on opposite sides of the canal. They each walk forwards at the same steady speed, keeping level with each other so that the distance between them is always 6 m . Jack and Jemima each pull a long light inextensible rope attached to the boat; initially they hold their ropes so the distance from each of them to the boat is 5 m , as shown in Fig. 5.1. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{3b808042-95b8-4862-8355-3979c1981089-4_417_1109_605_246} \captionsetup{labelformat=empty} \caption{Fig. 5.1}
\end{figure}
  1. Explain why the tension will be the same in each rope.
  2. Find the tension in each rope. Jemima then gradually releases more rope, so that the distance between her and the boat is 7 m . Jack and Jemima continue to walk at the same steady speed along the paths, but the position of the boat changes so that Jemima's rope makes an angle of \(\theta\) with the path and Jack's rope makes an angle of \(\phi\) with the path, as shown in Fig. 5.2. \begin{figure}[h]
    \includegraphics[alt={},max width=\textwidth]{3b808042-95b8-4862-8355-3979c1981089-4_513_1109_1610_246} \captionsetup{labelformat=empty} \caption{Fig. 5.2}
    \end{figure}
  3. - Show that \(\sin \phi = \frac { 1 } { 5 }\).
    • Show that \(\sin \theta = \frac { 5 } { 7 }\).
    • Find the tension in each rope in this new equilibrium position.
    • Without further calculation, state the effect on the tensions in the ropes if Jack now lengthens his rope to 7 m , the same length as Jemima's rope.
    • Suggest how the modelling assumption used in this question could be improved.
OCR MEI Further Mechanics Minor 2019 June Q6
6 A uniform solid cylinder, L, has base radius 5 cm , height 24 cm and mass 5 kg . L is placed on a rough plane inclined at an angle \(\alpha\) to the horizontal, as shown in Fig. 6. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{3b808042-95b8-4862-8355-3979c1981089-5_431_951_351_242} \captionsetup{labelformat=empty} \caption{Fig. 6}
\end{figure}
  1. On the copy of Fig. 6 in the Printed Answer Booklet mark the forces acting on L . The coefficient of friction between L and the plane is 0.3 . Initially \(\alpha\) is \(15 ^ { \circ }\).
  2. Show that L rests in equilibrium on the plane. A couple is applied to L . It is given that L will topple if the couple is applied in an anticlockwise sense, but L will not topple if the couple is applied in a clockwise sense.
  3. Find the range of possible values of the magnitude of the couple. The couple is now removed and the plane is slowly tilted so that \(\alpha\) increases.
  4. Determine whether L topples first without sliding or slides first without toppling.
OCR MEI Further Mechanics Minor 2022 June Q1
1 Newton's gravitational constant, \(G\), is approximately \(6.67 \times 10 ^ { - 11 } \mathrm {~N} \mathrm {~m} ^ { 2 } \mathrm {~kg} ^ { - 2 }\).
  1. Find the dimensions of \(G\). The escape velocity, \(v\), of a body from a planet's surface, is given by the formula \(\mathrm { v } = \mathrm { kG } ^ { \alpha } \mathrm { M } ^ { \beta } \mathrm { r } ^ { \gamma }\),
    where \(M\) is the planet's mass, \(r\) is the planet's radius and \(k\) is a dimensionless constant.
  2. Use dimensional analysis to find \(\alpha , \beta\) and \(\gamma\).
OCR MEI Further Mechanics Minor 2022 June Q2
2 The diagram below shows the cross-section through the centre of mass of a uniform block of weight \(W \mathrm {~N}\), resting on a slope inclined at an angle \(\alpha\) to the horizontal. The cross-section is a rectangle ABCD . The slope exerts a frictional force of magnitude \(F \mathrm {~N}\) and a normal contact force of magnitude \(R \mathrm {~N}\).
\includegraphics[max width=\textwidth, alt={}, center]{9b624694-edb6-4000-838f-3557e078952d-3_546_940_450_242}
  1. Explain why a triangle of forces may be used to model the scenario.
  2. In the space provided in the Printed Answer Booklet, draw such a triangle, fully annotated, including the angle \(\alpha\) in the correct position. The coefficient of friction between the block and the slope is \(\mu\).
  3. Given that the block is in limiting equilibrium, use your diagram in part (b) to show that \(\mu = \tan \alpha\). It is given that \(\mathrm { AB } = 8.9 \mathrm {~cm}\) and \(\mathrm { AD } = 11.6 \mathrm {~cm}\). The coefficient of friction between the slope and the block is 1.35 . The slope is slowly tilted so that \(\alpha\) increases.
  4. Determine whether the block topples first without sliding or slides first without toppling.
OCR MEI Further Mechanics Minor 2022 June Q3
3 A rough circular hoop, with centre O and radius 1 m , is fixed in a vertical plane. A , B and C are points on the hoop such that A and C are at the same horizontal level as O , and OB makes an angle of \(25 ^ { \circ }\) above the horizontal, as shown in the diagram.
\includegraphics[max width=\textwidth, alt={}, center]{9b624694-edb6-4000-838f-3557e078952d-4_650_729_404_251} A bead P of mass 0.3 kg is threaded onto the hoop. P is projected vertically downwards from A on two separate occasions.
  • The first time, when P is projected with a speed of \(4 \mathrm {~m} \mathrm {~s} ^ { - 1 }\), it first comes to rest at B .
  • The second time, when P is projected with a speed of \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\), it first comes to rest at C .
The situation is modelled by assuming that during the motion of P the magnitude of the frictional force exerted by the hoop on P is constant.
  1. Determine the value of \(v\).
  2. Comment on the validity of the modelling assumption used in this question.
OCR MEI Further Mechanics Minor 2022 June Q4
4 A uniform beam AB of mass 6 kg and length 5 m rests with its end A on smooth horizontal ground and its end B against a smooth vertical wall. The vertical distance between the ground and B is 4 m , and the angle between the beam and the downward vertical is \(\theta\). To prevent the beam from sliding, one end of a light taut rope of length 2 m is attached to the beam at C and the other end of the rope is attached to a point on the wall 2 m above the ground, as shown in the diagram.
\includegraphics[max width=\textwidth, alt={}, center]{9b624694-edb6-4000-838f-3557e078952d-5_558_556_500_251}
  1. By considering the value of \(\cos \theta\), determine the distance BC . An object of mass 75 kg is placed on the beam at a point which is \(x \mathrm {~m}\) from A . It is given that the tension in the rope is \(T \mathrm {~N}\) and the magnitude of the normal contact force between the ground and the beam is \(R \mathrm {~N}\).
  2. By taking moments about B for the beam, show that \(25 \mathrm { R } + 3675 \mathrm { x } - 16 \mathrm {~T} = 19110\).
  3. Given that the rope can withstand a maximum tension of 720 N , determine the largest possible value of \(x\).
OCR MEI Further Mechanics Minor 2022 June Q5
5 Point A lies 20 m vertically below a point B . A particle P of mass 4 m kg is projected upwards from A , at a speed of \(17.5 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). At the same time, a particle Q of mass \(m \mathrm {~kg}\) is released from rest at point B . The particles collide directly, and it is given that the coefficient of restitution in the collision between P and Q is 0.6 .
  1. Show that, immediately after the collision, P continues to travel upwards at \(0.7 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) and determine, at this time, the corresponding velocity of Q . In another situation, a particle of mass \(3 m \mathrm {~kg}\) is released from rest and falls vertically. After it has fallen 10 m , it explodes into two fragments. Immediately after the explosion, the lower fragment, of mass \(2 m \mathrm {~kg}\), moves vertically downwards with speed \(v _ { 1 } \mathrm {~m} \mathrm {~s} ^ { - 1 }\), and the upper fragment, of mass \(m \mathrm {~kg}\), moves vertically upwards with speed \(v _ { 2 } \mathrm {~m} \mathrm {~s} ^ { - 1 }\).
  2. Given that, in the explosion, the kinetic energy of the system increases by \(72 \%\), show that \(2 v _ { 1 } ^ { 2 } + v _ { 2 } ^ { 2 } = 1011.36\).
  3. By finding another equation connecting \(v _ { 1 }\) and \(v _ { 2 }\), determine the speeds of the fragments immediately after the explosion.
OCR MEI Further Mechanics Minor 2022 June Q6
6 Fig. 6.1 shows a light rod ABC , of length 60 cm , where B is the midpoint of AC . Particles of masses \(3.5 \mathrm {~kg} , 1.4 \mathrm {~kg}\) and 2.1 kg are attached to \(\mathrm { A } , \mathrm { B }\) and C respectively. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{9b624694-edb6-4000-838f-3557e078952d-7_241_1056_367_251} \captionsetup{labelformat=empty} \caption{Fig. 6.1}
\end{figure} The centre of mass is located at a point G along the rod.
  1. Determine the distance AG . Two light inextensible strings, each of length 40 cm , are attached to the rod, one at A , the other at C. The other ends of these strings are attached to a fixed point D. The rod is allowed to hang in equilibrium.
  2. Determine the angle AD makes with the vertical. The two strings are now replaced by a single light inextensible string of length 80 cm . One end of the string is attached to A and the other end of the string is attached to C. The string passes over a smooth peg fixed at D. The rod hangs in equilibrium, but is not vertical, as shown in Fig. 6.2. Fig. 6.2
  3. Explain why angle ADG and angle CDG must be equal.
  4. Determine the tension in the string.
OCR MEI Further Mechanics Minor 2023 June Q1
1
  1. State the dimensions of the following quantities.
    • Force
    • Velocity
    • Density
    A student investigating the drag force \(F\) experienced by an object moving through air conjectures the formula
    \(\mathrm { F } = \mathrm { ku } ^ { 2 } \left( \rho \mathrm {~m} ^ { 2 } \right) ^ { \frac { 1 } { 3 } }\),
    where
    • \(k\) is a dimensionless constant
    • \(u\) is the air velocity relative to the moving object
    • \(\rho\) is the air density
    • \(m\) is the mass of the object.
    • Show that the student's formula is dimensionally consistent.
    The student carries out experiments in an airflow tunnel. When the air density is doubled, the drag force is found to double as well, with all other conditions remaining the same.
  2. Show that the student's formula is inconsistent with the experimental observation. The student's teacher suggests revising the formula as
    \(\mathrm { F } = \mathrm { k } \rho ^ { \alpha } \mathrm { u } ^ { \beta } \mathrm { A } ^ { \gamma }\)
    where \(m\) has been replaced by \(A\), the cross-sectional area of the object. The constant \(k\) is still dimensionless.
  3. Use dimensional analysis to determine the values of \(\alpha , \beta\) and \(\gamma\).
OCR MEI Further Mechanics Minor 2023 June Q2
2 A car of mass 1400 kg , travels along a straight horizontal road AB , after which it descends a hill BC inclined at a constant angle of \(7 ^ { \circ }\) to the horizontal (see diagram). \(\mathrm { A } , \mathrm { B }\) and C all lie in the same vertical plane. Throughout the entire journey, the total resistance to the car's motion is constant.
\includegraphics[max width=\textwidth, alt={}, center]{cad8805d-59f6-4ed2-81f4-9e8c749461f5-3_232_1227_392_251} Between A and B, the car moves at a constant speed of \(12 \mathrm {~m} \mathrm {~s} ^ { - 1 }\), and the power developed by the car is a constant \(P \mathrm {~W}\). When the car reaches B , the engine is switched off and the car travels down a line of greatest slope from \(B\) to \(C\) with an acceleration of \(0.8 \mathrm {~m} \mathrm {~s} ^ { - 2 }\). The resistance to motion is unchanged.
  1. Determine the value of \(P\). When the car reaches C it turns round and travels back up the hill towards B at a constant speed of \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\). The power developed by the car between C and B is a constant 16 kW . The resistance to motion is unchanged.
  2. Determine the value of \(v\).
OCR MEI Further Mechanics Minor 2023 June Q3
3 The diagram shows two blocks P and Q of masses 0.5 kg and 2 kg respectively, on a horizontal surface. The points \(\mathrm { A } , \mathrm { B }\) and C lie on the surface in a straight line. There is a wall at C . The surface between B and C is smooth, and the surface between A and B is rough, such that the coefficient of friction between P and AB is \(\frac { 2 } { 3 }\).
\includegraphics[max width=\textwidth, alt={}, center]{cad8805d-59f6-4ed2-81f4-9e8c749461f5-3_229_1271_1601_278} P is projected with a speed of \(6 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) directly towards Q , which is at rest. As a result of the collision between P and Q, P changes direction and subsequently comes to rest at A. You may assume that P only collides with Q once.
  1. Determine the coefficient of restitution between P and Q .
  2. Calculate the impulse exerted on P by Q during their collision. After colliding with P , Q strikes the wall, which is perpendicular to the direction of the motion of Q , and comes to rest exactly halfway between A and B . The collision between Q and the wall is perfectly elastic.
  3. Determine the coefficient of friction between Q and AB .
OCR MEI Further Mechanics Minor 2023 June Q4
4 The diagram shows two particles P and Q , of masses 10 kg and 5 kg respectively, which are attached to the ends of a light inextensible string. The string is taut and passes over a small smooth pulley. The pulley is fixed at the highest point A on a smooth curved surface, the vertical cross-section of which is a quadrant of a circle with centre O and radius 2 m . Particle Q hangs vertically below the pulley and P is in contact with the surface, where the angle AOP is equal to \(\theta ^ { \circ }\). The pulley, P and Q all lie in the same vertical plane.
\includegraphics[max width=\textwidth, alt={}, center]{cad8805d-59f6-4ed2-81f4-9e8c749461f5-4_499_492_559_251} Throughout this question you may assume that there are no resistances to the motion of either P or Q and the force acting on P due to the tension in the string is tangential to the curved surface at P .
  1. Given that P is in equilibrium at the point where \(\theta = \alpha\), determine the value of \(\alpha\). Particle P is now released from rest at the point on the surface where \(\theta = 35\), and starts to move downwards on the surface. In the subsequent motion it is given that P does not leave the surface.
  2. By considering energy, determine the speed of P at the instant when \(\theta = 45\).
  3. State one modelling assumption you have made in determining the answer to part (b).
OCR MEI Further Mechanics Minor 2023 June Q5
5 Fig. 5.1 shows a particle P, of mass 5 kg , and a particle Q, of mass 11 kg , which are attached to the ends of a light, inextensible string. The string is taut and passes over a small smooth pulley fixed to the ceiling. \begin{figure}[h]
\captionsetup{labelformat=empty} \caption{Fig. 5.1} \includegraphics[alt={},max width=\textwidth]{cad8805d-59f6-4ed2-81f4-9e8c749461f5-5_367_707_495_251}
\end{figure} When a force of magnitude \(H \mathrm {~N}\), acting at an angle \(\theta\) to the upward vertical, is applied to Q the particles hang in equilibrium, with the part of the string connecting the pulley to Q making an angle of \(40 ^ { \circ }\) with the upward vertical. It is given that the force acts in the same vertical plane in which the string lies.
  1. Determine the values of \(H\) and \(\theta\). Particle Q is now removed. The string is instead attached to one end of a uniform beam B of length 3 m and mass 7 kg . The other end of B is in contact with a rough horizontal floor. The situation is shown in Fig. 5.2. \begin{figure}[h]
    \captionsetup{labelformat=empty} \caption{Fig. 5.2} \includegraphics[alt={},max width=\textwidth]{cad8805d-59f6-4ed2-81f4-9e8c749461f5-5_504_978_1557_251}
    \end{figure} With B in equilibrium, at an angle \(\phi\) to the horizontal, the part of the string connecting the pulley to B makes an angle of \(30 ^ { \circ }\) with the upward vertical. It is given that the string and B lie in the same vertical plane.
  2. Determine the smallest possible value for the coefficient of friction between B and the floor.
  3. Determine the value of \(\phi\).
OCR MEI Further Mechanics Minor 2023 June Q6
6 In this question you may use the fact that the volume of a sphere of radius \(r\) is \(\frac { 4 } { 3 } \pi r ^ { 3 }\).
Fig. 6.1 shows a container in the shape of an open-topped cylinder. The cylinder has height 18 cm and radius 4 cm . The curved surface and the base can be modelled as uniform laminae with the same mass per unit area. The container rests on a horizontal surface. \begin{figure}[h]
\captionsetup{labelformat=empty} \caption{Fig. 6.1} \includegraphics[alt={},max width=\textwidth]{cad8805d-59f6-4ed2-81f4-9e8c749461f5-6_506_342_621_255}
\end{figure}
  1. Show that the centre of mass of the container lies 8.1 cm above its base. The mass of the container is 400 grams. Water is poured into the container to reach a height of \(h \mathrm {~cm}\) above the base. The centre of mass of the combined container and water lies \(y \mathrm {~cm}\) above the base. Water has a density of 1 gram per \(\mathrm { cm } ^ { 3 }\).
  2. In this question you must show detailed reasoning. By formulating an expression for \(y\) in terms of \(h\), determine the value of \(h\) for which \(y\) is lowest. More water is now poured into the container. A sphere of radius 3 cm is placed into the container, where it sinks to the bottom. The surface of the water is now 4.5 cm from the top of the container, as shown in Fig. 6.2. \begin{figure}[h]
    \captionsetup{labelformat=empty} \caption{Fig. 6.2} \includegraphics[alt={},max width=\textwidth]{cad8805d-59f6-4ed2-81f4-9e8c749461f5-6_432_355_2001_255}
    \end{figure}
  3. Show that the centre of mass of the water in the container lies 7.5 cm above the base of the container. The sphere has a density of 4 grams per \(\mathrm { cm } ^ { 3 }\).
    The centre of mass of the combined container, water and sphere lies \(z \mathrm {~cm}\) above the base.
  4. Determine the value of \(z\). \section*{END OF QUESTION PAPER}
OCR MEI Further Mechanics Minor 2024 June Q1
1 A car of mass 1500 kg travels along a horizontal straight road. There are no resistances to the car's motion. The power developed by the car as it increases its speed from \(20 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) to \(30 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) over \(t\) seconds is a constant 5000 W .
  1. Determine the value of \(t\).
  2. Find the acceleration of the car when its speed is \(25 \mathrm {~m} \mathrm {~s} ^ { - 1 }\).
OCR MEI Further Mechanics Minor 2024 June Q2
2
  1. State the dimensions of force. Use the following metric-imperial conversion factors for the rest of this question.
    • \(1 \mathrm {~kg} = 2.2 \mathrm { lb }\) (pounds)
    • \(1 \mathrm {~m} = 39.4 \mathrm { in }\) (inches)
    A unit of force used in the imperial system is the pound-force (lbf). 1 lbf is defined as the gravitational force exerted on 1 lb on the surface of the Earth.
  2. Show that 1 lbf is approximately equal to 4.45 N . The pascal (Pa) is a unit of pressure equivalent to 1 Newton per square metre. Pressure can also be measured in pound-force per square inch (psi). A diver, at a depth of 40 m , experiences a typical pressure of \(5 \times 10 ^ { 5 } \mathrm {~Pa}\).
  3. Determine whether this is greater or less than the pressure in a bicycle tyre of 80 psi . In various physical contexts, energy density is the amount of energy stored in a given region of space per unit volume.
  4. Show that energy density and pressure are dimensionally equivalent.
OCR MEI Further Mechanics Minor 2024 June Q3
3 The diagram shows the three points A, B and C that lie along a line of greatest slope on a rough plane which is inclined at an angle of \(25 ^ { \circ }\) to the horizontal.
\includegraphics[max width=\textwidth, alt={}, center]{0a790ad0-7eda-40f1-9894-f156766ae46f-3_392_1136_383_242} A block of mass 6 kg is placed at B and is projected up the plane towards C with an initial speed of \(u \mathrm {~m} \mathrm {~s} ^ { - 1 }\). The block travels 3.5 m before coming instantaneously to rest at C , before sliding back down the plane. When the block is sliding back down the plane it attains its initial speed at A , which lies \(x \mathrm {~m}\) down the plane from B . It is given that the work done against resistance throughout the motion is 4 joules per metre.
  1. Use an energy method to determine the following.
    1. The value of \(u\)
    2. The value of \(x\) A student claims that half of the energy lost due to resistances is accounted for by friction between the block and the plane, and the other half by air resistance.
  2. Assuming that the student's claim is correct, determine the coefficient of friction between the block and the plane.
OCR MEI Further Mechanics Minor 2024 June Q4
4 Fig. 4.1 shows two spheres, A and B, on a smooth horizontal surface. Their masses are 3 kg and 1 kg respectively. \begin{figure}[h]
\captionsetup{labelformat=empty} \caption{Fig. 4.1} \includegraphics[alt={},max width=\textwidth]{0a790ad0-7eda-40f1-9894-f156766ae46f-4_158_1153_436_246}
\end{figure} Initially, sphere A travels at a speed of \(1 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) in a straight line towards B , which is at rest. The spheres collide and the coefficient of restitution between A and B is \(e\).
  1. Show that, after the collision, A has a speed of \(\frac { 1 } { 4 } ( 3 - e ) \mathrm { m } \mathrm { s } ^ { - 1 }\), and find an expression for the speed of B in terms of \(e\). During the collision, the kinetic energy of the system decreases by \(21 \%\).
  2. Determine the value of \(e\).
  3. State why in part (a) it was necessary to assume that A and B have equal radii. Fig. 4.2 shows two spheres, C and D , of equal radii on a smooth horizontal surface. Their masses are 1 kg and 2 kg respectively. \begin{figure}[h]
    \captionsetup{labelformat=empty} \caption{Fig. 4.2} \includegraphics[alt={},max width=\textwidth]{0a790ad0-7eda-40f1-9894-f156766ae46f-4_158_1155_1544_244}
    \end{figure} Spheres C and D travel towards each other along the same straight line, C with a speed of \(u \mathrm {~m} \mathrm {~s} ^ { - 1 }\) and D with a speed of \(1 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). The spheres collide and during the collision C exerts an impulse on D of magnitude \(\frac { 2 } { 3 } ( u + 1 ) \mathrm { Ns }\).
  4. Show that C and D have the same velocity after the collision.
  5. Determine the fraction of kinetic energy lost due to the collision between C and D as \(u \rightarrow \infty\).
OCR MEI Further Mechanics Minor 2024 June Q5
5 A uniform lamina OABC is in the shape of a trapezium where O is the origin of the coordinate system in which the points \(\mathrm { A } , \mathrm { B }\) and C have coordinates \(( 120,0 )\), \(( 60,90 )\) and \(( 30,90 )\) respectively (see diagram). The units of the axes are centimetres.
\includegraphics[max width=\textwidth, alt={}, center]{0a790ad0-7eda-40f1-9894-f156766ae46f-5_561_720_404_242} The centre of mass of the lamina lies at ( \(\mathrm { x } , \mathrm { y }\) ).
  1. Show that \(\bar { x } = 54\) and determine the value of \(\bar { y }\). The lamina is placed horizontally so that it rests on three supports, whose points of contact are at \(\mathrm { B } , \mathrm { C }\) and D , where D lies on the edge OA and has coordinates \(( d , 0 )\).
  2. Determine the range of values of \(d\) for the lamina to rest in equilibrium. It is now given that \(d = 63\), and that the lamina has a weight of 100 N .
  3. Determine the forces exerted on the lamina by each of the supports at \(\mathrm { B } , \mathrm { C }\) and D .
OCR MEI Further Mechanics Minor 2024 June Q6
6 Fig. 6.1 shows three forces of magnitude \(15 \mathrm {~N} , 15 \mathrm {~N}\) and 30 N acting on a rigid beam AB of length 6 m . One of the forces of magnitude 15 N acts at A, and the other force of magnitude 15 N acts at B. The force of magnitude 30 N acts at distance of \(x \mathrm {~m}\) from B. All three forces act in a direction perpendicular to the beam as shown in Fig. 6.1. The beam and the three forces all lie in the same horizontal plane. The three forces form a couple of magnitude 42 Nm in the clockwise direction. \begin{figure}[h]
\captionsetup{labelformat=empty} \caption{Fig. 6.1} \includegraphics[alt={},max width=\textwidth]{0a790ad0-7eda-40f1-9894-f156766ae46f-6_504_433_591_246}
\end{figure}
  1. Determine the value of \(x\). Fig. 6.2 shows the same beam, without the three forces from Fig. 6.1, resting in limiting equilibrium against a step. The point of contact, C , between the beam and the edge of the step lies 1.5 m from A. The other end of the beam rests on a horizontal floor. The contacts between the beam and both the step and the floor are rough. \begin{figure}[h]
    \captionsetup{labelformat=empty} \caption{Fig. 6.2} \includegraphics[alt={},max width=\textwidth]{0a790ad0-7eda-40f1-9894-f156766ae46f-6_348_412_1633_244}
    \end{figure} It is given that the beam is non-uniform, and that its centre of mass lies \(\sqrt { 3 } \mathrm {~m}\) from B .
  2. Draw a diagram to show all the forces acting on the beam. The coefficient of friction between the beam and the step and the coefficient of friction between the beam and the floor are the same, and are denoted by \(\mu\).
    1. Show that \(\mu ^ { 2 } - 6 \mu + 1 = 0\).
    2. Hence determine the value of \(\mu\).
OCR MEI Further Mechanics Minor 2020 November Q1
1 A uniform solid rectangular prism has cross-section with width \(w \mathrm {~cm}\) and height 24 cm . Another uniform solid prism has cross-section in the shape of an isosceles triangle with width \(w \mathrm {~cm}\) and height \(h \mathrm {~cm}\). The prisms are both placed with their axes vertical on a rough horizontal plane (see Fig. 1.1, which shows the cross-sections through the centres of mass of both solids). \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{6418c1b7-092a-4747-bc88-1b57815c6ad9-2_520_1123_520_246} \captionsetup{labelformat=empty} \caption{Fig. 1.1}
\end{figure} The plane is slowly tilted and both solids remain in equilibrium until the angle of inclination of the plane reaches \(\alpha\), when both solids topple simultaneously.
  1. Determine the value of \(h\). It is given that \(w = 12\).
  2. Determine the value of \(\alpha\). Both prisms are made from the same material and are of uniform density. The triangular prism is now placed on top of the rectangular prism to form a composite body C such that the base of the triangular prism coincides with the top of the rectangular prism. A cross-section of C is shown in Fig. 1.2. \begin{figure}[h]
    \includegraphics[alt={},max width=\textwidth]{6418c1b7-092a-4747-bc88-1b57815c6ad9-2_777_439_1784_258} \captionsetup{labelformat=empty} \caption{Fig. 1.2}
    \end{figure}
  3. Determine the height of the centre of mass of C from its base.