Questions M2 (1537 questions)

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OCR M2 2014 June Q7
12 marks Standard +0.8
7 \includegraphics[max width=\textwidth, alt={}, center]{5bfd0285-71cb-4dcb-8545-a379653f9a3e-4_529_403_264_829} A small smooth ring \(P\) of mass 0.4 kg is threaded onto a light inextensible string fixed at \(A\) and \(B\) as shown in the diagram, with \(A\) vertically above \(B\). The string is inclined to the vertical at angles of \(30 ^ { \circ }\) and \(45 ^ { \circ }\) at \(A\) and \(B\) respectively. \(P\) moves in a horizontal circle of radius 0.5 m about a point \(C\) vertically below \(B\).
  1. Calculate the tension in the string.
  2. Calculate the speed of \(P\). The end of the string at \(B\) is moved so both ends of the string are now fixed at \(A\).
  3. Show that, when the string is taut, \(A P\) is now 0.854 m correct to 3 significant figures. \(P\) moves in a horizontal circle with angular speed \(3.46 \mathrm { rad } \mathrm { s } ^ { - 1 }\).
  4. Find the tension in the string and the angle that the string now makes with the vertical.
OCR M2 2014 June Q8
12 marks Standard +0.3
8 A child is trying to throw a small stone to hit a target painted on a vertical wall. The child and the wall are on horizontal ground. The child is standing a horizontal distance of 8 m from the base of the wall. The child throws the stone from a height of 1 m with speed \(12 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) at an angle of \(20 ^ { \circ }\) above the horizontal.
  1. Find the direction of motion of the stone when it hits the wall. The child now throws the stone with a speed of \(\mathrm { Vm } \mathrm { s } ^ { - 1 }\) from the same initial position and still at an angle of \(20 ^ { \circ }\) above the horizontal. This time the stone hits the target which is 2.5 m above the ground.
  2. Find \(V\).
OCR M2 2015 June Q1
7 marks Moderate -0.3
1 A cyclist travels along a straight horizontal road. The total mass of the cyclist and her bicycle is 80 kg and the resistance to motion is a constant 60 N .
  1. The cyclist travels at a constant speed working at a constant rate of 480 W . Find the speed at which she travels.
  2. The cyclist now instantaneously increases her power to 600 W . After travelling at this power for 14.2 s her speed reaches \(9.4 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). Find the distance travelled at this power.
OCR M2 2015 June Q2
6 marks Moderate -0.8
2 A particle of mass 0.3 kg is attached to one end of a light inextensible string. The other end of the string is attached to a fixed point \(A\). The particle moves in a horizontal circle of radius 0.343 m , with centre vertically below \(A\), at a constant angular speed of \(6 \mathrm { rad } \mathrm { s } ^ { - 1 }\). Find the tension in the string and the angle at which the string is inclined to the vertical.
OCR M2 2015 June Q3
6 marks Standard +0.3
3 A car of mass 1500 kg travels along a straight horizontal road with its engine working at a constant rate of \(P \mathrm {~W}\). There is a constant resistance to motion of \(R \mathrm {~N}\). Points \(A\) and \(B\) are on the road. At point \(A\) the car's speed is \(16 \mathrm {~ms} ^ { - 1 }\) and its acceleration is \(0.3875 \mathrm {~ms} ^ { - 2 }\). At point \(B\) the car's speed is \(25 \mathrm {~ms} ^ { - 1 }\) and its acceleration is \(0.2 \mathrm {~ms} ^ { - 2 }\). Find the values of \(P\) and \(R\).
OCR M2 2015 June Q4
10 marks Standard +0.8
4 \includegraphics[max width=\textwidth, alt={}, center]{8492ec9b-3327-4d89-aaa4-bf98cdf0ebdc-2_721_513_1260_762} A uniform solid prism has cross-section \(A B C D E\) in the shape of a rectangle measuring 20 cm by 4 cm joined to a semicircle of radius 8 cm as shown in the diagram. The centre of mass of the solid lies in this cross-section.
  1. Find the distance of the centre of mass of the solid from \(A B\). The solid is placed with \(A E\) on rough horizontal ground (so the object does not slide) and is in equilibrium with a horizontal force of magnitude 4 N applied along \(C B\).
  2. Find the greatest and least possible values for the weight of the solid.
OCR M2 2015 June Q5
10 marks Standard +0.3
5 A small sphere of mass 0.2 kg is projected vertically downwards with a speed of \(5 \mathrm {~ms} ^ { - 1 }\) from a height of 1.6 m above horizontal ground. It hits the ground and rebounds vertically upwards coming to instantaneous rest at a height of \(h \mathrm {~m}\) above the ground. The coefficient of restitution between the sphere and the ground is 0.7 .
  1. Find \(h\).
  2. Find the magnitude and direction of the impulse exerted on the sphere by the ground.
  3. Find the loss of energy of the sphere between the instant of projection and the instant it comes to instantaneous rest at height \(h \mathrm {~m}\).
OCR M2 2015 June Q6
10 marks Standard +0.3
6 A particle is projected with speed \(v \mathrm {~ms} ^ { - 1 }\) from a point \(O\) on horizontal ground. The angle of projection is \(\theta ^ { \circ }\) above the horizontal. At time \(t\) seconds after the instant of projection the horizontal displacement of the particle from \(O\) is \(x \mathrm {~m}\) and the upward vertical displacement from \(O\) is \(y \mathrm {~m}\).
  1. Show that $$y = x \tan \theta - \frac { 4.9 x ^ { 2 } } { v ^ { 2 } \cos ^ { 2 } \theta } .$$ A stone is thrown from the top of a vertical cliff 100 m high. The initial speed of the stone is \(16 \mathrm {~ms} ^ { - 1 }\) and the angle of projection is \(\theta ^ { \circ }\) to the horizontal. The stone hits the sea 40 m from the foot of the cliff.
  2. Find the two possible values of \(\theta\). \includegraphics[max width=\textwidth, alt={}, center]{8492ec9b-3327-4d89-aaa4-bf98cdf0ebdc-3_623_995_1475_536} A uniform ladder \(A B\) of weight \(W \mathrm {~N}\) and length 4 m rests with its end \(A\) on rough horizontal ground and its end \(B\) against a smooth vertical wall. The ladder is inclined at an angle \(\theta\) to the horizontal where \(\tan \theta = \frac { 1 } { 2 }\) (see diagram). A small object \(S\) of weight \(2 W \mathrm {~N}\) is placed on the ladder at a point \(C\), which is 1 m from \(A\). The coefficient of friction between the ladder and the ground is \(\mu\) and the system is in limiting equilibrium.
OCR M2 2015 June Q8
12 marks Standard +0.3
8 \includegraphics[max width=\textwidth, alt={}, center]{8492ec9b-3327-4d89-aaa4-bf98cdf0ebdc-4_342_981_255_525} Two small spheres, \(A\) and \(B\), are free to move on the inside of a smooth hollow cylinder, in such a way that they remain in contact with both the curved surface of the cylinder and its horizontal base. The mass of \(A\) is 0.4 kg , the mass of \(B\) is 0.5 kg and the radius of the cylinder is 0.6 m (see diagram). The coefficient of restitution between \(A\) and \(B\) is 0.35 . Initially, \(A\) and \(B\) are at opposite ends of a diameter of the base of the cylinder with \(A\) travelling at a constant speed of \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\) and \(B\) stationary. The magnitude of the force exerted on \(A\) by the curved surface of the cylinder is 6 N .
  1. Show that \(v = 3\).
  2. Calculate the speeds of the particles after \(A\) 's first impact with \(B\). Sphere \(B\) is removed from the cylinder and sphere \(A\) is now set in motion with constant angular speed \(\omega \mathrm { rad } \mathrm { s } ^ { - 1 }\). The magnitude of the total force exerted on \(A\) by the cylinder is 4.9 N .
  3. Find \(\omega\). \section*{END OF QUESTION PAPER}
OCR M2 Specimen Q1
5 marks Moderate -0.8
1 \includegraphics[max width=\textwidth, alt={}, center]{b96a99a6-3df4-4000-9bf1-aab7ab954b4a-2_236_949_269_603} A barge \(B\) is pulled along a canal by a horse \(H\), which is on the tow-path. The barge and the horse move in parallel straight lines and the tow-rope makes a constant angle of \(15 ^ { \circ }\) with the direction of motion (see diagram). The tow-rope remains taut and horizontal, and has a constant tension of 500 N .
  1. Find the work done on the barge by the tow-rope, as the barge travels a distance of 400 m . The barge moves at a constant speed and takes 10 minutes to travel the 400 m .
  2. Find the power applied to the barge.
OCR M2 Specimen Q2
7 marks Standard +0.3
2 A uniform circular cylinder, of radius 6 cm and height 15 cm , is in equilibrium on a fixed inclined plane with one of its ends in contact with the plane.
  1. Given that the cylinder is on the point of toppling, find the angle the plane makes with the horizontal. The cylinder is now placed on a horizontal board with one of its ends in contact with the board. The board is then tilted so that the angle it makes with the horizontal gradually increases.
  2. Given that the coefficient of friction between the cylinder and the board is \(\frac { 3 } { 4 }\), determine whether or not the cylinder will slide before it topples, justifying your answer.
OCR M2 Specimen Q3
8 marks Standard +0.3
3 \includegraphics[max width=\textwidth, alt={}, center]{b96a99a6-3df4-4000-9bf1-aab7ab954b4a-2_389_698_1706_694} A uniform lamina \(A B C D\) has the shape of a square of side \(a\) adjoining a right-angled isosceles triangle whose equal sides are also of length \(a\). The weight of the lamina is \(W\). The lamina rests, in a vertical plane, on smooth supports at \(A\) and \(D\), with \(A D\) horizontal (see diagram).
  1. Show that the centre of mass of the lamina is at a horizontal distance of \(\frac { 11 } { 9 } a\) from \(A\).
  2. Find, in terms of \(W\), the magnitudes of the forces on the supports at \(A\) and \(D\).
OCR M2 Specimen Q4
8 marks Standard +0.3
4 \includegraphics[max width=\textwidth, alt={}, center]{b96a99a6-3df4-4000-9bf1-aab7ab954b4a-3_563_707_274_721} A rigid body \(A B C\) consists of two uniform rods \(A B\) and \(B C\), rigidly joined at \(B\). The lengths of \(A B\) and \(B C\) are 13 cm and 20 cm respectively, and their weights are 13 N and 20 N respectively. The distance of \(B\) from \(A C\) is 12 cm . The body hangs in equilibrium, with \(A C\) horizontal, from two vertical strings attached at \(A\) and \(C\). Find the tension in each string.
OCR M2 Specimen Q5
10 marks Standard +0.3
5 A cyclist and his machine have a combined mass of 80 kg . The cyclist ascends a straight hill \(A B\) of constant slope, starting from rest at \(A\) and reaching a speed of \(5 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) at \(B\). The level of \(B\) is 4 m above the level of \(A\).
  1. Find the gain in kinetic energy and the gain in gravitational potential energy of the cyclist and his machine. During the ascent the resistance to motion is constant and has magnitude 70 N .
  2. Given that the work done by the cyclist in ascending the hill is 8000 J , find the distance \(A B\). At \(B\) the cyclist is working at 720 watts and starts to move in a straight line along horizontal ground. The resistance to motion has the same magnitude of 70 N as before.
  3. Find the acceleration with which the cyclist starts to move horizontally.
OCR M2 Specimen Q6
10 marks Moderate -0.3
6 An athlete 'puts the shot' with an initial speed of \(19 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) at an angle of \(11 ^ { \circ }\) above the horizontal. At the instant of release the shot is 1.53 m above the horizontal ground. By treating the shot as a particle and ignoring air resistance, find
  1. the maximum height, above the ground, reached by the shot,
  2. the horizontal distance the shot has travelled when it hits the ground.
OCR M2 Specimen Q7
11 marks Standard +0.3
7 \includegraphics[max width=\textwidth, alt={}, center]{b96a99a6-3df4-4000-9bf1-aab7ab954b4a-4_314_757_285_708} A ball of mass 0.08 kg is attached by two strings to a fixed vertical post. The strings have lengths 2.5 m and 2.4 m , as shown in the diagram. The ball moves in a horizontal circle, of radius 2.4 m , with constant speed \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\). Each string is taut and the lower string is horizontal. The modelling assumptions made are that both strings are light and inextensible, and that there is no air resistance.
  1. Find the tension in each string when \(v = 10.5\).
  2. Find the least value of \(v\) for which the lower string is taut.
OCR M2 Specimen Q8
13 marks Standard +0.3
8 Two uniform smooth spheres, \(A\) and \(B\), have the same radius. The mass of \(A\) is 0.24 kg and the mass of \(B\) is \(m \mathrm {~kg}\). Sphere \(A\) is travelling in a straight line on a horizontal table, with speed \(8 \mathrm {~m} \mathrm {~s} ^ { - 1 }\), when it collides directly with sphere \(B\), which is at rest. As a result of the collision, sphere \(A\) continues in the same direction with a speed of \(6 \mathrm {~m} \mathrm {~s} ^ { - 1 }\).
  1. Find the magnitude of the impulse exerted by \(A\) on \(B\).
  2. Show that \(m \leqslant 0.08\). It is given that \(m = 0.06\).
  3. Find the coefficient of restitution between \(A\) and \(B\). On another occasion \(A\) and \(B\) are travelling towards each other, each with speed \(4 \mathrm {~m} \mathrm {~s} ^ { - 1 }\), when they collide directly.
  4. Find the speeds of \(A\) and \(B\) immediately after the collision.
OCR MEI M2 Q1
Moderate -0.5
1
  1. Roger of mass 70 kg and Sheuli of mass 50 kg are skating on a horizontal plane containing the standard unit vectors \(\mathbf { i }\) and \(\mathbf { j }\). The resistances to the motion of the skaters are negligible. The two skaters are locked in a close embrace and accelerate from rest until they reach a velocity of \(2 \mathrm { ims } ^ { - 1 }\), as shown in Fig. 1.1. \begin{figure}[h]
    \includegraphics[alt={},max width=\textwidth]{5dd6ba0d-e516-4b9e-ba19-6e90520b171b-002_191_181_543_740} \captionsetup{labelformat=empty} \caption{Fig. 1.1}
    \end{figure} \begin{figure}[h]
    \includegraphics[alt={},max width=\textwidth]{5dd6ba0d-e516-4b9e-ba19-6e90520b171b-002_177_359_589_1051} \captionsetup{labelformat=empty} \caption{Fig. 1.1}
    \end{figure}
    1. What impulse has acted on them? During a dance routine, the skaters separate on three occasions from their close embrace when travelling at a constant velocity of \(2 \mathrm { i } \mathrm { ms } ^ { - 1 }\).
    2. Calculate the velocity of Sheuli after the separation in the following cases.
      (A) Roger has velocity \(\mathrm { ims } ^ { - 1 }\) after the separation.
      (B) Roger and Sheuli have equal speeds in opposite senses after the separation, with Roger moving in the \(\mathbf { i }\) direction.
      (C) Roger has velocity \(4 ( \mathbf { i } + \mathbf { j } ) \mathrm { ms } ^ { - 1 }\) after the separation.
  2. Two discs with masses 2 kg and 3 kg collide directly in a horizontal plane. Their velocities just before the collision are shown in Fig. 1.2. The coefficient of restitution in the collision is 0.5. \begin{figure}[h]
    \includegraphics[alt={},max width=\textwidth]{5dd6ba0d-e516-4b9e-ba19-6e90520b171b-002_278_970_1759_594} \captionsetup{labelformat=empty} \caption{Fig. 1.2}
    \end{figure}
    1. Calculate the velocity of each disc after the collision. The disc of mass 3 kg moves freely after the collision and makes a perfectly elastic collision with a smooth wall inclined at \(60 ^ { \circ }\) to its direction of motion, as shown in Fig. 1.2.
    2. State with reasons the speed of the disc and the angle between its direction of motion and the wall after the collision.
OCR MEI M2 Q3
Standard +0.3
3 Fig. 3.1 shows an object made up as follows. ABCD is a uniform lamina of mass \(16 \mathrm {~kg} . \mathrm { BE } , \mathrm { EF }\), FG, HI, IJ and JD are each uniform rods of mass 2 kg . ABCD, BEFG and HIJD are squares lying in the same plane. The dimensions in metres are shown in the figure. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{5dd6ba0d-e516-4b9e-ba19-6e90520b171b-004_627_648_429_735} \captionsetup{labelformat=empty} \caption{Fig. 3.1}
\end{figure}
  1. Find the coordinates of the centre of mass of the object, referred to the axes shown in Fig.3.1. The rods are now re-positioned so that BEFG and HIJD are perpendicular to the lamina, as shown in Fig. 3.2. \begin{figure}[h]
    \includegraphics[alt={},max width=\textwidth]{5dd6ba0d-e516-4b9e-ba19-6e90520b171b-004_442_666_1510_722} \captionsetup{labelformat=empty} \caption{Fig. 3.2}
    \end{figure}
  2. Find the \(x\)-, \(y\)-and \(z\)-coordinates of the centre of mass of the object, referred to the axes shown in Fig. 3.2. Calculate the distance of the centre of mass from A . The object is now freely suspended from A and hangs in equilibrium with AC at \(\alpha ^ { \circ }\) to the vertical.
  3. Calculate \(\alpha\).
OCR MEI M2 2006 January Q1
17 marks Moderate -0.8
1 When a stationary firework P of mass 0.4 kg is set off, the explosion gives it an instantaneous impulse of 16 N s vertically upwards.
  1. Calculate the speed of projection of P . While travelling vertically upwards at \(32 \mathrm {~ms} ^ { - 1 } , P\) collides directly with another firework \(Q\), of mass 0.6 kg , that is moving directly downwards with speed \(u \mathrm {~m} \mathrm {~s} ^ { - 1 }\), as shown in Fig. 1. The coefficient of restitution in the collision is 0.1 and Q has a speed of \(4 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) vertically upwards immediately after the collision. \begin{figure}[h]
    \includegraphics[alt={},max width=\textwidth]{c1785fde-a6ce-4f8b-9948-4b4dd973ce84-2_520_422_753_817} \captionsetup{labelformat=empty} \caption{Fig. 1}
    \end{figure}
  2. Show that \(u = 18\) and calculate the speed and direction of motion of P immediately after the collision. Another firework of mass 0.5 kg has a velocity of \(( - 3.6 \mathbf { i } + 5.2 \mathbf { j } ) \mathrm { m } \mathrm { s } ^ { - 1 }\), where \(\mathbf { i }\) and \(\mathbf { j }\) are horizontal and vertical unit vectors, respectively. This firework explodes into two parts, C and D . Part C has mass 0.2 kg and velocity ( \(3 \mathbf { i } + 4 \mathbf { j }\) ) \(\mathrm { m } \mathrm { s } ^ { - 1 }\) immediately after the explosion.
  3. Calculate the velocity of D immediately after the explosion in the form \(a \mathbf { i } + b \mathbf { j }\). Show that C and D move off at \(90 ^ { \circ }\) to one another.
    [0pt] [8]
OCR MEI M2 2006 January Q2
19 marks Standard +0.3
2 A uniform beam, AB , is 6 m long and has a weight of 240 N .
Initially, the beam is in equilibrium on two supports at C and D, as shown in Fig. 2.1. The beam is horizontal. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{c1785fde-a6ce-4f8b-9948-4b4dd973ce84-3_200_687_486_689} \captionsetup{labelformat=empty} \caption{Fig. 2.1}
\end{figure}
  1. Calculate the forces acting on the beam from the supports at C and D . A workman tries to move the beam by applying a force \(T \mathrm {~N}\) at A at \(40 ^ { \circ }\) to the beam, as shown in Fig. 2.2. The beam remains in horizontal equilibrium but the reaction of support C on the beam is zero. \begin{figure}[h]
    \includegraphics[alt={},max width=\textwidth]{c1785fde-a6ce-4f8b-9948-4b4dd973ce84-3_318_691_1119_687} \captionsetup{labelformat=empty} \caption{Fig. 2.2}
    \end{figure}
  2. (A) Calculate the value of \(T\).
    (B) Explain why the support at D cannot be smooth. The beam is now supported by a light rope attached to the beam at A , with B on rough, horizontal ground. The rope is at \(90 ^ { \circ }\) to the beam and the beam is at \(30 ^ { \circ }\) to the horizontal, as shown in Fig. 2.3. The tension in the rope is \(P \mathrm {~N}\). The beam is in equilibrium on the point of sliding. \begin{figure}[h]
    \includegraphics[alt={},max width=\textwidth]{c1785fde-a6ce-4f8b-9948-4b4dd973ce84-3_438_633_1909_708} \captionsetup{labelformat=empty} \caption{Fig. 2.3}
    \end{figure}
  3. (A) Show that \(P = 60 \sqrt { 3 }\) and hence, or otherwise, find the frictional force between the beam and the ground.
    (B) Calculate the coefficient of friction between the beam and the ground.
OCR MEI M2 2006 January Q3
20 marks Standard +0.3
3
  1. A uniform lamina made from rectangular parts is shown in Fig. 3.1. All the dimensions are centimetres. All coordinates are referred to the axes shown in Fig. 3.1. \begin{figure}[h]
    \includegraphics[alt={},max width=\textwidth]{c1785fde-a6ce-4f8b-9948-4b4dd973ce84-4_691_529_427_762} \captionsetup{labelformat=empty} \caption{Fig. 3.1}
    \end{figure}
    1. Show that the \(x\)-coordinate of the centre of mass of the lamina is 6.5 and find the \(y\)-coordinate. A square of side 2 cm is to be cut from the lamina. The sides of the square are to be parallel to the coordinate axes and the centre of the square is to be chosen so that the \(x\)-coordinate of the centre of mass of the new shape is 6.4
    2. Calculate the \(x\)-coordinate of the centre of the square to be removed. The \(y\)-coordinate of the centre of the square to be removed is now chosen so that the \(y\)-coordinate of the centre of mass of the final shape is as large as possible.
    3. Calculate the \(y\)-coordinate of the centre of mass of the lamina with the square removed, giving your answer correct to three significant figures.
  2. Fig. 3.2 shows a framework made from light rods of length 2 m freely pin-jointed at \(\mathrm { A } , \mathrm { B } , \mathrm { C }\), D and E. The framework is in a vertical plane and is supported at A and C. There are loads of 120 N at B and at E . The force on the framework due to the support at A is \(R \mathrm {~N}\). \begin{figure}[h]
    \includegraphics[alt={},max width=\textwidth]{c1785fde-a6ce-4f8b-9948-4b4dd973ce84-5_448_741_459_662} \captionsetup{labelformat=empty} \caption{Fig. 3.2}
    \end{figure} each rod is 2 m long
    1. Show that \(R = 150\).
    2. Draw a diagram showing all the forces acting at the points \(\mathrm { A } , \mathrm { B } , \mathrm { D }\) and E , including the forces internal to the rods. Calculate the internal forces in rods AE and EB , and determine whether each is a tension or a thrust. [You may leave your answers in surd form.]
    3. Without any further calculation of the forces in the rods, explain briefly how you can tell that rod ED is in thrust.
OCR MEI M2 2006 January Q4
16 marks Standard +0.3
4 A block of mass 20 kg is pulled by a light, horizontal string over a rough, horizontal plane. During 6 seconds, the work done against resistances is 510 J and the speed of the block increases from \(5 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) to \(8 \mathrm {~m} \mathrm {~s} ^ { - 1 }\).
  1. Calculate the power of the pulling force. The block is now put on a rough plane that is at an angle \(\alpha\) to the horizontal, where \(\sin \alpha = \frac { 3 } { 5 }\). The frictional resistance to sliding is \(11 g \mathrm {~N}\). A light string parallel to the plane is connected to the block. The string passes over a smooth pulley and is connected to a freely hanging sphere of mass \(m \mathrm {~kg}\), as shown in Fig. 4. \begin{figure}[h]
    \includegraphics[alt={},max width=\textwidth]{c1785fde-a6ce-4f8b-9948-4b4dd973ce84-6_348_855_847_605} \captionsetup{labelformat=empty} \caption{Fig. 4}
    \end{figure} In parts (ii) and (iii), the sphere is pulled downwards and then released when travelling at a speed of \(4 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) vertically downwards. The block never reaches the pulley.
  2. Suppose that \(m = 5\) and that after the sphere is released the block moves \(x \mathrm {~m}\) up the plane before coming to rest.
    (A) Find an expression in terms of \(x\) for the change in gravitational potential energy of the system, stating whether this is a gain or a loss.
    (B) Find an expression in terms of \(x\) for the work done against friction.
    (C) Making use of your answers to parts (A) and (B), find the value of \(x\).
  3. Suppose instead that \(m = 15\). Calculate the speed of the sphere when it has fallen a distance 0.5 m from its point of release.
OCR MEI M2 2009 January Q2
17 marks Standard +0.3
2 One way to load a box into a van is to push the box so that it slides up a ramp. Some removal men are experimenting with the use of different ramps to load a box of mass 80 kg . \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{3865b4b3-97c7-412b-aabd-2705a954a847-3_345_1301_402_422} \captionsetup{labelformat=empty} \caption{Fig. 2}
\end{figure} Fig. 2 shows the general situation. The ramps are all uniformly rough with coefficient of friction 0.4 between the ramp and the box. The men push parallel to the ramp. As the box moves from one end of the ramp to the other it travels a vertical distance of 1.25 m .
  1. Find the limiting frictional force between the ramp and the box in terms of \(\theta\).
  2. From rest at the bottom, the box is pushed up the ramp and left at rest at the top. Show that the work done against friction is \(\frac { 392 } { \tan \theta } \mathrm {~J}\).
  3. Calculate the gain in the gravitational potential energy of the box when it is raised from the ground to the floor of the van. For the rest of the question take \(\theta = 35 ^ { \circ }\).
  4. Calculate the power required to slide the box up the ramp at a steady speed of \(1.5 \mathrm {~m} \mathrm {~s} ^ { - 1 }\).
  5. The box is given an initial speed of \(0.5 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) at the top of the ramp and then slides down without anyone pushing it. Determine whether it reaches a speed of \(3 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) while it is on the ramp.
OCR MEI M2 2009 January Q3
18 marks Standard +0.3
3 A fish slice consists of a blade and a handle as shown in Fig. 3.1. The rectangular blade ABCD is of mass 250 g and modelled as a lamina; this is 24 cm by 8 cm and is shown in the \(\mathrm { O } x y\) plane. The handle EF is of mass 125 g and is modelled as a thin rod; this is 30 cm long and E is attached to the mid-point of \(\mathrm { CD } . \mathrm { EF }\) is at right angles to CD and inclined at \(\alpha\) to the plane containing ABCD , where \(\sin \alpha = 0.6\) (and \(\cos \alpha = 0.8\) ). Coordinates refer to the axes shown in Fig. 3.1. Lengths are in centimetres. The \(y\) and \(z\)-coordinates of the centre of mass of the fish slice are \(\bar { y }\) and \(\bar { z }\) respectively. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{3865b4b3-97c7-412b-aabd-2705a954a847-4_517_1068_573_534} \captionsetup{labelformat=empty} \caption{Fig. 3.1}
\end{figure}
  1. Show that \(\bar { y } = 9 \frac { 1 } { 3 }\) and \(\bar { z } = 3\).
  2. Suppose that the plane \(\mathrm { O } x y\) in Fig. 3.1 is horizontal and represents a table top and that the fish slice is placed on it as shown. Determine whether the fish slice topples. The 'superior' version of the fish slice has an extra mass of 125 g uniformly distributed over the existing handle for 10 cm from F towards E , as shown in Fig. 3.2. This section of the handle may still be modelled as a thin rod. \begin{figure}[h]
    \includegraphics[alt={},max width=\textwidth]{3865b4b3-97c7-412b-aabd-2705a954a847-4_513_1065_1683_539} \captionsetup{labelformat=empty} \caption{Fig. 3.2}
    \end{figure}
  3. In this new situation show that \(\bar { y } = 14\) and \(\bar { z } = 6\). A sales feature of the 'superior' version is the ability to suspend it using a very small hole in the blade. This situation is modelled as the fish slice hanging in equilibrium when suspended freely about an axis through O .
  4. Indicate the position of the centre of mass on a diagram and calculate the angle of the line OE with the vertical.