Questions M2 (1391 questions)

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OCR M2 2016 June Q6
6 The masses of two particles \(A\) and \(B\) are 4 kg and 3 kg respectively. The particles are moving towards each other along a straight line on a smooth horizontal surface. \(A\) has speed \(8 \mathrm {~ms} ^ { - 1 }\) and \(B\) has speed \(10 \mathrm {~ms} ^ { - 1 }\) before they collide. The kinetic energy lost due to the collision is 121.5 J .
  1. Find the speed and direction of motion of each particle after the collision.
  2. Find the coefficient of restitution between \(A\) and \(B\).
    \includegraphics[max width=\textwidth, alt={}, center]{907be5c2-8a3c-482a-84fd-b45e8c36caa7-5_510_1504_653_271} A particle \(P\) is projected with speed \(32 \mathrm {~ms} ^ { - 1 }\) at an angle of elevation \(\alpha\), where \(\sin \alpha = \frac { 3 } { 5 }\), from a point \(A\) on horizontal ground. At the same instant a particle \(Q\) is projected with speed \(20 \mathrm {~ms} ^ { - 1 }\) at an angle of elevation \(\beta\), where \(\sin \beta = \frac { 24 } { 25 }\), from a point \(B\) on the same horizontal ground. The particles move freely under gravity in the same vertical plane and collide with each other at the point \(C\) at the instant when they are travelling horizontally (see diagram).
  3. Calculate the height of \(C\) above the ground and the distance \(A B\). Immediately after the collision \(P\) falls vertically. \(P\) hits the ground and rebounds vertically upwards, coming to instantaneous rest at a height 5 m above the ground.
  4. Given that the mass of \(P\) is 3 kg , find the magnitude and direction of the impulse exerted on \(P\) by the ground. The coefficient of restitution between the two particles is \(\frac { 1 } { 2 }\).
  5. Find the distance of \(Q\) from \(C\) at the instant when \(Q\) is travelling in a direction of \(25 ^ { \circ }\) below the horizontal.
OCR M2 Specimen Q1
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
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
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
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
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
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
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
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
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
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
8 marks
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
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
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
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 2007 January Q1
7 marks
1 A sledge and a child sitting on it have a combined mass of 29.5 kg . The sledge slides on horizontal ice with negligible resistance to its movement.
  1. While at rest, the sledge is hit directly from behind by a ball of mass 0.5 kg travelling horizontally at \(10 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). The coefficient of restitution in the collision is 0.8 . After the impact the speeds of the sledge and the ball are \(V _ { 1 } \mathrm {~ms} ^ { - 1 }\) and \(V _ { 2 } \mathrm {~ms} ^ { - 1 }\) respectively. Calculate \(V _ { 1 }\) and \(V _ { 2 }\) and state the direction in which the ball is travelling after the impact. [7]
  2. While at rest, the sledge is hit directly from behind by a snowball of mass 0.5 kg travelling horizontally at \(10 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). The snowball sticks to the sledge.
    (A) Calculate the velocity with which the combined sledge and snowball start to move.
    (B) The child scoops up the 0.5 kg of snow and drops it over the back of the sledge. What happens to the velocity of the sledge? Give a reason for your answer.
  3. In another situation, the sledge is travelling over the ice at \(2 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) with 10.5 kg of snow on it (giving a total mass of 40 kg ). The child throws a snowball of mass 0.5 kg from the sledge, parallel to the ground and in the positive direction of the motion of the sledge. Immediately after the snowball is thrown, the sledge has a speed of \(V \mathrm {~m} \mathrm {~s} ^ { - 1 }\) and the snowball and sledge are separating at a speed of \(10 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). Draw a diagram showing the velocities of the sledge and snowball before and after the snowball is thrown. Calculate \(V\). \begin{figure}[h]
    \includegraphics[alt={},max width=\textwidth]{26d30179-589e-462a-a38f-f9e4e5dec4f4-3_527_720_335_667} \captionsetup{labelformat=empty} \caption{Fig. 2}
    \end{figure} Fig. 2 shows a framework in a vertical plane made from the equal, light, rigid rods \(\mathrm { AB } , \mathrm { BC } , \mathrm { AD }\), \(\mathrm { BD } , \mathrm { BE } , \mathrm { CE }\) and DE . [The triangles \(\mathrm { ABD } , \mathrm { BDE }\) and BCE are all equilateral.] The rods \(\mathrm { AB } , \mathrm { BC }\) and DE are horizontal.
    The rods are freely pin-jointed to each other at A, B, C, D and E.
    The pin-joint at A is also fixed to an inclined plane. The plane is smooth and parallel to the rod AD . The pin-joint at D rests on this plane. The following external forces act on the framework: a vertical load of \(L \mathrm {~N}\) at C ; the normal reaction force \(R \mathrm {~N}\) of the plane on the framework at D ; the horizontal and vertical forces \(X \mathrm {~N}\) and \(Y \mathrm {~N}\), respectively, acting at A .
  4. Write down equations for the horizontal and vertical equilibrium of the framework.
  5. By considering moments, find the relationship between \(R\) and \(L\). Hence show that \(X = \sqrt { 3 } L\) and \(Y = 0\).
  6. Draw a diagram showing all the forces acting on the pin-joints, including the forces internal to the rods.
  7. Show that the internal force in the rod AD is zero.
  8. Find the forces internal to \(\mathrm { AB } , \mathrm { CE }\) and BC in terms of \(L\) and state whether each is a tension or a thrust (compression). [You may leave your answers in surd form.]
  9. Without calculating their values in terms of \(L\), show that the forces internal to the rods BD and BE have equal magnitude but one is a tension and the other a thrust.
OCR MEI M2 2007 January Q3
3 A box is to be assembled in the shape of the cuboid shown in Fig. 3.1. The lengths are in centimetres. All the faces are made of the same uniform, rigid and thin material. All coordinates refer to the axes shown in this figure. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{26d30179-589e-462a-a38f-f9e4e5dec4f4-4_410_911_443_577} \captionsetup{labelformat=empty} \caption{Fig. 3.1}
\end{figure}
  1. The four vertical faces \(\mathrm { OAED } , \mathrm { ABFE } , \mathrm { FGCB }\) and CODG are assembled first to make an open box without a base or a top. Write down the coordinates of the centre of mass of this open box. The base OABC is added to the vertical faces.
  2. Write down the \(x\) - and \(y\)-coordinates of the centre of mass of the box now. Show that the \(z\)-coordinate is now 1.875 . The top face FGDE is now added. This is a lid hinged to the rest of the box along the line FG. The lid is open so that it hangs in a vertical plane touching the face FGCB.
  3. Show that the coordinates of the centre of mass of the box in this situation are (10, 2.4, 2.1). The box, with the lid still touching face FGCB, is now put on a sloping plane with the edge OA horizontal and the base inclined at \(30 ^ { \circ }\) to the horizontal, as shown in Fig. 3.2. \begin{figure}[h]
    \includegraphics[alt={},max width=\textwidth]{26d30179-589e-462a-a38f-f9e4e5dec4f4-5_610_1091_370_477} \captionsetup{labelformat=empty} \caption{Fig. 3.2}
    \end{figure} The weight of the box is 40 N . A force \(P \mathrm {~N}\) acts parallel to the plane and is applied to the mid-point of FG at \(90 ^ { \circ }\) to FG. This force tends to push the box down the plane. The box does not slip and is on the point of toppling about the edge AO .
  4. Show that the clockwise moment of the weight of the box about the edge AO is about 0.411 Nm .
  5. Calculate the value of \(P\).
OCR MEI M2 2007 January Q4
4 Some tiles on a roof are being replaced. Each tile has a mass of 2 kg and the coefficient of friction between it and the existing roof is 0.75 . The roof is at \(30 ^ { \circ }\) to the horizontal and the bottom of the roof is 6 m above horizontal ground, as shown in Fig. 4. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{26d30179-589e-462a-a38f-f9e4e5dec4f4-6_494_622_413_719} \captionsetup{labelformat=empty} \caption{Fig. 4}
\end{figure}
  1. Calculate the limiting frictional force between a tile and the roof. A tile is placed on the roof. Does it slide? (Your answer should be supported by a calculation.)
  2. The tiles are raised 6 m from the ground, the only work done being against gravity. They are then slid 4 m up the roof and placed at the point A shown in Fig. 4.
    (A) Show that each tile gains 156.8 J of gravitational potential energy.
    (B) Calculate the work done against friction per tile.
    (C) What average power is required to raise 10 tiles per minute from the ground to A ?
  3. A tile is kicked from A directly down the roof. When the tile is at \(\mathrm { B } , x \mathrm {~m}\) from the edge of the roof, its speed is \(4 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). It subsequently hits the ground travelling at \(9 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). In the motion of the tile from B to the ground, the work done against sliding and other resistances is 90 J . Use an energy method to find \(x\).
OCR MEI M2 2008 January Q1
1
  1. A battering-ram consists of a wooden beam fixed to a trolley. The battering-ram runs along horizontal ground and collides directly with a vertical wall, as shown in Fig. 1.1. The batteringram has a mass of 4000 kg . \begin{figure}[h]
    \includegraphics[alt={},max width=\textwidth]{b2962c91-4739-4d1e-98f3-62d420f6dddf-2_310_793_424_717} \captionsetup{labelformat=empty} \caption{Fig. 1.1}
    \end{figure} Initially the battering-ram is at rest. Some men push it for 8 seconds and let go just as it is about to hit the wall. While the battering-ram is being pushed, the constant overall force on it in the direction of its motion is 1500 N .
    1. At what speed does the battering-ram hit the wall? The battering-ram hits a loose stone block of mass 500 kg in the wall. Linear momentum is conserved and the coefficient of restitution in the impact is 0.2 .
    2. Calculate the speeds of the stone block and of the battering-ram immediately after the impact.
    3. Calculate the energy lost in the impact.
  2. Small objects A and B are sliding on smooth, horizontal ice. Object A has mass 4 kg and speed \(18 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) in the \(\mathbf { i }\) direction. B has mass 8 kg and speed \(9 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) in the direction shown in Fig. 1.2, where \(\mathbf { i }\) and \(\mathbf { j }\) are the standard unit vectors. \begin{figure}[h]
    \includegraphics[alt={},max width=\textwidth]{b2962c91-4739-4d1e-98f3-62d420f6dddf-2_515_783_1637_721} \captionsetup{labelformat=empty} \caption{Fig. 1.2}
    \end{figure}
    1. Write down the linear momentum of A and show that the linear momentum of B is \(( 36 \mathbf { i } + 36 \sqrt { 3 } \mathbf { j } )\) Ns. After the objects meet they stick together (coalesce) and move with a common velocity of \(( u \mathbf { i } + v \mathbf { j } ) \mathrm { m } \mathrm { s } ^ { - 1 }\).
    2. Calculate \(u\) and \(v\).
    3. Find the angle between the direction of motion of the combined object and the \(\mathbf { i }\) direction. Make your method clear.
OCR MEI M2 2008 January Q2
2 A cyclist and her bicycle have a combined mass of 80 kg .
  1. Initially, the cyclist accelerates from rest to \(3 \mathrm {~ms} ^ { - 1 }\) against negligible resistances along a horizontal road.
    (A) How much energy is gained by the cyclist and bicycle?
    (B) The cyclist travels 12 m during this acceleration. What is the average driving force on the bicycle?
  2. While exerting no driving force, the cyclist free-wheels down a hill. Her speed increases from \(4 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) to \(10 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). During this motion, the total work done against friction is 1600 J and the drop in vertical height is \(h \mathrm {~m}\). Without assuming that the hill is uniform in either its angle or roughness, calculate \(h\).
  3. The cyclist reaches another horizontal stretch of road and there is now a constant resistance to motion of 40 N .
    (A) When the power of the driving force on the bicycle is a constant 200 W , what constant speed can the cyclist maintain?
    (B) Find the power of the driving force on the bicycle when travelling at a speed of \(0.5 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) with an acceleration of \(2 \mathrm {~m} \mathrm {~s} ^ { - 2 }\). \begin{figure}[h]
    \includegraphics[alt={},max width=\textwidth]{b2962c91-4739-4d1e-98f3-62d420f6dddf-4_671_760_296_310} \captionsetup{labelformat=empty} \caption{Fig. 3.1}
    \end{figure} \begin{figure}[h]
    \includegraphics[alt={},max width=\textwidth]{b2962c91-4739-4d1e-98f3-62d420f6dddf-4_703_622_264_1213} \captionsetup{labelformat=empty} \caption{Fig. 3.2}
    \end{figure} A lamina is made from uniform material in the shape shown in Fig.3.1. BCJA, DZOJ, ZEIO and FGHI are all rectangles. The lengths of the sides are shown in centimetres.
  4. Find the coordinates of the centre of mass of the lamina, referred to the axes shown in Fig. 3.1. The rectangles BCJA and FGHI are folded through \(90 ^ { \circ }\) about the lines CJ and FI respectively to give the fire-screen shown in Fig. 3.2.
  5. Show that the coordinates of the centre of mass of the fire-screen, referred to the axes shown in Fig. 3.2, are (2.5, 0, 57.5). The \(x\) - and \(y\)-axes are in a horizontal floor. The fire-screen has a weight of 72 N . A horizontal force \(P \mathrm {~N}\) is applied to the fire-screen at the point Z . This force is perpendicular to the line DE in the positive \(x\) direction. The fire-screen is on the point of tipping about the line AH .
  6. Calculate the value of \(P\). The coefficient of friction between the fire-screen and the floor is \(\mu\).
  7. For what values of \(\mu\) does the fire-screen slide before it tips?
OCR MEI M2 2008 January Q4
4 Fig. 4.1 shows a uniform beam, CE, of weight 2200 N and length 4.5 m . The beam is freely pivoted on a fixed support at D and is supported at C . The distance CD is 2.75 m . \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{b2962c91-4739-4d1e-98f3-62d420f6dddf-5_172_631_406_315} \captionsetup{labelformat=empty} \caption{Fig. 4.1}
\end{figure} \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{b2962c91-4739-4d1e-98f3-62d420f6dddf-5_310_643_406_1190} \captionsetup{labelformat=empty} \caption{Fig. 4.2}
\end{figure} The beam is horizontal and in equilibrium.
  1. Show that the anticlockwise moment of the weight of the beam about D is 1100 Nm . Find the value of the normal reaction on the beam of the support at C . The support at C is removed and spheres at P and Q are suspended from the beam by light strings attached to the points C and R . The sphere at P has weight 440 N and the sphere at Q has weight \(W \mathrm {~N}\). The point R of the beam is 1.5 m from D . This situation is shown in Fig. 4.2.
  2. The beam is horizontal and in equilibrium. Show that \(W = 1540\). The sphere at P is changed for a lighter one with weight 400 N . The sphere at Q is unchanged. The beam is now held in equilibrium at an angle of \(20 ^ { \circ }\) to the horizontal by means of a light rope attached to the beam at E. This situation (but without the rope at E) is shown in Fig. 4.3. \begin{figure}[h]
    \includegraphics[alt={},max width=\textwidth]{b2962c91-4739-4d1e-98f3-62d420f6dddf-5_451_611_1635_767} \captionsetup{labelformat=empty} \caption{Fig. 4.3}
    \end{figure}
  3. Calculate the tension in the rope when it is
    (A) at \(90 ^ { \circ }\) to the beam,
    \(( B )\) horizontal.
OCR MEI M2 2009 January Q2
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
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.
OCR MEI M2 2009 January Q4
4
  1. A uniform, rigid beam, AB , has a weight of 600 N . It is horizontal and in equilibrium resting on two small smooth pegs at P and Q . Fig. 4.1 shows the positions of the pegs; lengths are in metres. \begin{figure}[h]
    \includegraphics[alt={},max width=\textwidth]{3865b4b3-97c7-412b-aabd-2705a954a847-5_229_647_404_790} \captionsetup{labelformat=empty} \caption{Fig. 4.1}
    \end{figure}
    1. Calculate the forces exerted by the pegs on the beam. A force of \(L \mathrm {~N}\) is applied vertically downwards at B . The beam is in equilibrium but is now on the point of tipping.
    2. Calculate the value of \(L\).
  2. Fig. 4.2 shows a framework in a vertical plane constructed of light, rigid rods \(\mathrm { AB } , \mathrm { BC }\) and CA . The rods are freely pin-jointed to each other at \(\mathrm { A } , \mathrm { B }\) and C and to a fixed point at A . The pin-joint at C rests on a smooth, horizontal support. The dimensions of the framework are shown in metres. There is a force of 340 N acting at B in the plane of the framework. This force and the \(\operatorname { rod } \mathrm { BC }\) are both inclined to the vertical at an angle \(\alpha\), which is defined in triangle BCX . The force on the framework exerted by the support at C is \(R \mathrm {~N}\). \begin{figure}[h]
    \includegraphics[alt={},max width=\textwidth]{3865b4b3-97c7-412b-aabd-2705a954a847-5_675_869_1434_678} \captionsetup{labelformat=empty} \caption{Fig. 4.2}
    \end{figure}
    1. Show that \(R = 600\).
    2. Draw a diagram showing all the forces acting on the framework and also the internal forces in the rods.
      [0pt]
    3. Calculate the internal forces in the three rods, indicating whether each rod is in tension or in compression (thrust). [Your working in this part should correspond to your diagram in part (ii).]
OCR MEI M2 2010 January Q1
1
  1. An object P , with mass 6 kg and speed \(1 \mathrm {~m} \mathrm {~s} ^ { - 1 }\), is sliding on a smooth horizontal table. Object P explodes into two small parts, Q and \(\mathrm { R } . \mathrm { Q }\) has mass 4 kg and R has mass 2 kg and speed \(4 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) in the direction of motion of P before the explosion. This information is shown in Fig. 1.1. \begin{figure}[h]
    \includegraphics[alt={},max width=\textwidth]{f2aaae62-a5f3-47da-afa5-1dd4b37ea2d6-2_346_1267_429_479} \captionsetup{labelformat=empty} \caption{Fig. 1.1}
    \end{figure}
    1. Calculate the velocity of Q . Just as object R reaches the edge of the table, it collides directly with a small object S of mass 3 kg that is travelling horizontally towards R with a speed of \(1 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). This information is shown in Fig. 1.2. The coefficient of restitution in this collision is 0.1 . \begin{figure}[h]
      \includegraphics[alt={},max width=\textwidth]{f2aaae62-a5f3-47da-afa5-1dd4b37ea2d6-2_506_647_1215_790} \captionsetup{labelformat=empty} \caption{Fig. 1.2}
      \end{figure}
    2. Calculate the velocities of R and S immediately after the collision. The table is 0.4 m above a horizontal floor. After the collision, R and S have no contact with the table.
    3. Calculate the distance apart of R and S when they reach the floor.
  2. A particle of mass \(m \mathrm {~kg}\) bounces off a smooth horizontal plane. The components of velocity of the particle just before the impact are \(u \mathrm {~m} \mathrm {~s} ^ { - 1 }\) parallel to the plane and \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\) perpendicular to the plane. The coefficient of restitution is \(e\). Show that the mechanical energy lost in the impact is \(\frac { 1 } { 2 } m v ^ { 2 } \left( 1 - e ^ { 2 } \right) \mathrm { J }\).