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CAIE M2 2014 November Q4
7 marks Standard +0.8
4 \includegraphics[max width=\textwidth, alt={}, center]{84a2b2eb-a750-4047-864b-4a165fc66b2a-3_611_977_260_584} \(A B C D E F\) is the cross-section through the centre of mass of a uniform solid prism. \(A B C F\) is a rectangle in which \(A B = C F = 1.6 \mathrm {~m}\), and \(B C = A F = 0.4 \mathrm {~m}\). \(C D E\) is a triangle in which \(C D = 1.8 \mathrm {~m}\), \(C E = 0.4 \mathrm {~m}\), and angle \(D C E = 90 ^ { \circ }\). The prism stands on a rough horizontal surface. A horizontal force of magnitude \(T \mathrm {~N}\) acts at \(B\) in the direction \(C B\) (see diagram). The prism is in equilibrium.
  1. Show that the distance of the centre of mass of the prism from \(A B\) is 0.488 m .
  2. Given that the weight of the prism is 100 N , find the greatest and least possible values of \(T\).
CAIE M2 2014 November Q5
9 marks Standard +0.3
5 The equation of the trajectory of a small ball \(B\) projected from a fixed point \(O\) is $$y = - 0.05 x ^ { 2 }$$ where \(x\) and \(y\) are, respectively, the displacements in metres of \(B\) from \(O\) in the horizontal and vertically upwards directions.
  1. Show that \(B\) is projected horizontally, and find its speed of projection.
  2. Find the value of \(y\) when the direction of motion of \(B\) is \(60 ^ { \circ }\) below the horizontal, and find the corresponding speed of \(B\). \(6 \quad O , A\) and \(B\) are three points in a straight line on a smooth horizontal surface. A particle \(P\) of mass 0.6 kg moves along the line. At time \(t \mathrm {~s}\) the particle has displacement \(x \mathrm {~m}\) from \(O\) and speed \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\). The only horizontal force acting on \(P\) has magnitude \(0.4 v ^ { \frac { 1 } { 2 } } \mathrm {~N}\) and acts in the direction \(O A\). Initially the particle is at \(A\), where \(x = 1\) and \(v = 1\).
  3. Show that \(3 v ^ { \frac { 1 } { 2 } } \frac { \mathrm {~d} v } { \mathrm {~d} x } = 2\).
  4. Express \(v\) in terms of \(x\).
  5. Given that \(A B = 7 \mathrm {~m}\), find the value of \(t\) when \(P\) passes through \(B\).
CAIE M2 2014 November Q7
12 marks Challenging +1.8
7 \includegraphics[max width=\textwidth, alt={}, center]{84a2b2eb-a750-4047-864b-4a165fc66b2a-4_558_857_260_644} One end of a light elastic string with modulus of elasticity 15 N is attached to a fixed point \(A\) which is 2 m vertically above a fixed small smooth ring \(R\). The string has natural length 2 m and it passes through \(R\). The other end of the string is attached to a particle \(P\) of mass \(m \mathrm {~kg}\) which moves with constant angular speed \(\omega \mathrm { rad } \mathrm { s } ^ { - 1 }\) in a horizontal circle which has its centre 0.4 m vertically below the ring. \(P R\) makes an acute angle \(\theta\) with the vertical (see diagram).
  1. Show that the tension in the string is \(\frac { 3 } { \cos \theta } \mathrm {~N}\) and hence find the value of \(m\).
  2. Show that the value of \(\omega\) does not depend on \(\theta\). It is given that for one value of \(\theta\) the elastic potential energy stored in the string is twice the kinetic energy of \(P\).
  3. Find this value of \(\theta\).
CAIE M2 2014 November Q1
3 marks Moderate -0.5
1 A golf ball \(B\) is projected from a point \(O\) on horizontal ground. \(B\) hits the ground for the first time at a point 48 m away from \(O\) at time 2.4 s after projection. Calculate the angle of projection.
CAIE M2 2014 November Q2
6 marks Standard +0.3
2 A particle \(P\) of mass 0.2 kg is attached to one end of a light elastic string of natural length 0.8 m and modulus of elasticity 64 N . The other end of the string is attached to a fixed point \(A\) on a smooth horizontal surface. \(P\) is placed on the surface at a point 0.8 m from \(A\). The particle \(P\) is then projected with speed \(10 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) directly away from \(A\).
  1. Calculate the distance \(A P\) when \(P\) is at instantaneous rest.
  2. Calculate the speed of \(P\) when it is 1.0 m from \(A\).
CAIE M2 2014 November Q3
6 marks Standard +0.8
3 A small ball of mass \(m \mathrm {~kg}\) is projected vertically upwards with speed \(14 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). The ball has velocity \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\) upwards when it is \(x \mathrm {~m}\) above the point of projection. A resisting force of magnitude \(0.02 m v \mathrm {~N}\) acts on the ball during its upward motion.
  1. Show that, while the ball is moving upwards, \(\left( \frac { 500 } { v + 500 } - 1 \right) \frac { \mathrm { d } v } { \mathrm {~d} x } = 0.02\).
  2. Find the greatest height of the ball above its point of projection.
CAIE M2 2014 November Q4
7 marks Moderate -0.3
4 A particle \(P\) is projected with speed \(50 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) at an angle of \(30 ^ { \circ }\) above the horizontal from a point \(O\) on a horizontal plane.
  1. Calculate the speed of \(P\) when it has been in motion for 4 s , and calculate another time at which \(P\) has this speed.
  2. Find the distance \(O P\) when \(P\) has been in motion for 4 s .
CAIE M2 2014 November Q5
8 marks Challenging +1.2
5 \includegraphics[max width=\textwidth, alt={}, center]{81be887c-ab01-4327-a5df-f25c68a6fdb6-2_337_517_1749_813} Two light elastic strings each have one end attached to a fixed horizontal beam. One string has natural length 0.6 m and modulus of elasticity 12 N ; the other string has natural length 0.7 m and modulus of elasticity 21 N . The other ends of the strings are attached to a small block \(B\) of weight \(W \mathrm {~N}\). The block hangs in equilibrium \(d \mathrm {~m}\) below the beam, with both strings vertical (see diagram).
  1. Given that the tensions in the strings are equal, find \(d\) and \(W\). The small block is now raised vertically to the point 0.7 m below the beam, and then released from rest.
  2. Find the greatest speed of the block in its subsequent motion.
CAIE M2 2014 November Q6
9 marks Standard +0.3
6 A horizontal disc with a rough surface rotates about a fixed vertical axis which passes through the centre of the disc. A particle \(P\) of mass 0.2 kg is in contact with the surface and rotates with the disc, without slipping, at a distance 0.5 m from the axis. The greatest speed of \(P\) for which this motion is possible is \(1.5 \mathrm {~m} \mathrm {~s} ^ { - 1 }\).
  1. Calculate the coefficient of friction between the disc and \(P\). \(P\) is now attached to one end of a light elastic string, which is connected at its other end to a point on the vertical axis above the disc. The tension in the string is equal to half the weight of \(P\). The disc rotates with constant angular speed \(\omega \mathrm { rad } \mathrm { s } ^ { - 1 }\) and \(P\) rotates with the disc without slipping. \(P\) moves in a circle of radius 0.5 m , and the taut string makes an angle of \(30 ^ { \circ }\) with the horizontal.
  2. Find the greatest and least values of \(\omega\) for which this motion is possible.
  3. Calculate the value of \(\omega\) for which the disc exerts no frictional force on \(P\).
CAIE M2 2014 November Q7
11 marks Challenging +1.2
7 \includegraphics[max width=\textwidth, alt={}, center]{81be887c-ab01-4327-a5df-f25c68a6fdb6-3_586_527_1030_810} A uniform lamina \(A B C\) is in the form of a major segment of a circle with centre \(O\) and radius 0.35 m . The straight edge of the lamina is \(A B\), and angle \(A O B = \frac { 2 } { 3 } \pi\) radians (see diagram).
  1. Show that the centre of mass of the lamina is 0.0600 m from \(O\), correct to 3 significant figures. The weight of the lamina is 14 N . It is placed on a rough horizontal surface with \(A\) vertically above \(B\) and the lowest point of the arc \(B C\) in contact with the surface. The lamina is held in equilibrium in a vertical plane by a force of magnitude \(F \mathrm {~N}\) acting at \(A\).
  2. Find \(F\) in each of the following cases:
    (a) the force of magnitude \(F \mathrm {~N}\) acts along \(A B\);
    (b) the force of magnitude \(F \mathrm {~N}\) acts along the tangent to the circular arc at \(A\).
CAIE M2 2015 November Q1
4 marks Challenging +1.2
1 A particle \(P\) moves in a straight line and passes through a point \(O\) of the line with velocity \(2 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). At time \(t \mathrm {~s}\) after passing through \(O\), the velocity of \(P\) is \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\) and the acceleration of \(P\) is given by \(\mathrm { e } ^ { - 0.5 v } \mathrm {~m} \mathrm {~s} ^ { - 2 }\). Calculate the velocity of \(P\) when \(t = 1.2\).
CAIE M2 2015 November Q2
5 marks Challenging +1.2
2 \includegraphics[max width=\textwidth, alt={}, center]{8d128de7-f6ca-4187-ae8b-cbd8dfcdfc93-2_476_716_486_715} A uniform rigid rod \(A B\) of length 1.2 m and weight 8 N has a particle of weight 2 N attached at the end \(B\). The end \(A\) of the rod is freely hinged to a fixed point. One end of a light elastic string of natural length 0.8 m and modulus of elasticity 20 N is attached to the hinge. The string passes over a small smooth pulley \(P\) fixed 0.8 m vertically above the hinge. The other end of the string is attached to a small light smooth ring \(R\) which can slide on the rod. The system is in equilibrium with the rod inclined at an angle \(\theta ^ { \circ }\) to the vertical (see diagram).
  1. Show that the tension in the string is \(20 \sin \theta \mathrm {~N}\).
  2. Explain why the part of the string attached to the ring is perpendicular to the rod.
  3. Find \(\theta\).
CAIE M2 2015 November Q3
5 marks Standard +0.3
3 A particle \(P\) of mass 0.3 kg moves in a straight line on a smooth horizontal surface. \(P\) passes through a fixed point \(O\) of the line with velocity \(8 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). A force of magnitude \(2 x \mathrm {~N}\) acts on \(P\) in the direction \(P O\), where \(x \mathrm {~m}\) is the displacement of \(P\) from \(O\).
  1. Show that \(v \frac { \mathrm {~d} v } { \mathrm {~d} x } = k x\) and state the value of the constant \(k\).
  2. Find the value of \(x\) at the instant when \(P\) comes to instantaneous rest. \includegraphics[max width=\textwidth, alt={}, center]{8d128de7-f6ca-4187-ae8b-cbd8dfcdfc93-3_921_672_258_733} One end of a light inextensible string is attached to a fixed point \(A\). The string passes through a smooth bead \(B\) of mass 0.3 kg and the other end of the string is attached to a fixed point \(C\) vertically below \(A\). The bead \(B\) moves with constant speed in a horizontal circle of radius 0.6 m which has its centre between \(A\) and \(C\). The string makes an angle of \(30 ^ { \circ }\) with the vertical at \(A\) and an angle of \(45 ^ { \circ }\) with the vertical at \(C\) (see diagram).
  3. Calculate the speed of \(B\). The lower end of the string is detached from \(C\), and \(B\) is now attached to this end of the string. The other end of the string remains attached to \(A\). The bead is set in motion so that it moves with angular speed \(3 \mathrm { rad } \mathrm { s } ^ { - 1 }\) in a horizontal circle which has its centre vertically below \(A\).
  4. Calculate the tension in the string.
CAIE M2 2015 November Q5
8 marks Challenging +1.2
5 A particle \(P\) of mass 0.2 kg is attached to one end of a light elastic string of natural length 0.75 m and modulus of elasticity 21 N . The other end of the string is attached to a fixed point \(A\) which is 0.8 m vertically above a smooth horizontal surface. \(P\) rests in equilibrium on the surface.
  1. Find the magnitude of the force exerted on \(P\) by the surface. \(P\) is now projected horizontally along the surface with speed \(3 \mathrm {~ms} ^ { - 1 }\).
  2. Calculate the extension of the string at the instant when \(P\) leaves the surface.
  3. Hence find the speed of \(P\) at the instant when it leaves the surface.
CAIE M2 2015 November Q6
9 marks Standard +0.3
6 \includegraphics[max width=\textwidth, alt={}, center]{8d128de7-f6ca-4187-ae8b-cbd8dfcdfc93-4_647_641_260_751} A uniform circular disc has centre \(O\) and radius 1.2 m . The centre of the disc is at the origin of \(x\)-and \(y\)-axes. Two circular holes with centres at \(A\) and \(B\) are made in the disc (see diagram). The point \(A\) is on the negative \(x\)-axis with \(O A = 0.5 \mathrm {~m}\). The point \(B\) is on the negative \(y\)-axis with \(O B = 0.7 \mathrm {~m}\). The hole with centre \(A\) has radius 0.3 m and the hole with centre \(B\) has radius 0.4 m . Find the distance of the centre of mass of the object from
  1. the \(x\)-axis,
  2. the \(y\)-axis. The object can rotate freely in a vertical plane about a horizontal axis through \(O\).
  3. Calculate the angle which \(O A\) makes with the vertical when the object rests in equilibrium.
CAIE M2 2015 November Q7
11 marks Challenging +1.2
7 A particle \(P\) is projected with speed \(V \mathrm {~m} \mathrm {~s} ^ { - 1 }\) at an angle of \(60 ^ { \circ }\) above the horizontal from a point \(O\). At the instant 1 s later a particle \(Q\) is projected from \(O\) with the same initial speed at an angle of \(45 ^ { \circ }\) above the horizontal. The two particles collide when \(Q\) has been in motion for \(t \mathrm {~s}\).
  1. Show that \(t = 2.414\), correct to 3 decimal places.
  2. Find the value of \(V\). The collision occurs after \(P\) has passed through the highest point of its trajectory.
  3. Calculate the vertical distance of \(P\) below its greatest height when \(P\) and \(Q\) collide. \footnotetext{Permission to reproduce items where third-party owned material protected by copyright is included has been sought and cleared where possible. Every reasonable effort has been made by the publisher (UCLES) to trace copyright holders, but if any items requiring clearance have unwittingly been included, the publisher will be pleased to make amends at the earliest possible opportunity.
    To avoid the issue of disclosure of answer-related information to candidates, all copyright acknowledgements are reproduced online in the Cambridge International Examinations Copyright Acknowledgements Booklet. This is produced for each series of examinations and is freely available to download at \href{http://www.cie.org.uk}{www.cie.org.uk} after the live examination series. Cambridge International Examinations is part of the Cambridge Assessment Group. Cambridge Assessment is the brand name of University of Cambridge Local Examinations Syndicate (UCLES), which is itself a department of the University of Cambridge. }
CAIE M2 2015 November Q2
5 marks Challenging +1.2
2 \includegraphics[max width=\textwidth, alt={}, center]{e39fcb6b-7bd2-4dd5-bbd6-5de6e13b30b3-2_476_716_486_715} A uniform rigid rod \(A B\) of length 1.2 m and weight 8 N has a particle of weight 2 N attached at the end \(B\). The end \(A\) of the rod is freely hinged to a fixed point. One end of a light elastic string of natural length 0.8 m and modulus of elasticity 20 N is attached to the hinge. The string passes over a small smooth pulley \(P\) fixed 0.8 m vertically above the hinge. The other end of the string is attached to a small light smooth ring \(R\) which can slide on the rod. The system is in equilibrium with the rod inclined at an angle \(\theta ^ { \circ }\) to the vertical (see diagram).
  1. Show that the tension in the string is \(20 \sin \theta \mathrm {~N}\).
  2. Explain why the part of the string attached to the ring is perpendicular to the rod.
  3. Find \(\theta\).
CAIE M2 2015 November Q3
5 marks Standard +0.3
3 A particle \(P\) of mass 0.3 kg moves in a straight line on a smooth horizontal surface. \(P\) passes through a fixed point \(O\) of the line with velocity \(8 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). A force of magnitude \(2 x \mathrm {~N}\) acts on \(P\) in the direction \(P O\), where \(x \mathrm {~m}\) is the displacement of \(P\) from \(O\).
  1. Show that \(v \frac { \mathrm {~d} v } { \mathrm {~d} x } = k x\) and state the value of the constant \(k\).
  2. Find the value of \(x\) at the instant when \(P\) comes to instantaneous rest. \includegraphics[max width=\textwidth, alt={}, center]{e39fcb6b-7bd2-4dd5-bbd6-5de6e13b30b3-3_921_672_258_733} One end of a light inextensible string is attached to a fixed point \(A\). The string passes through a smooth bead \(B\) of mass 0.3 kg and the other end of the string is attached to a fixed point \(C\) vertically below \(A\). The bead \(B\) moves with constant speed in a horizontal circle of radius 0.6 m which has its centre between \(A\) and \(C\). The string makes an angle of \(30 ^ { \circ }\) with the vertical at \(A\) and an angle of \(45 ^ { \circ }\) with the vertical at \(C\) (see diagram).
  3. Calculate the speed of \(B\). The lower end of the string is detached from \(C\), and \(B\) is now attached to this end of the string. The other end of the string remains attached to \(A\). The bead is set in motion so that it moves with angular speed \(3 \mathrm { rad } \mathrm { s } ^ { - 1 }\) in a horizontal circle which has its centre vertically below \(A\).
  4. Calculate the tension in the string.
CAIE M2 2015 November Q6
9 marks Standard +0.8
6 \includegraphics[max width=\textwidth, alt={}, center]{e39fcb6b-7bd2-4dd5-bbd6-5de6e13b30b3-4_647_641_260_751} A uniform circular disc has centre \(O\) and radius 1.2 m . The centre of the disc is at the origin of \(x\)-and \(y\)-axes. Two circular holes with centres at \(A\) and \(B\) are made in the disc (see diagram). The point \(A\) is on the negative \(x\)-axis with \(O A = 0.5 \mathrm {~m}\). The point \(B\) is on the negative \(y\)-axis with \(O B = 0.7 \mathrm {~m}\). The hole with centre \(A\) has radius 0.3 m and the hole with centre \(B\) has radius 0.4 m . Find the distance of the centre of mass of the object from
  1. the \(x\)-axis,
  2. the \(y\)-axis. The object can rotate freely in a vertical plane about a horizontal axis through \(O\).
  3. Calculate the angle which \(O A\) makes with the vertical when the object rests in equilibrium.
CAIE M2 2015 November Q1
4 marks Standard +0.3
1 A particle is projected with speed \(25 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) at an angle of \(50 ^ { \circ }\) above the horizontal. Calculate the time after projection when the particle has speed \(18 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) and is rising.
CAIE M2 2015 November Q2
5 marks Moderate -0.8
2 One end of a light inextensible string of length 0.5 m is attached to a fixed point \(A\). A particle \(P\) of mass 0.2 kg is attached to the other end of the string. \(P\) moves with constant speed in a horizontal circle with centre \(O\) which is 0.4 m vertically below \(A\).
  1. Show that the tension in the string is 2.5 N .
  2. Find the speed of \(P\).
CAIE M2 2015 November Q3
5 marks Standard +0.3
3 A particle \(P\) is projected with speed \(V \mathrm {~m} \mathrm {~s} ^ { - 1 }\) at an angle of \(\theta ^ { \circ }\) above the horizontal from a point \(O\) on horizontal ground. At the instant 4 s after projection the particle passes through the point \(A\), where \(O A = 40 \mathrm {~m}\) and the line \(O A\) makes an angle of \(30 ^ { \circ }\) with the horizontal. Calculate \(V\) and \(\theta\).
CAIE M2 2015 November Q4
7 marks Standard +0.3
4 \includegraphics[max width=\textwidth, alt={}, center]{727412ec-d783-4392-8b84-e7d5435a3f4e-2_387_613_1073_767} A particle \(P\) of mass 0.4 kg moves with constant speed in a horizontal circle on the smooth inner surface of a fixed hollow hemisphere with centre \(O\) and radius 0.5 m (see diagram).
  1. Given that the speed of the particle is \(4 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) and its angular speed is \(10 \mathrm { rad } \mathrm { s } ^ { - 1 }\), calculate the angle between \(O P\) and the vertical.
  2. Given instead that the magnitude of the force exerted on \(P\) by the hemisphere is 6 N , calculate
    (a) the angle between \(O P\) and the vertical,
    (b) the angular speed of \(P\).
CAIE M2 2015 November Q5
9 marks Challenging +1.2
5 A particle \(P\) of mass 0.5 kg is projected vertically upwards from a point on a horizontal surface. A resisting force of magnitude \(0.02 v ^ { 2 } \mathrm {~N}\) acts on \(P\), where \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\) is the upward velocity of \(P\) when it is a height of \(x \mathrm {~m}\) above the surface. The initial speed of \(P\) is \(8 \mathrm {~m} \mathrm {~s} ^ { - 1 }\).
  1. Show that, while \(P\) is moving upwards, \(v \frac { \mathrm {~d} v } { \mathrm {~d} x } = - 10 - 0.04 v ^ { 2 }\).
  2. Find the greatest height of \(P\) above the surface.
  3. Find the speed of \(P\) immediately before it strikes the surface after descending.
CAIE M2 2015 November Q6
9 marks Challenging +1.2
6 \includegraphics[max width=\textwidth, alt={}, center]{727412ec-d783-4392-8b84-e7d5435a3f4e-3_424_953_255_596} An object is formed by joining a hemispherical shell of radius 0.2 m and a solid cone with base radius 0.2 m and height \(h \mathrm {~m}\) along their circumferences. The centre of mass, \(G\), of the object is \(d \mathrm {~m}\) from the vertex of the cone on the axis of symmetry of the object. The object rests in equilibrium on a horizontal plane, with the curved surface of the cone in contact with the plane (see diagram). The object is on the point of toppling.
  1. Show that \(d = h + \frac { 0.04 } { h }\).
  2. It is given that the cone is uniform and of weight 4 N , and that the hemispherical shell is uniform and of weight \(W \mathrm {~N}\). Given also that \(h = 0.8\), find \(W\).