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CAIE M2 2013 June Q2
6 marks Standard +0.3
2 A uniform semicircular lamina of radius 0.25 m has diameter \(A B\). It is freely suspended at \(A\) from a fixed point and hangs in equilibrium.
  1. Find the distance of the centre of mass of the lamina from the diameter \(A B\).
  2. Calculate the angle which the diameter \(A B\) makes with the vertical. The lamina is now held in equilibrium with the diameter \(A B\) vertical by means of a force applied at \(B\). This force has magnitude 6 N and acts at \(45 ^ { \circ }\) to the upward vertical in the plane of the lamina.
  3. Calculate the weight of the lamina.
CAIE M2 2013 June Q3
7 marks Challenging +1.2
3 A particle \(P\) of mass 0.2 kg is attached to one end of a light elastic string of natural length 1.6 m and modulus of elasticity 18 N . The other end of the string is attached to a fixed point \(O\) which is 1.6 m above a smooth horizontal surface. \(P\) is placed on the surface vertically below \(O\) and then projected horizontally. \(P\) moves with initial speed \(1.5 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) in a straight line on the surface. Show that, when \(O P = 1.8 \mathrm {~m}\),
  1. \(P\) is at instantaneous rest,
  2. \(P\) is on the point of losing contact with the surface.
CAIE M2 2013 June Q4
6 marks Moderate -0.8
4 A ball \(B\) is projected from a point \(O\) on horizontal ground at an angle of \(40 ^ { \circ }\) above the horizontal. \(B\) hits the ground 1.8 s after the instant of projection. Calculate
  1. the speed of projection of \(B\),
  2. the greatest height of \(B\),
  3. the distance from \(O\) of the point at which \(B\) hits the ground.
CAIE M2 2013 June Q5
8 marks Challenging +1.2
5 \includegraphics[max width=\textwidth, alt={}, center]{c85aa042-7b8c-44cc-b579-a5deef91e7e5-3_341_529_260_808} A block \(B\) of mass 3 kg is attached to one end of a light elastic string of modulus of elasticity 70 N and natural length 1.4 m . The other end of the string is attached to a particle \(P\) of mass 0.3 kg . \(B\) is at rest 0.9 m from the edge of a horizontal table and the string passes over a small smooth pulley at the edge of the table. \(P\) is released from rest at a point next to the pulley and falls vertically. At the first instant when \(P\) is 0.8 m below the pulley and descending, \(B\) is in limiting equilibrium with the part of the string attached to \(B\) horizontal (see diagram).
  1. Calculate the speed of \(P\) when \(B\) is first in limiting equilibrium.
  2. Find the coefficient of friction between \(B\) and the table.
CAIE M2 2013 June Q6
9 marks Standard +0.3
6 \includegraphics[max width=\textwidth, alt={}, center]{c85aa042-7b8c-44cc-b579-a5deef91e7e5-3_291_993_1238_575} A uniform solid cone of height 0.6 m and mass 0.5 kg has its axis of symmetry vertical and its vertex \(V\) uppermost. The semi-vertical angle of the cone is \(60 ^ { \circ }\) and the surface is smooth. The cone is fixed to a horizontal surface. A particle \(P\) of mass 0.2 kg is connected to \(V\) by a light inextensible string of length 0.4 m (see diagram).
  1. Calculate the height, above the horizontal surface, of the centre of mass of the cone with the particle. \(P\) is set in motion, and moves with angular speed \(4 \mathrm { rad } \mathrm { s } ^ { - 1 }\) in a circular path on the surface of the cone.
  2. Show that the tension in the string is 1.96 N , and calculate the magnitude of the force exerted on \(P\) by the cone.
  3. Find the speed of \(P\).
CAIE M2 2013 June Q7
10 marks Standard +0.8
7 A particle \(P\) of mass 0.5 kg moves in a straight line on a smooth horizontal surface. The velocity of \(P\) is \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\) when the displacement of \(P\) from \(O\) is \(x \mathrm {~m}\). A single horizontal force of magnitude \(0.16 \mathrm { e } ^ { x } \mathrm {~N}\) acts on \(P\) in the direction \(O P\). The velocity of \(P\) when it is at \(O\) is \(0.8 \mathrm {~m} \mathrm {~s} ^ { - 1 }\).
  1. Show that \(v = 0.8 \mathrm { e } ^ { \frac { 1 } { 2 } x }\).
  2. Find the time taken by \(P\) to travel 1.4 m from \(O\).
CAIE M2 2013 June Q4
Standard +0.3
4 \includegraphics[max width=\textwidth, alt={}, center]{10abedc3-c814-47c0-8ed4-849ef325feca-2_631_531_1117_806} A smooth hollow cylinder of internal radius 0.3 m is fixed with its axis vertical. One end of a light inextensible string of length 0.5 m is fixed to a point \(A\) on the axis. The other end of the string is attached to a particle \(P\) of mass 0.2 kg which moves in a horizontal circle on the surface of the cylinder (see diagram).
  1. Find the tension in the string.
  2. Find the least angular speed of \(P\) for which the motion is possible.
  3. Calculate the magnitude of the force exerted on \(P\) by the cylinder given that the speed of \(P\) is \(1.8 \mathrm {~m} \mathrm {~s} ^ { - 1 }\).
CAIE M2 2013 June Q5
Standard +0.3
5 One end of a light elastic string \(S _ { 1 }\) of modulus of elasticity 20 N and natural length 0.5 m is attached to a fixed point \(O\). The other end of \(S _ { 1 }\) is attached to a particle \(P\) of mass \(0.4 \mathrm {~kg} . P\) hangs in equilibrium vertically below \(O\).
  1. Find the distance \(O P\). The opposite ends of a light inextensible string \(S _ { 2 }\) of length \(l \mathrm {~m}\) are now attached to \(O\) and \(P\) respectively. The elastic string \(S _ { 1 }\) remains attached to \(O\) and \(P\). The particle \(P\) hangs in equilibrium vertically below \(O\).
  2. Find the tension in the inextensible string \(S _ { 2 }\) for each of the following cases:
    1. \(l < 0.5\);
    2. \(l > 0.6\);
    3. \(l = 0.54\). In the case \(l = 0.54\), the inextensible string \(S _ { 2 }\) suddenly breaks and \(P\) begins to descend vertically.
    4. Calculate the greatest speed of \(P\) in the subsequent motion.
CAIE M2 2013 June Q6
Easy -4.0
6 \includegraphics[max width=\textwidth, alt={}, center]{10abedc3-c814-47c0-8ed4-849ef325feca-3_474_860_1288_644} A uniform solid cone of height 1.2 m and semi-vertical angle \(\theta ^ { \circ }\) is divided into two parts by a cut parallel to and 0.4 m from the circular base. The upper conical part, \(C\), has weight 16 N , and the lower part, \(L\), has weight 38 N . The two parts of the solid rest in equilibrium with the larger plane face of \(L\) on a horizontal surface and the smaller plane face of \(L\) covered by the base of \(C\) (see diagram).
  1. Calculate the distance of the centre of mass of \(L\) from its larger plane face. An increasing horizontal force is applied to the vertex of \(C\). Equilibrium is broken when the magnitude of this force first exceeds 4 N , and \(C\) begins to slide on \(L\).
  2. By considering the forces on \(C\),
    1. find the coefficient of friction between \(C\) and \(L\),
    2. show that \(\theta > 14.0\), correct to 3 significant figures. \(C\) is removed and \(L\) is placed with its curved surface on the horizontal surface.
    3. Given that \(L\) is on the point of toppling, calculate \(\theta\).
CAIE M2 2014 June Q2
Moderate -0.3
2 \includegraphics[max width=\textwidth, alt={}, center]{5998f4b1-21da-4c25-8b09-91a1cb1eee42-2_565_549_438_797} A non-uniform rod \(A B\) of weight 6 N rests in limiting equilibrium with the end \(A\) in contact with a rough vertical wall. \(A B = 1.2 \mathrm {~m}\), the centre of mass of the rod is 0.8 m from \(A\), and the angle between \(A B\) and the downward vertical is \(\theta ^ { \circ }\). A force of magnitude 10 N acting at an angle of \(30 ^ { \circ }\) to the upwards vertical is applied to the rod at \(B\) (see diagram). The rod and the line of action of the 10 N force lie in a vertical plane perpendicular to the wall. Calculate
  1. the value of \(\theta\),
  2. the coefficient of friction between the rod and the wall.
CAIE M2 2014 June Q5
Standard +0.3
5 \includegraphics[max width=\textwidth, alt={}, center]{5998f4b1-21da-4c25-8b09-91a1cb1eee42-3_365_679_264_733} A uniform metal frame \(O A B C\) is made from a semicircular \(\operatorname { arc } A B C\) of radius 1.8 m , and a straight \(\operatorname { rod } A O C\) with \(A O = O C = 1.8 \mathrm {~m}\) (see diagram).
  1. Calculate the distance of the centre of mass of the frame from \(O\). A uniform semicircular lamina of radius 1.8 m has weight 27.5 N . A non-uniform object is formed by attaching the frame \(O A B C\) around the perimeter of the lamina. The object is freely suspended from a fixed point at \(A\) and hangs in equilibrium. The diameter \(A O C\) of the object makes an angle of \(22 ^ { \circ }\) with the vertical.
  2. Calculate the weight of the frame.
CAIE M2 2014 June Q6
Moderate -0.5
6 A particle \(P\) of mass 0.6 kg is released from rest at a point above ground level and falls vertically. The motion of \(P\) is opposed by a force of magnitude \(3 v \mathrm {~N}\), where \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\) is the speed of \(P\). Immediately before \(P\) reaches the ground, \(v = 1.95\).
  1. Calculate the time after its release when \(P\) reaches the ground. \(P\) is now projected horizontally with speed \(1.95 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) across a smooth horizontal surface. The motion of \(P\) is again opposed by a force of magnitude \(3 v \mathrm {~N}\), where \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\) is the speed of \(P\).
  2. Calculate the distance \(P\) travels after projection before coming to rest.
CAIE M2 2014 June Q7
Standard +0.3
7 \includegraphics[max width=\textwidth, alt={}, center]{5998f4b1-21da-4c25-8b09-91a1cb1eee42-4_357_776_260_680} A small bead \(B\) of mass \(m \mathrm {~kg}\) moves with constant speed in a horizontal circle on a fixed smooth wire. The wire is in the form of a circle with centre \(O\) and radius 0.4 m . One end of a light elastic string of natural length 0.4 m and modulus of elasticity \(42 m \mathrm {~N}\) is attached to \(B\). The other end of the string is attached to a fixed point \(A\) which is 0.3 m vertically above \(O\) (see diagram).
  1. Show that the vertical component of the contact force exerted by the wire on the bead is 3.7 mN upwards.
  2. Given that the contact force has zero horizontal component, find the angular speed of \(B\).
  3. Given instead that the horizontal component of the contact force has magnitude \(2 m \mathrm {~N}\), find the two possible speeds of \(B\). The string is now removed. \(B\) again moves on the wire in a horizontal circle with constant speed. It is given that the vertical and horizontal components of the contact force exerted by the wire on the bead have equal magnitudes.
  4. Find the speed of \(B\). \end{document}
CAIE M2 2014 June Q1
3 marks Standard +0.3
1 A light elastic string has modulus of elasticity 5 N and natural length 1.5 m . One end of the string is attached to a fixed point \(O\) and a particle \(P\) of mass 0.1 kg is attached to the other end of the string. \(P\) is released from rest at the point 2.4 m vertically below \(O\). Calculate the speed of \(P\) at the instant the string first becomes slack.
CAIE M2 2014 June Q2
3 marks Standard +0.3
2 \includegraphics[max width=\textwidth, alt={}, center]{9c82b387-8e5e-48b9-973d-5337b4e56a66-2_536_905_520_621} A uniform lamina \(A B C\) in the shape of an isosceles triangle has weight 24 N . The perpendicular distance from \(A\) to \(B C\) is 12 cm . The lamina rests in a vertical plane in equilibrium, with the vertex \(A\) in contact with a horizontal surface. Angle \(B A C = 100 ^ { \circ }\) and \(A B\) makes an angle of \(10 ^ { \circ }\) with the horizontal. Equilibrium is maintained by a force of magnitude \(F \mathrm {~N}\) acting along \(B C\) (see diagram). Show that \(F = 8\).
CAIE M2 2014 June Q3
8 marks Standard +0.3
3 A small block \(B\) of mass 0.2 kg is placed at a fixed point \(O\) on a smooth horizontal surface. A horizontal force of magnitude 0.42 N is applied to \(B\). At time \(t \mathrm {~s}\) after the force is first applied, the velocity of \(B\) away from \(O\) is \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\).
  1. Find the value of \(v\) when \(t = 1\). For \(t > 1\) an additional force, of magnitude \(0.32 t \mathrm {~N}\) and directed towards \(O\), is applied to \(B\). The force of magnitude 0.42 N continues to act as before.
  2. Find the value of \(v\) when \(t = 2\). For \(t > 2\) a third force, of magnitude \(0.06 t ^ { 2 } \mathrm {~N}\) and directed away from \(O\), is applied to \(B\). The other two forces continue to act as before.
  3. Show that the velocity of \(B\) is the same when \(t = 2\) and when \(t = 3\).
CAIE M2 2014 June Q4
8 marks Standard +0.3
4 One end of a light inextensible string of length 2.4 m is attached to a fixed point \(A\). The other end of the string is attached to a particle \(P\) of mass \(0.2 \mathrm {~kg} . P\) moves with constant speed in a horizontal circle which has its centre vertically below \(A\), with the string taut and making an angle of \(60 ^ { \circ }\) with the vertical.
  1. Find the speed of \(P\). The string of length 2.4 m is removed, and \(P\) is now connected to \(A\) by a light inextensible string of length 1.2 m . The particle \(P\) moves with angular speed \(4 \mathrm { rad } \mathrm { s } ^ { - 1 }\) in a horizontal circle with its centre vertically below \(A\).
  2. Calculate the angle between the string and the vertical.
CAIE M2 2014 June Q5
9 marks Moderate -0.8
5 A small ball is thrown horizontally with speed \(5 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) from a point \(O\) on the roof of a building. At time \(t \mathrm {~s}\) after projection, the horizontal and vertically downwards displacements of the ball from \(O\) are \(x \mathrm {~m}\) and \(y \mathrm {~m}\) respectively.
  1. Express \(x\) and \(y\) in terms of \(t\), and hence show that the equation of the trajectory of the ball is \(y = 0.2 x ^ { 2 }\). The ball strikes the horizontal ground which surrounds the building at a point \(A\).
  2. Given that \(O A = 18 \mathrm {~m}\), calculate the value of \(x\) at \(A\), and the speed of the ball immediately before it strikes the ground at \(A\).
CAIE M2 2014 June Q6
9 marks Challenging +1.2
6 \includegraphics[max width=\textwidth, alt={}, center]{9c82b387-8e5e-48b9-973d-5337b4e56a66-3_652_618_849_762} A particle \(P\) of mass 0.6 kg is attached to one end of a light elastic string of natural length 1.5 m and modulus of elasticity 9 N . The string passes through a small smooth ring \(R\) fixed at a height of 0.4 m above a rough horizontal surface. The other end of the string is attached to a fixed point \(O\) which is 1.5 m vertically above \(R\). The points \(A\) and \(B\) are on the horizontal surface, and \(B\) is vertically below \(R\). When \(P\) is on the surface between \(A\) and \(B , R P\) makes an acute angle \(\theta ^ { \circ }\) with the horizontal (see diagram).
  1. Show that the normal force exerted on \(P\) by the surface has magnitude 3.6 N , for all values of \(\theta\). \(P\) is projected with speed \(2.5 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) towards \(B\) from its initial position at \(A\) where \(\theta = 30\). The speed of \(P\) when it passes through \(B\) is \(3 \mathrm {~m} \mathrm {~s} ^ { - 1 }\).
  2. Find the work done against friction as \(P\) moves from \(A\) to \(B\).
  3. Calculate the value of the coefficient of friction between \(P\) and the surface.
CAIE M2 2014 June Q7
10 marks Challenging +1.2
7 \includegraphics[max width=\textwidth, alt={}, center]{9c82b387-8e5e-48b9-973d-5337b4e56a66-4_553_630_258_753} The diagram shows a container which consists of a bowl of weight 14 N and a handle of weight 8 N . The bowl of the container is in the form of a uniform hemispherical shell with centre \(O\) and radius 0.3 m . The handle is in the form of a uniform semicircular arc of radius 0.3 m and is freely hinged to the bowl at \(A\) and \(B\), where \(A B\) is a diameter of the bowl.
  1. Calculate the distance of the centre of mass of the container from \(O\) for the position indicated in the diagram, where the handle is perpendicular to the rim of the bowl.
  2. Show that the distance of the centre of mass of the container from \(O\) when the handle lies on the rim of the bowl is 0.118 m , correct to 3 significant figures. In the case when the handle lies on the rim of the bowl, the container rests in equilibrium with the curved surface of the bowl on a horizontal table.
  3. Find the angle which the plane containing the rim of the bowl makes with the horizontal.
CAIE M2 2015 June Q1
3 marks Moderate -0.8
1 A particle \(P\) of mass 0.6 kg is on the rough surface of a horizontal disc with centre \(O\). The distance \(O P\) is 0.4 m . The disc and \(P\) rotate with angular speed \(3 \mathrm { rad } \mathrm { s } ^ { - 1 }\) about a vertical axis which passes through \(O\). Find the magnitude of the frictional force which the disc exerts on the particle, and state the direction of this force.
CAIE M2 2015 June Q2
5 marks Standard +0.3
2 One end of a light elastic string of natural length 0.5 m and modulus of elasticity 30 N is attached to a fixed point \(O\). The other end of the string is attached to a particle \(P\) which hangs in equilibrium vertically below \(O\), with \(O P = 0.8 \mathrm {~m}\).
  1. Show that the mass of \(P\) is 1.8 kg . The particle is pulled vertically downwards and released from rest from the point where \(O P = 1.2 \mathrm {~m}\).
  2. Find the speed of \(P\) at the instant when the string first becomes slack.
CAIE M2 2015 June Q3
6 marks Standard +0.3
3 A triangular frame \(A B C\) consists of two uniform rigid rods each of length 0.8 m and weight 3 N , and a longer uniform rod of weight 4 N . The triangular frame has \(A B = B C\), and angle \(B A C =\) angle \(B C A = 30 ^ { \circ }\).
  1. Calculate the distance of the centre of mass of the frame from \(A C\). \includegraphics[max width=\textwidth, alt={}, center]{a03ad6c1-b4a3-4007-8d3b-ce289a998a55-2_722_335_1302_904} The vertex \(A\) of the frame is attached to a smooth hinge at a fixed point. The frame is held in equilibrium with \(A C\) vertical by a vertical force of magnitude \(F \mathrm {~N}\) applied to the frame at \(B\) (see diagram).
  2. Calculate \(F\), and state the magnitude and direction of the force acting on the frame at the hinge.
CAIE M2 2015 June Q4
8 marks Challenging +1.2
4 One end of a light inextensible string of length 0.5 m is attached to a fixed point \(A\). The other end of the string is attached to a particle \(P\) of weight 6 N . Another light inextensible string of length 0.5 m connects \(P\) to a fixed point \(B\) which is 0.8 m vertically below \(A\). The particle \(P\) moves with constant speed in a horizontal circle with centre at the mid-point of \(A B\). Both strings are taut.
  1. Calculate the speed of \(P\) when the tension in the string \(B P\) is 2 N .
  2. Show that the angular speed of \(P\) must exceed \(5 \mathrm { rad } \mathrm { s } ^ { - 1 }\).
CAIE M2 2015 June Q5
7 marks Challenging +1.2
5 \includegraphics[max width=\textwidth, alt={}, center]{a03ad6c1-b4a3-4007-8d3b-ce289a998a55-3_499_721_715_712} A uniform solid cube with edges of length 0.4 m rests in equilibrium on a rough plane inclined at an angle of \(30 ^ { \circ }\) to the horizontal. \(A B C D\) is a cross-section through the centre of mass of the cube, with \(A B\) along a line of greatest slope. \(B\) lies below the level of \(A\). One end of a light elastic string with modulus of elasticity 12 N and natural length 0.4 m is attached to \(C\). The other end of the string is attached to a point below the level of \(B\) on the same line of greatest slope, such that the string makes an angle of \(30 ^ { \circ }\) with the plane (see diagram). The cube is on the point of toppling. Find
  1. the tension in the string,
  2. the weight of the cube.