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CAIE Further Paper 3 2020 June Q1
2 marks Standard +0.3
1 A particle \(P\) of mass \(m\) is attached to one end of a light inextensible string of length \(a\). The other end of the string is attached to a fixed point \(O\) on a smooth horizontal plane. The particle \(P\) moves in horizontal circles about \(O\). The tension in the string is \(4 m g\). Find, in terms of \(a\) and \(g\), the time that \(P\) takes to make one complete revolution.
CAIE Further Paper 3 2020 June Q2
6 marks Standard +0.3
2 A particle \(Q\) of mass \(m \mathrm {~kg}\) falls from rest under gravity. The motion of \(Q\) is resisted by a force of magnitude \(m k v \mathrm {~N}\), where \(v \mathrm {~ms} ^ { - 1 }\) is the speed of \(Q\) at time \(t \mathrm {~s}\) and \(k\) is a positive constant. Find an expression for \(v\) in terms of \(g , k\) and \(t\).
CAIE Further Paper 3 2020 June Q3
6 marks Standard +0.8
3 A particle \(Q\) of mass \(m\) is attached to a fixed point \(O\) by a light inextensible string of length \(a\). The particle moves in complete vertical circles about \(O\). The points \(A\) and \(B\) are on the path of \(Q\) with \(A B\) a diameter of the circle. \(O A\) makes an angle of \(60 ^ { \circ }\) with the downward vertical through \(O\) and \(O B\) makes an angle of \(60 ^ { \circ }\) with the upward vertical through \(O\). The speed of \(Q\) when it is at \(A\) is \(2 \sqrt { \mathrm { ag } }\). Given that \(T _ { A }\) and \(T _ { B }\) are the tensions in the string at \(A\) and \(B\) respectively, find the ratio \(T _ { A } : T _ { B }\). \includegraphics[max width=\textwidth, alt={}, center]{5cc14ffc-e957-4582-b9d0-182fd89b3df5-06_880_428_260_817} A uniform solid circular cone, of vertical height \(4 r\) and radius \(2 r\), is attached to a uniform solid cylinder, of height \(3 r\) and radius \(k r\), where \(k\) is a constant less than 2 . The base of the cone is joined to one of the circular faces of the cylinder so that the axes of symmetry of the two solids coincide (see diagram). The cone and the cylinder are made of the same material.
  1. Show that the distance of the centre of mass of the combined solid from the vertex of the cone is \(\frac { \left( 99 \mathrm { k } ^ { 2 } + 96 \right) \mathrm { r } } { 18 \mathrm { k } ^ { 2 } + 32 }\).
    The point \(C\) is on the circumference of the base of the cone. When the combined solid is freely suspended from \(C\) and hanging in equilibrium, the diameter through \(C\) makes an angle \(\alpha\) with the downward vertical, where \(\tan \alpha = \frac { 1 } { 8 }\).
  2. Given that the centre of mass of the combined solid is within the cylinder, find the value of \(k\). [4]
CAIE Further Paper 3 2020 June Q5
8 marks Challenging +1.8
5 \includegraphics[max width=\textwidth, alt={}, center]{5cc14ffc-e957-4582-b9d0-182fd89b3df5-08_561_1068_255_500} Two uniform smooth spheres \(A\) and \(B\) of equal radii each have mass \(m\). The two spheres are each moving with speed \(u\) on a horizontal surface when they collide. Immediately before the collision A's direction of motion makes an angle of \(\alpha ^ { \circ }\) with the line of centres, and \(B\) 's direction of motion is perpendicular to that of \(A\) (see diagram). The coefficient of restitution between the spheres is \(e\). Immediately after the collision, \(B\) moves in a direction at right angles to the line of centres.
  1. Show that \(\tan \alpha = \frac { 1 + e } { 1 - e }\).
  2. Given that \(\tan \alpha = 2\), find the speed of \(A\) after the collision.
CAIE Further Paper 3 2020 June Q6
10 marks Challenging +1.2
6 A particle \(P\) is projected with speed \(u\) at an angle \(\theta\) above the horizontal from a point \(O\) on a horizontal plane and moves freely under gravity. The direction of motion of \(P\) makes an angle \(\alpha\) above the horizontal when \(P\) first reaches three-quarters of its greatest height.
  1. Show that \(\tan \alpha = \frac { 1 } { 2 } \tan \theta\).
  2. Given that \(\tan \theta = \frac { 4 } { 3 }\), find the horizontal distance travelled by \(P\) when it first reaches three-quarters of its greatest height. Give your answer in terms of \(u\) and \(g\). \includegraphics[max width=\textwidth, alt={}, center]{5cc14ffc-e957-4582-b9d0-182fd89b3df5-12_241_1009_269_529} One end of a light spring of natural length \(a\) and modulus of elasticity \(4 m g\) is attached to a fixed point \(O\). The other end of the spring is attached to a particle \(A\) of mass \(k m\), where \(k\) is a constant. Initially the spring lies at rest on a smooth horizontal surface and has length \(a\). A second particle \(B\), of mass \(m\), is moving towards \(A\) with speed \(\sqrt { \frac { 4 } { 3 } \mathrm { ga } }\) along the line of the spring from the opposite direction to \(O\) (see diagram). The particles \(A\) and \(B\) collide and coalesce. At a point \(C\) in the subsequent motion, the length of the spring is \(\frac { 3 } { 4 } a\) and the speed of the combined particle is half of its initial speed.
  3. Find the value of \(k\).
    At the point \(C\) the horizontal surface becomes rough, with coefficient of friction \(\mu\) between the combined particle and the surface. The deceleration of the combined particle at \(C\) is \(\frac { 9 } { 20 } \mathrm {~g}\).
  4. Find the value of \(\mu\).
    If you use the following lined page to complete the answer(s) to any question(s), the question number(s) must be clearly shown.
CAIE Further Paper 3 2021 June Q1
5 marks Challenging +1.8
1 A particle \(P\) of mass 1 kg is moving along a straight line against a resistive force of magnitude \(\frac { 10 \sqrt { \mathrm { v } } } { ( \mathrm { t } + 1 ) ^ { 2 } } \mathrm {~N}\), where \(\mathrm { vms } ^ { - 1 }\) is the speed of \(P\) at time \(t \mathrm {~s}\). When \(\mathrm { t } = 0 , \mathrm { v } = 25\). Find an expression for \(v\) in terms of \(t\).
CAIE Further Paper 3 2021 June Q2
6 marks Standard +0.8
2 A hollow hemispherical bowl of radius \(a\) has a smooth inner surface and is fixed with its axis vertical. A particle \(P\) of mass \(m\) moves in horizontal circles on the inner surface of the bowl, at a height \(x\) above the lowest point of the bowl. The speed of \(P\) is \(\sqrt { \frac { 8 } { 3 } } \mathrm { ga }\). Find \(x\) in terms of \(a\).
CAIE Further Paper 3 2021 June Q3
7 marks Challenging +1.2
3 One end of a light elastic string, of natural length \(a\) and modulus of elasticity \(k m g\), is attached to a fixed point \(A\). The other end of the string is attached to a particle \(P\) of mass \(4 m\). The particle \(P\) hangs in equilibrium a distance \(x\) vertically below \(A\).
  1. Show that \(\mathrm { k } = \frac { 4 \mathrm { a } } { \mathrm { x } - \mathrm { a } }\).
    An additional particle, of mass \(2 m\), is now attached to \(P\) and the combined particle is released from rest at the original equilibrium position of \(P\). When the combined particle has descended a distance \(\frac { 1 } { 3 } a\), its speed is \(\frac { 1 } { 3 } \sqrt { \mathrm { ga } }\).
  2. Find \(x\) in terms of \(a\). \includegraphics[max width=\textwidth, alt={}, center]{cef5eb87-1760-4aed-8e4d-e076d5d10252-06_602_520_264_753} A uniform solid circular cone has vertical height \(k h\) and radius \(r\). A uniform solid cylinder has height \(h\) and radius \(r\). The base of the cone is joined to one of the circular faces of the cylinder so that the axes of symmetry of the two solids coincide (see diagram, which shows a cross-section). The cone and the cylinder are made of the same material.
  3. Show that the distance of the centre of mass of the combined solid from the base of the cylinder is \(\frac { \mathrm { h } \left( \mathrm { k } ^ { 2 } + 4 \mathrm { k } + 6 \right) } { 4 ( 3 + \mathrm { k } ) }\).
    The solid is placed on a plane that is inclined to the horizontal at an angle \(\theta\). The base of the cylinder is in contact with the plane. The plane is sufficiently rough to prevent sliding. It is given that \(3 h = 2 r\) and that the solid is on the point of toppling when \(\tan \theta = \frac { 4 } { 3 }\).
  4. Find the value of \(k\).
CAIE Further Paper 3 2021 June Q5
8 marks
5 A particle \(P\) of mass \(m\) is attached to one end of a light inextensible string of length \(a\). The other end of the string is attached to a fixed point \(O\). The particle completes vertical circles with centre \(O\). The points \(A\) and \(B\) are on the path of \(P\), both on the same side of the vertical through \(O\). OA makes an angle \(\theta\) with the downward vertical through \(O\) and \(O B\) makes an angle \(\theta\) with the upward vertical through \(O\). The speed of \(P\) when it is at \(A\) is \(u\) and the speed of \(P\) when it is at \(B\) is \(\sqrt { \mathrm { ag } }\). The tensions in the string at \(A\) and \(B\) are \(T _ { A }\) and \(T _ { B }\) respectively. It is given that \(T _ { A } = 7 T _ { B }\). Find the value of \(\theta\) and find an expression for \(u\) in terms of \(a\) and \(g\). \includegraphics[max width=\textwidth, alt={}, center]{cef5eb87-1760-4aed-8e4d-e076d5d10252-10_558_1040_258_516} Two uniform smooth spheres \(A\) and \(B\) of equal radii each have mass \(m\). The two spheres are each moving with speed \(u\) on a horizontal surface when they collide. Immediately before the collision, A's direction of motion makes an angle \(\alpha\) with the line of centres, and \(B\) 's direction of motion makes an angle \(\beta\) with the line of centres (see diagram). The coefficient of restitution between the spheres is \(\frac { 1 } { 3 }\) and \(2 \cos \beta = \cos \alpha\).
  1. Show that the direction of motion of \(A\) after the collision is perpendicular to the line of centres.
    The total kinetic energy of the spheres after the collision is \(\frac { 3 } { 4 } \mathrm { mu } ^ { 2 }\).
  2. Find the value of \(\alpha\).
CAIE Further Paper 3 2021 June Q7
9 marks Standard +0.3
7 A particle \(P\) is projected from a point \(O\) on a horizontal plane and moves freely under gravity. The initial velocity of \(P\) is \(100 \mathrm {~ms} ^ { - 1 }\) at an angle \(\theta\) above the horizontal, where \(\tan \theta = \frac { 4 } { 3 }\). The two times at which \(P\) 's height above the plane is \(H \mathrm {~m}\) differ by 10 s .
  1. Find the value of \(H\). \includegraphics[max width=\textwidth, alt={}]{cef5eb87-1760-4aed-8e4d-e076d5d10252-12_72_1569_463_328} ........................................................................................................................................ ........................................................................................................................................ \includegraphics[max width=\textwidth, alt={}, center]{cef5eb87-1760-4aed-8e4d-e076d5d10252-12_72_1573_735_324} \includegraphics[max width=\textwidth, alt={}, center]{cef5eb87-1760-4aed-8e4d-e076d5d10252-12_72_1572_826_322} \includegraphics[max width=\textwidth, alt={}]{cef5eb87-1760-4aed-8e4d-e076d5d10252-12_74_1572_916_322} ........................................................................................................................................ . ......................................................................................................................................... \includegraphics[max width=\textwidth, alt={}, center]{cef5eb87-1760-4aed-8e4d-e076d5d10252-12_70_1570_1187_324} \includegraphics[max width=\textwidth, alt={}, center]{cef5eb87-1760-4aed-8e4d-e076d5d10252-12_67_1570_1279_324} \includegraphics[max width=\textwidth, alt={}, center]{cef5eb87-1760-4aed-8e4d-e076d5d10252-12_67_1570_1370_324} \includegraphics[max width=\textwidth, alt={}, center]{cef5eb87-1760-4aed-8e4d-e076d5d10252-12_64_1570_1462_324} \includegraphics[max width=\textwidth, alt={}]{cef5eb87-1760-4aed-8e4d-e076d5d10252-12_63_1570_1553_324} ......................................................................................................................................... . .......................................................................................................................................... ......................................................................................................................................... . \includegraphics[max width=\textwidth, alt={}, center]{cef5eb87-1760-4aed-8e4d-e076d5d10252-12_71_1570_1905_324} \includegraphics[max width=\textwidth, alt={}, center]{cef5eb87-1760-4aed-8e4d-e076d5d10252-12_74_1570_1994_324} \includegraphics[max width=\textwidth, alt={}]{cef5eb87-1760-4aed-8e4d-e076d5d10252-12_76_1570_2083_324} ......................................................................................................................................... .
  2. Find the magnitude and direction of the velocity of \(P\) one second before it strikes the plane.
    If you use the following lined page to complete the answer(s) to any question(s), the question number(s) must be clearly shown.
CAIE Further Paper 3 2021 June Q3
7 marks Challenging +1.2
3 One end of a light elastic string, of natural length \(a\) and modulus of elasticity \(k m g\), is attached to a fixed point \(A\). The other end of the string is attached to a particle \(P\) of mass \(4 m\). The particle \(P\) hangs in equilibrium a distance \(x\) vertically below \(A\).
  1. Show that \(\mathrm { k } = \frac { 4 \mathrm { a } } { \mathrm { x } - \mathrm { a } }\).
    An additional particle, of mass \(2 m\), is now attached to \(P\) and the combined particle is released from rest at the original equilibrium position of \(P\). When the combined particle has descended a distance \(\frac { 1 } { 3 } a\), its speed is \(\frac { 1 } { 3 } \sqrt { \mathrm { ga } }\).
  2. Find \(x\) in terms of \(a\). \includegraphics[max width=\textwidth, alt={}, center]{0671bcad-5f74-46d2-823b-48f62f5954ce-06_602_520_264_753} A uniform solid circular cone has vertical height \(k h\) and radius \(r\). A uniform solid cylinder has height \(h\) and radius \(r\). The base of the cone is joined to one of the circular faces of the cylinder so that the axes of symmetry of the two solids coincide (see diagram, which shows a cross-section). The cone and the cylinder are made of the same material.
  3. Show that the distance of the centre of mass of the combined solid from the base of the cylinder is \(\frac { \mathrm { h } \left( \mathrm { k } ^ { 2 } + 4 \mathrm { k } + 6 \right) } { 4 ( 3 + \mathrm { k } ) }\).
    The solid is placed on a plane that is inclined to the horizontal at an angle \(\theta\). The base of the cylinder is in contact with the plane. The plane is sufficiently rough to prevent sliding. It is given that \(3 h = 2 r\) and that the solid is on the point of toppling when \(\tan \theta = \frac { 4 } { 3 }\).
  4. Find the value of \(k\).
CAIE Further Paper 3 2021 June Q5
8 marks Challenging +1.2
5 A particle \(P\) of mass \(m\) is attached to one end of a light inextensible string of length \(a\). The other end of the string is attached to a fixed point \(O\). The particle completes vertical circles with centre \(O\). The points \(A\) and \(B\) are on the path of \(P\), both on the same side of the vertical through \(O\). OA makes an angle \(\theta\) with the downward vertical through \(O\) and \(O B\) makes an angle \(\theta\) with the upward vertical through \(O\). The speed of \(P\) when it is at \(A\) is \(u\) and the speed of \(P\) when it is at \(B\) is \(\sqrt { \mathrm { ag } }\). The tensions in the string at \(A\) and \(B\) are \(T _ { A }\) and \(T _ { B }\) respectively. It is given that \(T _ { A } = 7 T _ { B }\). Find the value of \(\theta\) and find an expression for \(u\) in terms of \(a\) and \(g\). \includegraphics[max width=\textwidth, alt={}, center]{0671bcad-5f74-46d2-823b-48f62f5954ce-10_558_1040_258_516} Two uniform smooth spheres \(A\) and \(B\) of equal radii each have mass \(m\). The two spheres are each moving with speed \(u\) on a horizontal surface when they collide. Immediately before the collision, A's direction of motion makes an angle \(\alpha\) with the line of centres, and \(B\) 's direction of motion makes an angle \(\beta\) with the line of centres (see diagram). The coefficient of restitution between the spheres is \(\frac { 1 } { 3 }\) and \(2 \cos \beta = \cos \alpha\).
  1. Show that the direction of motion of \(A\) after the collision is perpendicular to the line of centres.
    The total kinetic energy of the spheres after the collision is \(\frac { 3 } { 4 } \mathrm { mu } ^ { 2 }\).
  2. Find the value of \(\alpha\).
CAIE Further Paper 3 2021 June Q7
9 marks Standard +0.3
7 A particle \(P\) is projected from a point \(O\) on a horizontal plane and moves freely under gravity. The initial velocity of \(P\) is \(100 \mathrm {~ms} ^ { - 1 }\) at an angle \(\theta\) above the horizontal, where \(\tan \theta = \frac { 4 } { 3 }\). The two times at which \(P\) 's height above the plane is \(H \mathrm {~m}\) differ by 10 s .
  1. Find the value of \(H\). \includegraphics[max width=\textwidth, alt={}]{0671bcad-5f74-46d2-823b-48f62f5954ce-12_72_1569_463_328} ........................................................................................................................................ ........................................................................................................................................ \includegraphics[max width=\textwidth, alt={}, center]{0671bcad-5f74-46d2-823b-48f62f5954ce-12_72_1573_735_324} \includegraphics[max width=\textwidth, alt={}, center]{0671bcad-5f74-46d2-823b-48f62f5954ce-12_72_1575_826_322} \includegraphics[max width=\textwidth, alt={}]{0671bcad-5f74-46d2-823b-48f62f5954ce-12_74_1570_916_324} ........................................................................................................................................ . ......................................................................................................................................... \includegraphics[max width=\textwidth, alt={}, center]{0671bcad-5f74-46d2-823b-48f62f5954ce-12_70_1570_1187_324} \includegraphics[max width=\textwidth, alt={}, center]{0671bcad-5f74-46d2-823b-48f62f5954ce-12_67_1570_1279_324} \includegraphics[max width=\textwidth, alt={}, center]{0671bcad-5f74-46d2-823b-48f62f5954ce-12_67_1570_1370_324} \includegraphics[max width=\textwidth, alt={}, center]{0671bcad-5f74-46d2-823b-48f62f5954ce-12_64_1570_1462_324} \includegraphics[max width=\textwidth, alt={}]{0671bcad-5f74-46d2-823b-48f62f5954ce-12_63_1570_1553_324} ......................................................................................................................................... . .......................................................................................................................................... ......................................................................................................................................... . \includegraphics[max width=\textwidth, alt={}, center]{0671bcad-5f74-46d2-823b-48f62f5954ce-12_71_1570_1905_324} \includegraphics[max width=\textwidth, alt={}, center]{0671bcad-5f74-46d2-823b-48f62f5954ce-12_74_1570_1994_324} \includegraphics[max width=\textwidth, alt={}]{0671bcad-5f74-46d2-823b-48f62f5954ce-12_76_1570_2083_324} ......................................................................................................................................... .
  2. Find the magnitude and direction of the velocity of \(P\) one second before it strikes the plane.
    If you use the following lined page to complete the answer(s) to any question(s), the question number(s) must be clearly shown.
CAIE Further Paper 3 2021 June Q2
5 marks Standard +0.8
2 One end of a light elastic string of natural length 0.8 m and modulus of elasticity 36 N is attached to a fixed point \(O\) on a smooth plane. The plane is inclined at an angle \(\alpha\) to the horizontal, where \(\sin \alpha = \frac { 3 } { 5 }\). A particle \(P\) of mass 2 kg is attached to the other end of the string. The string lies along a line of greatest slope of the plane with the particle below the level of \(O\). The particle is projected with speed \(\sqrt { 2 } \mathrm {~ms} ^ { - 1 }\) directly down the plane from the position where \(O P\) is equal to the natural length of the string. Find the maximum extension of the string during the subsequent motion. \includegraphics[max width=\textwidth, alt={}, center]{6dcce6fe-7a19-4c5f-9361-20e7acda458f-04_380_1173_267_447} Particles \(A\) and \(B\), of masses \(3 m\) and \(m\) respectively, are connected by a light inextensible string of length \(a\) that passes through a fixed smooth ring \(R\). Particle \(B\) hangs in equilibrium vertically below the ring. Particle \(A\) moves in horizontal circles on a smooth horizontal surface with speed \(\frac { 2 } { 5 } \sqrt { \text { ga } }\). The angle between \(A R\) and \(B R\) is \(\theta\) (see diagram). The normal reaction between \(A\) and the surface is \(\frac { 12 } { 5 } \mathrm { mg }\).
  1. Find \(\cos \theta\).
  2. Find, in terms of \(a\), the distance of \(B\) below the ring. \includegraphics[max width=\textwidth, alt={}, center]{6dcce6fe-7a19-4c5f-9361-20e7acda458f-06_703_481_264_785} A particle of mass \(m\) is attached to one end of a light inextensible string of length \(a\). The other end of the string is attached to a fixed point \(O\). The particle is initially held with the string taut at the point \(A\), where \(O A\) makes an angle \(\theta\) with the downward vertical through \(O\). The particle is then projected with speed \(u\) perpendicular to \(O A\) and begins to move upwards in part of a vertical circle. The string goes slack when the particle is at the point \(B\) where angle \(A O B\) is a right angle. The speed of the particle when it is at \(B\) is \(\frac { 1 } { 2 } u\) (see diagram). Find the tension in the string at \(A\), giving your answer in terms of \(m\) and \(g\).
CAIE Further Paper 3 2021 June Q5
10 marks Standard +0.8
5 A particle \(P\) of mass \(m \mathrm {~kg}\) is projected vertically upwards from a point \(O\), with speed \(20 \mathrm {~ms} ^ { - 1 }\), and moves under gravity. There is a resistive force of magnitude \(2 m v \mathrm {~N}\), where \(v \mathrm {~ms} ^ { - 1 }\) is the speed of \(P\) at time \(t \mathrm {~s}\) after projection.
  1. Find an expression for \(v\) in terms of \(t\), while \(P\) is moving upwards.
    The displacement of \(P\) from \(O\) is \(x \mathrm {~m}\) at time \(t \mathrm {~s}\).
  2. Find an expression for \(x\) in terms of \(t\), while \(P\) is moving upwards.
  3. Find, correct to 3 significant figures, the greatest height above \(O\) reached by \(P\).
CAIE Further Paper 3 2021 June Q6
9 marks Challenging +1.2
6 \includegraphics[max width=\textwidth, alt={}, center]{6dcce6fe-7a19-4c5f-9361-20e7acda458f-10_339_983_258_541} Two uniform smooth spheres \(A\) and \(B\) of equal radii have masses \(m\) and \(k m\) respectively. Sphere \(A\) is moving with speed \(u\) on a smooth horizontal surface when it collides with sphere \(B\) which is at rest. Immediately before the collision, \(A\) 's direction of motion makes an angle \(\theta\) with the line of centres (see diagram). The coefficient of restitution between the spheres is \(\frac { 1 } { 3 }\).
  1. Show that the speed of \(B\) after the collision is \(\frac { 4 \mathrm { u } \cos \theta } { 3 ( 1 + \mathrm { k } ) }\).
    70\% of the total kinetic energy of the spheres is lost as a result of the collision.
  2. Given that \(\tan \theta = \frac { 1 } { 3 }\), find the value of \(k\).
CAIE Further Paper 3 2021 June Q7
9 marks Standard +0.3
7 A particle \(P\) is projected with speed \(u\) at an angle \(\theta\) above the horizontal from a point \(O\) on a horizontal plane and moves freely under gravity. The horizontal and vertical displacements of \(P\) from \(O\) at a subsequent time \(t\) are denoted by \(x\) and \(y\) respectively.
  1. Use the equation of the trajectory given in the List of formulae (MF19), together with the condition \(y = 0\), to establish an expression for the range \(R\) in terms of \(u , \theta\) and \(g\).
  2. Deduce an expression for the maximum height \(H\), in terms of \(u , \theta\) and \(g\).
    It is given that \(\mathrm { R } = \frac { 4 \mathrm { H } } { \sqrt { 3 } }\).
  3. Show that \(\theta = 60 ^ { \circ }\).
    It is given also that \(u = \sqrt { 40 } \mathrm {~ms} ^ { - 1 }\).
  4. Find, by differentiating the equation of the trajectory or otherwise, the set of values of \(x\) for which the direction of motion makes an angle of less than \(45 ^ { \circ }\) with the horizontal.
    If you use the following lined page to complete the answer(s) to any question(s), the question number(s) must be clearly shown.
CAIE Further Paper 3 2022 June Q2
5 marks Challenging +1.2
2 One end of a light inextensible string of length \(a\) is attached to a fixed point \(O\). A particle of mass \(m\) is attached to the other end of the string. The particle is held at the point \(A\) with the string taut. The angle between \(O A\) and the downward vertical is equal to \(\alpha\), where \(\cos \alpha = \frac { 4 } { 5 }\). The particle is projected from \(A\), perpendicular to the string in an upwards direction, with a speed \(\sqrt { 3 \text { ga } }\). It then moves along a circular path in a vertical plane. The string first goes slack when it makes an angle \(\theta\) with the upward vertical through \(O\). Find the value of \(\cos \theta\).
CAIE Further Paper 3 2022 June Q3
5 marks Standard +0.3
3 A particle \(P\) is moving in a horizontal straight line. Initially \(P\) is at the point \(O\) on the line and is moving with velocity \(25 \mathrm {~ms} ^ { - 1 }\). At time \(t \mathrm {~s}\) after passing through \(O\), the acceleration of \(P\) is \(\frac { 4000 } { ( 5 t + 4 ) ^ { 3 } } \mathrm {~ms} ^ { - 2 }\) in the direction \(P O\). The displacement of \(P\) from \(O\) at time \(t\) is \(x \mathrm {~m}\). Find an expression for \(x\) in terms of \(t\). \includegraphics[max width=\textwidth, alt={}, center]{9067c549-00d7-4078-b47d-87b28396e2ab-06_894_809_260_628} An object is composed of a hemispherical shell of radius \(2 a\) attached to a closed hollow circular cylinder of height \(h\) and base radius \(a\). The hemispherical shell and the hollow cylinder are made of the same uniform material. The axes of symmetry of the shell and the cylinder coincide. \(A B\) is a diameter of the lower end of the cylinder (see diagram).
  1. Find, in terms of \(a\) and \(h\), an expression for the distance of the centre of mass of the object from \(A B\). [4]
    The object is placed on a rough plane which is inclined to the horizontal at an angle \(\theta\), where \(\tan \theta = \frac { 2 } { 3 }\). The object is in equilibrium with \(A B\) in contact with the plane and lying along a line of greatest slope of the plane.
  2. Find the set of possible values of \(h\), in terms of \(a\). \includegraphics[max width=\textwidth, alt={}, center]{9067c549-00d7-4078-b47d-87b28396e2ab-08_629_1358_269_367} A light inextensible string \(A B\) passes through two small holes \(C\) and \(D\) in a smooth horizontal table where \(A C = 3 a\) and \(D B = a\). A particle of mass \(m\) is attached at the end \(A\) and moves in a horizontal circle with angular velocity \(\omega\). A particle of mass \(\frac { 3 } { 4 } m\) is attached to the end \(B\) and moves in a horizontal circle with angular velocity \(k \omega\). \(A C\) makes an angle \(\theta\) with the downward vertical and \(D B\) makes an angle \(\theta\) with the horizontal (see diagram). Find the value of \(k\).
CAIE Further Paper 3 2022 June Q6
9 marks Challenging +1.8
6 Two uniform smooth spheres \(A\) and \(B\) of equal radii have masses \(m\) and \(k m\) respectively. The two spheres are on a horizontal surface. Sphere \(A\) is travelling with speed \(u\) towards sphere \(B\) which is at rest. The spheres collide. Immediately before the collision, the direction of motion of \(A\) makes an angle \(\alpha\) with the line of centres. The coefficient of restitution between the spheres is \(\frac { 1 } { 2 }\).
  1. Show that the speed of \(B\) after the collision is \(\frac { 3 \mathrm { u } \cos \alpha } { 2 ( 1 + \mathrm { k } ) }\) and find also an expression for the speed of \(A\) along the line of centres after the collision, in terms of \(k , u\) and \(\alpha\).
    After the collision, the kinetic energy of \(A\) is equal to the kinetic energy of \(B\).
  2. Given that \(\tan \alpha = \frac { 2 } { 3 }\), find the possible values of \(k\).
CAIE Further Paper 3 2022 June Q7
11 marks Challenging +1.2
7 Particles \(P\) and \(Q\) are projected in the same vertical plane from a point \(O\) at the top of a cliff. The height of the cliff exceeds 50 m . Both particles move freely under gravity. Particle \(P\) is projected with speed \(\frac { 35 } { 2 } \mathrm {~ms} ^ { - 1 }\) at an angle \(\alpha\) above the horizontal, where \(\tan \alpha = \frac { 4 } { 3 }\). Particle \(Q\) is projected with speed \(u \mathrm {~ms} ^ { - 1 }\) at an angle \(\beta\) above the horizontal, where \(\tan \beta = \frac { 1 } { 2 }\). Particle \(Q\) is projected one second after the projection of particle \(P\). The particles collide \(T\) s after the projection of particle \(Q\).
  1. Write down expressions, in terms of \(T\), for the horizontal displacements of \(P\) and \(Q\) from \(O\) when they collide and hence show that \(4 \mathrm { uT } = 21 \sqrt { 5 } ( \mathrm {~T} + 1 )\).
  2. Find the value of \(T\).
  3. Find the horizontal and vertical displacements of the particles from \(O\) when they collide.
    If you use the following page to complete the answer to any question, the question number must be clearly shown.
CAIE Further Paper 3 2022 June Q3
4 marks Standard +0.3
3 A particle \(P\) is moving in a horizontal straight line. Initially \(P\) is at the point \(O\) on the line and is moving with velocity \(25 \mathrm {~ms} ^ { - 1 }\). At time \(t \mathrm {~s}\) after passing through \(O\), the acceleration of \(P\) is \(\frac { 4000 } { ( 5 t + 4 ) ^ { 3 } } \mathrm {~ms} ^ { - 2 }\) in the direction \(P O\). The displacement of \(P\) from \(O\) at time \(t\) is \(x \mathrm {~m}\). Find an expression for \(x\) in terms of \(t\). \includegraphics[max width=\textwidth, alt={}, center]{c486c59a-2493-4dd3-bf1e-dde57fe744d9-06_894_809_260_628} An object is composed of a hemispherical shell of radius \(2 a\) attached to a closed hollow circular cylinder of height \(h\) and base radius \(a\). The hemispherical shell and the hollow cylinder are made of the same uniform material. The axes of symmetry of the shell and the cylinder coincide. \(A B\) is a diameter of the lower end of the cylinder (see diagram).
  1. Find, in terms of \(a\) and \(h\), an expression for the distance of the centre of mass of the object from \(A B\). [4]
    The object is placed on a rough plane which is inclined to the horizontal at an angle \(\theta\), where \(\tan \theta = \frac { 2 } { 3 }\). The object is in equilibrium with \(A B\) in contact with the plane and lying along a line of greatest slope of the plane.
  2. Find the set of possible values of \(h\), in terms of \(a\). \includegraphics[max width=\textwidth, alt={}, center]{c486c59a-2493-4dd3-bf1e-dde57fe744d9-08_629_1358_269_367} A light inextensible string \(A B\) passes through two small holes \(C\) and \(D\) in a smooth horizontal table where \(A C = 3 a\) and \(D B = a\). A particle of mass \(m\) is attached at the end \(A\) and moves in a horizontal circle with angular velocity \(\omega\). A particle of mass \(\frac { 3 } { 4 } m\) is attached to the end \(B\) and moves in a horizontal circle with angular velocity \(k \omega\). \(A C\) makes an angle \(\theta\) with the downward vertical and \(D B\) makes an angle \(\theta\) with the horizontal (see diagram). Find the value of \(k\).
CAIE Further Paper 3 2022 June Q1
4 marks Standard +0.3
1 A uniform lamina \(O A B C\) is a trapezium whose vertices can be represented by coordinates in the \(x - y\) plane. The coordinates of the vertices are \(O ( 0,0 ) , A ( 15,0 ) , B ( 9,4 )\) and \(C ( 3,4 )\). Find the \(x\)-coordinate of the centre of mass of the lamina.
CAIE Further Paper 3 2022 June Q2
5 marks Challenging +1.2
2 A particle \(P\) of mass \(m\) is attached to one end of a light elastic string of natural length \(a\) and modulus of elasticity \(\frac { 4 } { 3 } \mathrm { mg }\). The other end of the string is attached to a fixed point \(O\) on a rough horizontal surface. The particle is at rest on the surface with the string at its natural length. The coefficient of friction between \(P\) and the surface is \(\frac { 1 } { 3 }\). The particle is projected along the surface in the direction \(O P\) with a speed of \(\frac { 1 } { 2 } \sqrt { \mathrm { ga } }\). Find the greatest extension of the string during the subsequent motion.
CAIE Further Paper 3 2022 June Q3
8 marks Standard +0.3
3 A particle \(P\) is projected with speed \(25 \mathrm {~ms} ^ { - 1 }\) at an angle \(\theta\) above the horizontal from a point \(O\) on a horizontal plane and moves freely under gravity. After 2 s the speed of \(P\) is \(15 \mathrm {~ms} ^ { - 1 }\).
  1. Find the value of \(\sin \theta\).
  2. Find the range of the flight.