Questions M2 (1537 questions)

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CAIE M2 2012 November Q2
8 marks Challenging +1.2
2 A light elastic string has natural length 4 m and modulus of elasticity 60 N . A particle \(P\) of mass 0.6 kg is attached to the mid-point of the string. The ends of the string are attached to fixed points \(A\) and \(B\) which lie in the same vertical line with \(A\) at a distance of 6 m above \(B\). \(P\) is projected vertically upwards from the point 2 m vertically above \(B\). In the subsequent motion, \(P\) comes to instantaneous rest at a distance of 2 m below \(A\).
  1. Calculate the speed of projection of \(P\).
  2. Calculate the distance of \(P\) from \(A\) at an instant when \(P\) has its greatest kinetic energy, and calculate this kinetic energy.
CAIE M2 2012 November Q3
8 marks Standard +0.3
3 \includegraphics[max width=\textwidth, alt={}, center]{e30ba526-db21-4904-96dc-c12a1f67c81a-2_397_1303_1790_422} The point \(O\) is 1.2 m below rough horizontal ground \(A B C\). A ball is projected from \(O\) with speed \(25 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) at an angle of \(70 ^ { \circ }\) to the horizontal. The ball passes over the point \(A\) after travelling a horizontal distance of 2 m . The ball subsequently bounces once on the ground at \(B\). The ball leaves \(B\) with speed \(15 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) and travels a further horizontal distance of 20 m before landing at \(C\) (see diagram).
  1. Calculate the height above the level of \(O\) of the ball when it is vertically above \(A\).
  2. Calculate the time after the instant of projection when the ball reaches \(B\).
  3. Find the angle which the trajectory of the ball makes with the horizontal immediately after it bounces at \(B\). \includegraphics[max width=\textwidth, alt={}, center]{e30ba526-db21-4904-96dc-c12a1f67c81a-3_663_695_258_726} A cylinder of height 0.9 m and radius 0.9 m is placed symmetrically on top of a cylinder of height \(h \mathrm {~m}\) and radius \(r \mathrm {~m}\), where \(r < 0.9\), with plane faces in contact and axes in the same vertical line \(A B\), where \(A\) and \(B\) are centres of plane faces of the cylinders (see diagram). Both cylinders are uniform and made of the same material. The lower cylinder is gradually tilted and when the axis of symmetry is inclined at \(45 ^ { \circ }\) to the horizontal the upper cylinder is on the point of toppling without sliding.
CAIE M2 2012 November Q5
9 marks Standard +0.3
5 A small ball \(B\) of mass 0.2 kg is attached to fixed points \(P\) and \(Q\) by two light inextensible strings of equal length. \(P\) is vertically above \(Q\), the strings are taut and each is inclined at \(60 ^ { \circ }\) to the vertical. \(B\) moves with constant speed in a horizontal circle of radius 0.6 m .
  1. Given that the tension in the string \(P B\) is 7 N , calculate
    1. the tension in string \(Q B\),
    2. the speed of \(B\).
    3. Given instead that \(B\) is moving with angular speed \(7 \mathrm { rad } \mathrm { s } ^ { - 1 }\), calculate the tension in the string \(Q B\).
CAIE M2 2012 November Q6
11 marks Standard +0.8
6 \includegraphics[max width=\textwidth, alt={}, center]{e30ba526-db21-4904-96dc-c12a1f67c81a-4_238_725_258_712} Two particles \(P\) and \(Q\), of masses 0.4 kg and 0.2 kg respectively, are attached to opposite ends of a light inextensible string. \(P\) is placed on a horizontal table and the string passes over a small smooth pulley at the edge of the table. The string is taut and the part of the string attached to \(Q\) is vertical (see diagram). The coefficient of friction between \(P\) and the table is 0.5 . \(Q\) is projected vertically downwards with speed \(5 \mathrm {~m} \mathrm {~s} ^ { - 1 }\), and at time \(t \mathrm {~s}\) after the instant of projection the speed of the particles is \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\). The motion of each particle is opposed by a resisting force of magnitude \(0.9 v \mathrm {~N}\). The particle \(P\) does not reach the pulley.
  1. Show that \(\frac { \mathrm { d } v } { \mathrm {~d} t } = - 3 v\).
  2. Find the value of \(t\) when the particles have speed \(2.5 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) and the distance that each particle has travelled in this time.
CAIE M2 2013 November Q1
2 marks Easy -1.2
1 A particle \(P\) of mass 0.3 kg is attached to one end of a light inextensible string of length 0.6 m . The other end of the string is attached to a fixed point \(O\) of a smooth horizontal plane. \(P\) moves on the plane at constant speed \(5 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) in a circle with centre \(O\). Calculate the tension in the string.
CAIE M2 2013 November Q2
6 marks Standard +0.3
2 \includegraphics[max width=\textwidth, alt={}, center]{6503ebb1-5649-4ca5-9500-da4fb28009dd-2_359_686_484_731} A uniform frame consists of a semicircular arc \(A B C\) of radius 0.6 m together with its diameter \(A O C\), where \(O\) is the centre of the semicircle (see diagram).
  1. Calculate the distance of the centre of mass of the frame from \(O\). The frame is freely suspended at \(A\) and hangs in equilibrium.
  2. Calculate the angle between \(A C\) and the vertical.
CAIE M2 2013 November Q3
7 marks Standard +0.3
3 A particle \(P\) of mass 0.8 kg moves along the \(x\)-axis on a horizontal surface. When the displacement of \(P\) from the origin \(O\) is \(x \mathrm {~m}\) the velocity of \(P\) is \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\) in the positive \(x\)-direction. Two horizontal forces act on \(P\). One force has magnitude \(4 \mathrm { e } ^ { - x } \mathrm {~N}\) and acts in the positive \(x\)-direction. The other force has magnitude \(2.4 x ^ { 2 } \mathrm {~N}\) and acts in the negative \(x\)-direction.
  1. Show that \(v \frac { \mathrm {~d} v } { \mathrm {~d} x } = 5 \mathrm { e } ^ { - x } - 3 x ^ { 2 }\).
  2. The velocity of \(P\) as it passes through \(O\) is \(6 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). Find the velocity of \(P\) when \(x = 2\).
CAIE M2 2013 November Q4
8 marks Standard +0.3
4 A small ball \(B\) is projected from a point \(O\) with speed \(14 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) at an angle of \(60 ^ { \circ }\) above the horizontal.
  1. Calculate the speed and direction of motion of \(B\) for the instant 1.8 s after projection. The point \(O\) is 2 m above a horizontal plane.
  2. Calculate the time after projection when \(B\) reaches the plane.
CAIE M2 2013 November Q5
8 marks Challenging +1.2
5 \includegraphics[max width=\textwidth, alt={}, center]{6503ebb1-5649-4ca5-9500-da4fb28009dd-3_540_537_255_804} A particle \(P\) of mass 0.2 kg is attached to a fixed point \(A\) by a light inextensible string of length 0.4 m . A second light inextensible string of length 0.3 m connects \(P\) to a fixed point \(B\) which is vertically below \(A\). The particle \(P\) moves in a horizontal circle, which has its centre on the line \(A B\), with the angle \(A P B = 90 ^ { \circ }\) (see diagram).
  1. Given that the tensions in the two strings are equal, calculate the speed of \(P\).
  2. It is given instead that \(P\) moves with its least possible angular speed for motion in this circle. Find this angular speed.
CAIE M2 2013 November Q6
9 marks Standard +0.3
6 \includegraphics[max width=\textwidth, alt={}, center]{6503ebb1-5649-4ca5-9500-da4fb28009dd-3_454_1029_1379_557} \(A B C D\) is the cross-section through the centre of mass of a uniform rectangular block of weight 260 N . The lengths \(A B\) and \(B C\) are 1.5 m and 0.8 m respectively. The block rests in equilibrium with the point \(D\) on a rough horizontal floor. Equilibrium is maintained by a light rope attached to the point \(A\) on the block and the point \(E\) on the floor. The points \(E , A\) and \(B\) lie in a straight line inclined at \(30 ^ { \circ }\) to the horizontal (see diagram).
  1. By taking moments about \(D\), show that the tension in the rope is 146 N , correct to 3 significant figures.
  2. Given that the block is in limiting equilibrium, calculate the coefficient of friction between the block and the floor.
CAIE M2 2013 November Q7
10 marks Standard +0.8
7 A particle \(P\) of mass 0.4 kg is attached to one end of a light elastic string of natural length 0.8 m and modulus of elasticity 32 N . The other end of the string is attached to a fixed point \(O\). The particle is released from rest at \(O\).
  1. Calculate the distance \(O P\) at the instant when \(P\) first comes to instantaneous rest. A horizontal plane is fixed at a distance 1 m below \(O\). The particle \(P\) is again released from rest at \(O\).
  2. Calculate the speed of \(P\) immediately before it collides with the plane.
  3. In the collision with the plane, \(P\) loses \(96 \%\) of its kinetic energy. Calculate the distance \(O P\) at the instant when \(P\) first comes to instantaneous rest above the plane, given that this occurs when the string is slack.
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:
    1. the force of magnitude \(F \mathrm {~N}\) acts along \(A B\);
    2. the force of magnitude \(F \mathrm {~N}\) acts along the tangent to the circular arc at \(A\).
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
    1. the angle between \(O P\) and the vertical,
    2. 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 \(W\).
CAIE M2 2015 November Q7
11 marks Challenging +1.2
7 A particle \(P\) of mass \(M \mathrm {~kg}\) is attached to one end of a light elastic string of natural length 0.8 m and modulus of elasticity 12.5 N . The other end of the string is attached to a fixed point \(A\). The particle is released from rest at \(A\) and falls vertically until it comes to instantaneous rest at the point \(B\). The greatest speed of \(P\) during its descent is \(4.4 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) when the extension of the string is \(e \mathrm {~m}\).
  1. Show that \(e = 0.64 M\).
  2. Find a second equation in \(e\) and \(M\), and hence find \(M\).
  3. Calculate the distance \(A B\).