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CAIE M2 2010 June Q7
11 marks Standard +0.3
7 A particle \(P\) of mass 0.25 kg moves in a straight line on a smooth horizontal surface. \(P\) starts at the point \(O\) with speed \(10 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) and moves towards a fixed point \(A\) on the line. At time \(t \mathrm {~s}\) the displacement of \(P\) from \(O\) is \(x \mathrm {~m}\) and the velocity of \(P\) is \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\). A resistive force of magnitude (5-x) N acts on \(P\) in the direction towards \(O\).
  1. Form a differential equation in \(v\) and \(x\). By solving this differential equation, show that \(v = 10 - 2 x\).
  2. Find \(x\) in terms of \(t\), and hence show that the particle is always less than 5 m from \(O\). \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. University of 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 2010 June Q1
4 marks Standard +0.3
1 \includegraphics[max width=\textwidth, alt={}, center]{d3609757-50ae-4c68-8992-4304867a2d84-2_618_441_253_852} A frame consists of a uniform semicircular wire of radius 20 cm and mass 2 kg , and a uniform straight wire of length 40 cm and mass 0.9 kg . The ends of the semicircular wire are attached to the ends of the straight wire (see diagram). Find the distance of the centre of mass of the frame from the straight wire.
CAIE M2 2010 June Q2
5 marks Standard +0.8
2 \includegraphics[max width=\textwidth, alt={}, center]{d3609757-50ae-4c68-8992-4304867a2d84-2_551_519_1231_813} A uniform solid cone has height 30 cm and base radius \(r \mathrm {~cm}\). The cone is placed with its axis vertical on a rough horizontal plane. The plane is slowly tilted and the cone remains in equilibrium until the angle of inclination of the plane reaches \(35 ^ { \circ }\), when the cone topples. The diagram shows a cross-section of the cone.
  1. Find the value of \(r\).
  2. Show that the coefficient of friction between the cone and the plane is greater than 0.7 .
CAIE M2 2010 June Q3
6 marks Moderate -0.5
3 \includegraphics[max width=\textwidth, alt={}, center]{d3609757-50ae-4c68-8992-4304867a2d84-3_456_511_260_817} A particle of mass 0.24 kg is attached to one end of a light inextensible string of length 2 m . The other end of the string is attached to a fixed point. The particle moves with constant speed in a horizontal circle. The string makes an angle \(\theta\) with the vertical (see diagram), and the tension in the string is \(T \mathrm {~N}\). The acceleration of the particle has magnitude \(7.5 \mathrm {~m} \mathrm {~s} ^ { - 2 }\).
  1. Show that \(\tan \theta = 0.75\) and find the value of \(T\).
  2. Find the speed of the particle.
CAIE M2 2010 June Q4
5 marks Standard +0.3
4 \includegraphics[max width=\textwidth, alt={}, center]{d3609757-50ae-4c68-8992-4304867a2d84-3_727_565_1256_790} A uniform lamina of weight 15 N is in the form of a trapezium \(A B C D\) with dimensions as shown in the diagram. The lamina is freely hinged at \(A\) to a fixed point. One end of a light inextensible string is attached to the lamina at \(B\). The lamina is in equilibrium with \(A B\) horizontal; the string is taut and in the same vertical plane as the lamina, and makes an angle of \(30 ^ { \circ }\) upwards from the horizontal (see diagram). Find the tension in the string.
CAIE M2 2010 June Q6
10 marks Challenging +1.2
6 \includegraphics[max width=\textwidth, alt={}, center]{d3609757-50ae-4c68-8992-4304867a2d84-4_324_1267_794_440} A particle \(P\) of mass 0.35 kg is attached to the mid-point of a light elastic string of natural length 4 m . The ends of the string are attached to fixed points \(A\) and \(B\) which are 4.8 m apart at the same horizontal level. \(P\) hangs in equilibrium at a point 0.7 m vertically below the mid-point \(M\) of \(A B\) (see diagram).
  1. Find the tension in the string and hence show that the modulus of elasticity of the string is 25 N . \(P\) is now held at rest at a point 1.8 m vertically below \(M\), and is then released.
  2. Find the speed with which \(P\) passes through \(M\).
CAIE M2 2010 June Q1
3 marks Moderate -0.8
1 A particle is projected horizontally with speed \(12 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) from the top of a high cliff. Find the direction of motion of the particle after 2 s .
CAIE M2 2010 June Q2
4 marks Standard +0.8
2 \includegraphics[max width=\textwidth, alt={}, center]{5a2248f6-3ef9-4e69-90cf-4d6a2351be14-2_319_908_438_616} A uniform solid cone has height 20 cm and base radius \(4 \mathrm {~cm} . P Q\) is a diameter of the base of the cone. The cone is held in equilibrium, with \(P\) in contact with a horizontal surface and \(P Q\) vertical, by a force applied at \(Q\). This force has magnitude 3 N and acts parallel to the axis of the cone (see diagram). Calculate the mass of the cone.
CAIE M2 2010 June Q3
8 marks Standard +0.8
3 Two particles \(P\) and \(Q\) are projected simultaneously with speed \(40 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) from a point \(O\) on a horizontal plane. Both particles subsequently pass at different times through the point \(A\) which has horizontal and vertically upward displacements from \(O\) of 40 m and 15 m respectively.
  1. By considering the equation of the trajectory of a projectile, show that each angle of projection satisfies the equation \(\tan ^ { 2 } \theta - 8 \tan \theta + 4 = 0\).
  2. Calculate the distance between the points at which \(P\) and \(Q\) strike the plane.
CAIE M2 2010 June Q4
7 marks Standard +0.8
4 \(A B\) is the diameter of a uniform semicircular lamina which has radius 0.3 m and mass 0.4 kg . The lamina is hinged to a vertical wall at \(A\) with \(A B\) inclined at \(30 ^ { \circ }\) to the vertical. One end of a light inextensible string is attached to the lamina at \(B\) and the other end of the string is attached to the wall vertically above \(A\). The lamina is in equilibrium in a vertical plane perpendicular to the wall with the string horizontal (see diagram).
  1. Show that the tension in the string is 2.00 N correct to 3 significant figures.
  2. Find the magnitude and direction of the force exerted on the lamina by the hinge. \includegraphics[max width=\textwidth, alt={}, center]{5a2248f6-3ef9-4e69-90cf-4d6a2351be14-3_956_540_258_804} A small ball \(B\) of mass 0.4 kg is attached to fixed points \(P\) and \(Q\) on a vertical axis by two light inextensible strings of equal length. Both strings are taut and each is inclined at \(30 ^ { \circ }\) to the vertical. The ball moves in a horizontal circle (see diagram).
CAIE M2 2010 June Q6
9 marks Standard +0.8
6 A particle \(P\) of mass 0.5 kg moves in a straight line on a smooth horizontal surface. At time \(t \mathrm {~s}\), the displacement of \(P\) from a fixed point on the line is \(x \mathrm {~m}\) and the velocity of \(P\) is \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\). It is given that when \(t = 0 , x = 0\) and \(v = 9\). The motion of \(P\) is opposed by a force of magnitude \(3 \sqrt { } v \mathrm {~N}\).
  1. By solving an appropriate differential equation, show that \(v = ( 27 - 9 x ) ^ { \frac { 2 } { 3 } }\).
  2. Calculate the value of \(x\) when \(t = 0.5\).
CAIE M2 2010 June Q7
11 marks Standard +0.3
7 One end of a light elastic string of natural length 3 m and modulus of elasticity 24 N is attached to a fixed point \(O\). A particle \(P\) of mass 0.4 kg is attached to the other end of the string. \(P\) is projected vertically downwards from \(O\) with initial speed \(2 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). When the extension of the string is \(x \mathrm {~m}\) the speed of \(P\) is \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\).
  1. Show that \(v ^ { 2 } = 64 + 20 x - 20 x ^ { 2 }\).
  2. Find the greatest speed of the particle.
  3. Calculate the greatest tension in the string.
CAIE M2 2011 June Q1
2 marks Easy -1.2
1 A particle is projected with speed \(15 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) at an angle of \(40 ^ { \circ }\) above the horizontal from a point on horizontal ground. Calculate the time taken for the particle to hit the ground.
CAIE M2 2011 June Q2
6 marks Standard +0.3
2 \includegraphics[max width=\textwidth, alt={}, center]{18398d27-15eb-4515-8210-4f0f614d5b28-2_406_483_431_829} \(A O B\) is a uniform lamina in the shape of a quadrant of a circle with centre \(O\) and radius 0.6 m (see diagram).
  1. Calculate the distance of the centre of mass of the lamina from \(A\). The lamina is freely suspended at \(A\) and hangs in equilibrium.
  2. Find the angle between the vertical and the side \(A O\) of the lamina.
CAIE M2 2011 June Q3
6 marks Standard +0.8
3 \includegraphics[max width=\textwidth, alt={}, center]{18398d27-15eb-4515-8210-4f0f614d5b28-2_247_839_1375_653} A light elastic string of natural length 1.2 m and modulus of elasticity 24 N is attached to fixed points \(A\) and \(B\) on a smooth horizontal surface, where \(A B = 1.2 \mathrm {~m}\). A particle \(P\) is attached to the mid-point of the string. \(P\) is projected with speed \(0.5 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) along the surface in a direction perpendicular to \(A B\) (see diagram). \(P\) comes to instantaneous rest at a distance 0.25 m from \(A B\).
  1. Show that the mass of \(P\) is 0.8 kg .
  2. Calculate the greatest deceleration of \(P\).
CAIE M2 2011 June Q4
7 marks Standard +0.8
4 A particle \(P\) starts from rest at a point \(O\) and travels in a straight line. The acceleration of \(P\) is \(( 15 - 6 x ) \mathrm { m } \mathrm { s } ^ { - 2 }\), where \(x \mathrm {~m}\) is the displacement of \(P\) from \(O\).
  1. Find the value of \(x\) for which \(P\) reaches its maximum velocity, and calculate this maximum velocity.
  2. Calculate the acceleration of \(P\) when it is at instantaneous rest and \(x > 0\).
CAIE M2 2011 June Q5
8 marks Challenging +1.2
5 \includegraphics[max width=\textwidth, alt={}, center]{18398d27-15eb-4515-8210-4f0f614d5b28-3_348_1205_251_470} \(A B C\) is a uniform triangular lamina of weight 19 N , with \(A B = 0.22 \mathrm {~m}\) and \(A C = B C = 0.61 \mathrm {~m}\). The plane of the lamina is vertical. \(A\) rests on a rough horizontal surface, and \(A B\) is vertical. The equilibrium of the lamina is maintained by a light elastic string of natural length 0.7 m which passes over a small smooth peg \(P\) and is attached to \(B\) and \(C\). The portion of the string attached to \(B\) is horizontal, and the portion of the string attached to \(C\) is vertical (see diagram).
  1. Show that the tension in the string is 10 N .
  2. Calculate the modulus of elasticity of the string.
  3. Find the magnitude and direction of the force exerted by the surface on the lamina at \(A\).
CAIE M2 2011 June Q6
9 marks Moderate -0.3
6 A particle \(P\) is projected from a point \(O\) on horizontal ground. 0.4 s after the instant of projection, \(P\) is 5 m above the ground and a horizontal distance of 12 m from \(O\).
  1. Calculate the initial speed and the angle of projection of \(P\).
  2. Find the direction of motion of the particle 0.4 s after the instant of projection.
CAIE M2 2011 June Q7
12 marks Challenging +1.2
7 \includegraphics[max width=\textwidth, alt={}, center]{18398d27-15eb-4515-8210-4f0f614d5b28-4_713_933_258_605} A narrow groove is cut along a diameter in the surface of a horizontal disc with centre \(O\). Particles \(P\) and \(Q\), of masses 0.2 kg and 0.3 kg respectively, lie in the groove, and the coefficient of friction between each of the particles and the groove is \(\mu\). The particles are attached to opposite ends of a light inextensible string of length 1 m . The disc rotates with angular velocity \(\omega \mathrm { rad } \mathrm { s } ^ { - 1 }\) about a vertical axis passing through \(O\) and the particles move in horizontal circles (see diagram).
  1. Given that \(\mu = 0.36\) and that both \(P\) and \(Q\) move in the same horizontal circle of radius 0.5 m , calculate the greatest possible value of \(\omega\) and the corresponding tension in the string.
  2. Given instead that \(\mu = 0\) and that the tension in the string is 0.48 N , calculate
    (a) the radius of the circle in which \(P\) moves and the radius of the circle in which \(Q\) moves,
    (b) the speeds of the particles.
CAIE M2 2011 June Q1
6 marks Moderate -0.3
1 \includegraphics[max width=\textwidth, alt={}, center]{1d2e8f3a-dab6-4306-bc4a-d47805947cd2-2_518_609_255_769} A uniform \(\operatorname { rod } A B\) of weight 16 N is freely hinged at \(A\) to a fixed point. A force of magnitude 4 N acting perpendicular to the rod is applied at \(B\) (see diagram). Given that the rod is in equilibrium,
  1. calculate the angle the rod makes with the horizontal,
  2. find the magnitude and direction of the force exerted on the rod at \(A\).
CAIE M2 2011 June Q2
7 marks Challenging +1.2
2 A uniform lamina \(A B C D\) consists of a semicircle \(B C D\) with centre \(O\) and diameter 0.4 m , and an isosceles triangle \(A B D\) with base \(B D = 0.4 \mathrm {~m}\) and perpendicular height \(h \mathrm {~m}\). The centre of mass of the lamina is at \(O\).
  1. Find the value of \(h\).
  2. \includegraphics[max width=\textwidth, alt={}, center]{1d2e8f3a-dab6-4306-bc4a-d47805947cd2-2_680_627_1466_797} The lamina is suspended from a vertical string attached to a point \(X\) on the side \(A D\) of the triangle (see diagram). Given the lamina is in equilibrium with \(A D\) horizontal, calculate \(X D\).
CAIE M2 2011 June Q3
8 marks Standard +0.3
3 \includegraphics[max width=\textwidth, alt={}, center]{1d2e8f3a-dab6-4306-bc4a-d47805947cd2-3_385_1154_253_497} A particle \(P\) of mass 0.5 kg is attached to the vertex \(V\) of a fixed solid cone by a light inextensible string. \(P\) lies on the smooth curved surface of the cone and moves in a horizontal circle of radius 0.1 m with centre on the axis of the cone. The cone has semi-vertical angle \(60 ^ { \circ }\) (see diagram).
  1. Calculate the speed of \(P\), given that the tension in the string and the contact force between the cone and \(P\) have the same magnitude.
  2. Calculate the greatest angular speed at which \(P\) can move on the surface of the cone.
CAIE M2 2011 June Q4
9 marks Standard +0.8
4 One end of a light elastic string of natural length 0.5 m and modulus of elasticity 12 N is attached to a fixed point \(O\). The other end of the string is attached to a particle \(P\) of mass \(0.24 \mathrm {~kg} . P\) is projected vertically upwards with speed \(3 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) from a position 0.8 m vertically below \(O\).
  1. Calculate the speed of the particle when it is moving upwards with zero acceleration.
  2. Show that the particle moves 0.6 m while it is moving upwards with constant acceleration.
CAIE M2 2011 June Q5
9 marks Standard +0.8
5 A particle \(P\) of mass 0.4 kg moves in a straight line on a horizontal surface and has velocity \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\) at time \(t \mathrm {~s}\). A horizontal force of magnitude \(k \sqrt { } v \mathrm {~N}\) opposes the motion of \(P\). When \(t = 0 , v = 9\) and when \(t = 2 , v = 4\).
  1. Express \(\frac { \mathrm { d } v } { \mathrm {~d} t }\) in terms of \(k\) and \(v\), and hence show that \(v = \frac { 1 } { 4 } ( t - 6 ) ^ { 2 }\).
  2. Find the distance travelled by \(P\) in the first 3 seconds of its motion.
CAIE M2 2011 June Q6
11 marks Standard +0.3
6 A particle \(P\) is projected with speed \(26 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) at an angle of \(30 ^ { \circ }\) below the horizontal, from a point \(O\) which is 80 m above horizontal ground.
  1. Calculate the distance from \(O\) of the particle 2.3 s after projection.
  2. Find the horizontal distance travelled by \(P\) before it reaches the ground.
  3. Calculate the speed and direction of motion of \(P\) immediately before it reaches the ground.