Questions M2 (1537 questions)

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CAIE M2 2017 November Q5
8 marks Standard +0.3
5 \includegraphics[max width=\textwidth, alt={}, center]{6b220343-1d64-4dbc-a42d-77967eef9c6d-08_449_890_262_630} \(O A B\) is a uniform lamina in the shape of a quadrant of a circle with centre \(O\) and radius 0.8 m which has its centre of mass at \(G\). The lamina is smoothly hinged at \(A\) to a fixed point and is free to rotate in a vertical plane. A horizontal force of magnitude 12 N acting in the plane of the lamina is applied to the lamina at \(B\). The lamina is in equilibrium with \(A G\) horizontal (see diagram).
  1. Calculate the length \(A G\).
  2. Find the weight of the lamina.
CAIE M2 2017 November Q6
9 marks Standard +0.3
6 One end of a light elastic string of natural length 0.4 m and modulus of elasticity 8 N is attached to a fixed point \(O\) on a smooth horizontal plane. The other end of the string is attached to a particle \(P\) of mass 0.2 kg which moves on the plane in a circular path with centre \(O\). The speed of \(P\) is \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\) and the extension of the string is \(x \mathrm {~m}\).
  1. Given that \(v = 2.5\), find \(x\).
    It is given instead that the kinetic energy of \(P\) is twice the elastic potential energy stored in the string.
  2. Form two simultaneous equations and hence find \(x\) and \(v\).
CAIE M2 2017 November Q7
11 marks Standard +0.8
7 A small ball \(B\) is projected from a point \(O\) which is \(h \mathrm {~m}\) above a horizontal plane. At time 2 s after projection \(B\) has speed \(18 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) and is moving in the direction \(30 ^ { \circ }\) above the horizontal.
  1. Find the initial speed and the angle of projection of \(B\). \(B\) has speed \(38 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) immediately before it strikes the plane.
  2. Calculate \(h\). \(B\) bounces when it strikes the plane, and leaves the plane with speed \(20 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) but with its horizontal component of velocity unchanged.
  3. Find the total time which elapses between the initial projection of \(B\) and the instant when it strikes the plane for the second time.
CAIE M2 2017 November Q3
7 marks Standard +0.3
3 \includegraphics[max width=\textwidth, alt={}, center]{2e4e6e32-eafc-4196-aaa8-42909cc2078e-04_305_510_264_813} One end of a light inextensible string of length 0.4 m is attached to a fixed point \(A\) which is above a smooth horizontal surface. A particle \(P\) of mass 0.6 kg is attached to the other end of the string. \(P\) moves in a circle on the surface with constant speed \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\), with the string taut and making an angle of \(60 ^ { \circ }\) with the horizontal (see diagram).
  1. Given that \(v = 0.5\), calculate the magnitude of the force that the surface exerts on \(P\).
  2. Find the greatest possible value of \(v\) for which \(P\) remains in contact with the surface.
CAIE M2 2018 November Q1
3 marks Standard +0.3
1 A small ball \(B\) is projected with speed \(38 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) at an angle of \(30 ^ { \circ }\) to the horizontal from a point on horizontal ground. Find the speed of \(B\) when the path of \(B\) makes an angle of \(20 ^ { \circ }\) above the horizontal.
CAIE M2 2018 November Q2
7 marks Standard +0.3
2 A uniform solid object is made by attaching a cone to a cylinder so that the circumferences of the base of the cone and a plane face of the cylinder coincide. The cone and the cylinder each have radius 0.3 m and height 0.4 m .
  1. Calculate the distance of the centre of mass of the object from the vertex of the cone.
    [0pt] [The volume of a cone is \(\frac { 1 } { 3 } \pi r ^ { 2 } h\).]
    The object has weight \(W \mathrm {~N}\) and is placed with its plane circular face on a rough horizontal surface. A force of magnitude \(k W \mathrm {~N}\) acting at \(30 ^ { \circ }\) to the upward vertical is applied to the vertex of the cone. The object does not slip.
  2. Find the greatest possible value of \(k\) for which the object does not topple.
CAIE M2 2018 November Q3
7 marks Challenging +1.8
3 A particle \(P\) of mass 0.4 kg is projected horizontally along a smooth horizontal plane from a point \(O\). After projection the velocity of \(P\) is \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\) and its displacement from \(O\) is \(x \mathrm {~m}\). A force of magnitude \(8 x \mathrm {~N}\) directed away from \(O\) acts on \(P\) and a force of magnitude ( \(2 \mathrm { e } ^ { - x } + 4\) ) N opposes the motion of \(P\). One end of a light elastic string of natural length 0.5 m is attached to \(O\) and the other end of the string is attached to \(P\).
  1. Show that \(v \frac { \mathrm {~d} v } { \mathrm {~d} x } = 20 x - 10 - 5 \mathrm { e } ^ { - x }\) before the elastic string becomes taut.
  2. Given that the initial velocity of \(P\) is \(6 \mathrm {~m} \mathrm {~s} ^ { - 1 }\), find \(v\) when the string first becomes taut.
    When the string is taut, the acceleration of \(P\) is proportional to \(\mathrm { e } ^ { - x }\).
  3. Find the modulus of elasticity of the string.
CAIE M2 2018 November Q4
8 marks Moderate -0.3
4 A small object is projected horizontally with speed \(V \mathrm {~m} \mathrm {~s} ^ { - 1 }\) from a point \(O\) above horizontal ground. At time \(t \mathrm {~s}\) after projection, the horizontal and vertically upwards displacements of the object 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 path of the object is \(y = - \frac { 5 x ^ { 2 } } { V ^ { 2 } }\).
    The object passes through points with coordinates \(( a , - a )\) and \(\left( a ^ { 2 } , - 16 a \right)\), where \(a\) is a positive constant.
  2. Find the value of \(a\).
  3. Given that the object strikes the ground at the point where \(x = 5 a\), find the height of \(O\) above the ground .
CAIE M2 2018 November Q5
8 marks Standard +0.3
5 A particle \(P\) of mass 0.7 kg is attached to a fixed point \(O\) by a light elastic string of natural length 0.6 m and modulus of elasticity 15 N . The particle \(P\) is projected vertically downwards from the point \(A , 0.8 \mathrm {~m}\) vertically below \(O\). The initial speed of \(P\) is \(2 \mathrm {~m} \mathrm {~s} ^ { - 1 }\).
  1. Find the distance below \(A\) of the point at which \(P\) comes to instantaneous rest.
  2. Find the greatest speed of \(P\) in the motion. \includegraphics[max width=\textwidth, alt={}, center]{f922bf53-94a0-4ccc-8c38-959d2f795629-10_478_652_260_751} The diagram shows a uniform lamina \(A B C D E F G H\). The lamina consists of a quarter-circle \(O A B\) of radius \(r \mathrm {~m}\), a rectangle \(D E F G\) and two isosceles right-angled triangles \(C O D\) and \(G O H\). The rectangle has \(D G = E F = r \mathrm {~m}\) and \(D E = F G = x \mathrm {~m}\).
  3. Given that the centre of mass of the lamina is at \(O\), express \(x\) in terms of \(r\).
  4. Given instead that the rectangle \(D E F G\) is a square with edges of length \(r \mathrm {~m}\), state with a reason whether the centre of mass of the lamina lies within the square or the quarter-circle. \includegraphics[max width=\textwidth, alt={}, center]{f922bf53-94a0-4ccc-8c38-959d2f795629-12_384_693_258_726} A rough horizontal rod \(A B\) of length 0.45 m rotates with constant angular velocity \(6 \mathrm { rad } \mathrm { s } ^ { - 1 }\) about a vertical axis through \(A\). A small ring \(R\) of mass 0.2 kg can slide on the rod. A particle \(P\) of mass 0.1 kg is attached to the mid-point of a light inextensible string of length 0.6 m . One end of the string is attached to \(R\) and the other end of the string is attached to \(B\), with angle \(R P B = 60 ^ { \circ }\) (see diagram). \(R\) and \(P\) move in horizontal circles as the system rotates. \(R\) is in limiting equilibrium.
  5. Show that the tension in the portion \(P R\) of the string is 1.66 N , correct to 3 significant figures.
  6. Find the coefficient of friction between the ring and the rod.
    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 M2 2019 November Q2
6 marks Moderate -0.3
2 A particle is projected from a point on horizontal ground with speed \(15 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) at an angle of \(\theta ^ { \circ }\) above the horizontal. The particle strikes the ground 2 s after projection.
  1. Find \(\theta\). \includegraphics[max width=\textwidth, alt={}, center]{4cd525d5-d59b-4ab9-85a3-fc3d97fd09fc-03_67_1571_438_328}
  2. Calculate the time after projection at which the direction of motion of the particle is \(20 ^ { \circ }\) below the horizontal.
CAIE M2 2019 November Q3
6 marks Challenging +1.2
3 A smooth horizontal surface has two fixed points \(O\) and \(A\) which are 0.8 m apart. A particle \(P\) of mass 0.25 kg is projected with velocity \(3 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) horizontally from \(A\) in the direction away from \(O\). The velocity of \(P\) is \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\) when the displacement of \(P\) from \(O\) is \(x \mathrm {~m}\). A force of magnitude \(k v ^ { 2 } x ^ { - 2 } \mathrm {~N}\) opposes the motion of \(P\).
  1. Show that \(v \frac { \mathrm {~d} v } { \mathrm {~d} x } = - 4 k v ^ { 2 } x ^ { - 2 }\).
  2. Express \(v\) in terms of \(k\) and \(x\).
CAIE M2 2019 November Q4
7 marks Moderate -0.3
4 A small ball \(B\) is projected with speed \(30 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) at an angle of \(60 ^ { \circ }\) above the horizontal from a point \(O\). At time \(t \mathrm {~s}\) after projection the horizontal and vertically upwards displacements of \(B\) from \(O\) are \(x \mathrm {~m}\) and \(y \mathrm {~m}\) respectively.
  1. Express \(x\) and \(y\) in terms of \(t\) and hence find the equation of the trajectory of the ball.
  2. Find the value of \(x\) for which \(O B\) makes an angle of \(45 ^ { \circ }\) above the horizontal.
CAIE M2 2019 November Q5
9 marks Standard +0.3
5 A particle \(P\) of mass 0.3 kg is attached to one end of a light elastic string of natural length 0.6 m and modulus of elasticity 9 N . The other end of the string is attached to a fixed point \(O\) on a smooth plane inclined at \(30 ^ { \circ }\) to the horizontal. \(O A\) is a line of greatest slope of the plane with \(A\) below the level of \(O\) and \(O A = 0.8 \mathrm {~m}\). The particle \(P\) is released from rest at \(A\).
  1. Find the initial acceleration of \(P\).
  2. Find the greatest speed of \(P\). \(6 \quad A\) and \(B\) are two fixed points on a vertical axis with \(A 0.6 \mathrm {~m}\) above \(B\). A particle \(P\) of mass 0.3 kg is attached to \(A\) by a light inextensible string of length 0.5 m . The particle \(P\) is attached to \(B\) by a light elastic string with modulus of elasticity 46 N . The particle \(P\) moves with constant angular speed \(8 \mathrm { rad } \mathrm { s } ^ { - 1 }\) in a horizontal circle with centre at the mid-point of \(A B\).
  3. Find the speed of \(P\).
  4. Calculate the tension in the string \(B P\) and hence find the natural length of this string. \includegraphics[max width=\textwidth, alt={}, center]{4cd525d5-d59b-4ab9-85a3-fc3d97fd09fc-10_540_574_260_781} \(A B C\) is the cross-section through the centre of mass of a uniform prism which rests with \(A B\) on a rough horizontal surface. \(A B = 0.4 \mathrm {~m}\) and \(C\) is 0.9 m above the surface (see diagram). The prism is on the point of toppling about its edge through \(B\).
  5. Show that angle \(B A C = 48.4 ^ { \circ }\), correct to 3 significant figures.
    A force of magnitude 18 N acting in the plane of the cross-section and perpendicular to \(A C\) is now applied to the prism at \(C\). The prism is on the point of rotating about its edge through \(A\).
  6. Calculate the weight of the prism.
  7. Given also that the prism is on the point of slipping, calculate the coefficient of friction between the prism and the surface.
    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 M2 2019 November Q1
3 marks Standard +0.3
1 A particle of mass 0.3 kg is attached to one end of a light elastic string of natural length 0.6 m and modulus of elasticity 9 N . The other end of the string is attached to a fixed point \(O\) on a smooth horizontal surface. The particle is projected horizontally from \(O\) with speed \(4 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). Find the greatest distance of the particle from \(O\).
CAIE M2 2019 November Q2
5 marks Moderate -0.5
2 A small ball is projected from a point \(O\) on horizontal ground at an angle of \(30 ^ { \circ }\) above the horizontal. At time \(t \mathrm {~s}\) after projection the horizontal and vertically upwards displacements of the ball from \(O\) are \(x \mathrm {~m}\) and \(y \mathrm {~m}\) respectively. It is given that \(x = 40 t\).
  1. Calculate the initial speed of the ball, and express \(y\) in terms of \(t\).
  2. Hence find the equation of the trajectory of the ball.
CAIE M2 2019 November Q3
6 marks Standard +0.8
3 A particle \(P\) of mass 0.5 kg is attached to one end of a light elastic string of natural length 0.6 m and modulus of elasticity 12 N . The other end of the string is attached to a fixed point \(O\). The particle \(P\) is projected vertically downwards with speed \(2 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) from the point 0.5 m vertically below \(O\). For an instant when the acceleration of \(P\) is \(4 \mathrm {~m} \mathrm {~s} ^ { - 2 }\) downwards, find the extension of the string and the speed of \(P\).
CAIE M2 2019 November Q4
6 marks Standard +0.3
4 A particle is projected from a point \(O\) on horizontal ground with speed \(V \mathrm {~m} \mathrm {~s} ^ { - 1 }\) at an angle of \(60 ^ { \circ }\) above the horizontal. At the instant 3 s after projection the direction of motion of the particle is \(30 ^ { \circ }\) below the horizontal.
  1. Find \(V\).
    ..................................................................................................................................
  2. Calculate the distance of the particle from \(O\) at the instant 3 s after projection.
CAIE M2 2019 November Q5
10 marks Standard +0.8
5 \includegraphics[max width=\textwidth, alt={}, center]{9daebcbe-826e-4eda-afa7-c935c6ea2bfc-06_671_504_255_824} \(A\) and \(B\) are two fixed points on a vertical axis with \(A\) above \(B\). A particle \(P\) of mass 0.4 kg is attached to \(A\) by a light inextensible string of length 0.5 m . The particle \(P\) is attached to \(B\) by another light inextensible string. \(P\) moves with constant speed in a horizontal circle with centre \(O\) between \(A\) and \(B\). Angle \(B A P = 30 ^ { \circ }\) and angle \(A B P = 70 ^ { \circ }\) (see diagram).
  1. Given that the tensions in the two strings are equal, find the speed of \(P\).
  2. Given instead that the angular speed of \(P\) is \(12 \mathrm { rad } \mathrm { s } ^ { - 1 }\), find the tensions in the strings.
CAIE M2 2019 November Q6
9 marks Standard +0.8
6 A particle \(P\) of mass 0.2 kg is projected horizontally from a fixed point \(O\) on a smooth horizontal surface. When the displacement of \(P\) from \(O\) is \(x \mathrm {~m}\) the velocity of \(P\) is \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\). A horizontal force of variable magnitude \(0.09 \sqrt { } x \mathrm {~N}\) directed away from \(O\) acts on \(P\). An additional force of constant magnitude 0.3 N directed towards \(O\) acts on \(P\).
  1. Show that \(v \frac { \mathrm {~d} v } { \mathrm {~d} x } = 0.45 \sqrt { } x - 1.5\).
  2. Find the value of \(x\) for which the acceleration of \(P\) is zero.
  3. Given that the minimum value of \(v\) is positive, find the set of possible values for the speed of projection.
CAIE M2 2019 November Q7
11 marks Challenging +1.2
7 \includegraphics[max width=\textwidth, alt={}, center]{9daebcbe-826e-4eda-afa7-c935c6ea2bfc-10_451_574_258_781} \(A B C D\) is a uniform lamina in the shape of a trapezium which has centre of mass \(G\). The sides \(A D\) and \(B C\) are parallel and 1.8 m apart, with \(A D = 2.4 \mathrm {~m}\) and \(B C = 1.2 \mathrm {~m}\) (see diagram).
  1. Show that the distance of \(G\) from \(A D\) is 0.8 m .
    The lamina is freely suspended at \(A\) and hangs in equilibrium with \(A D\) making an angle of \(30 ^ { \circ }\) with the vertical.
  2. Calculate the distance \(A G\).
    With the lamina still freely suspended at \(A\) a horizontal force of magnitude 7 N acting in the plane of the lamina is applied at \(D\). The lamina is in equilibrium with \(A G\) making an angle of \(10 ^ { \circ }\) with the downward vertical.
  3. Find the two possible values for the weight of the lamina.
    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 M2 2019 November Q2
6 marks Moderate -0.3
2 A particle is projected from a point on horizontal ground with speed \(15 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) at an angle of \(\theta ^ { \circ }\) above the horizontal. The particle strikes the ground 2 s after projection.
  1. Find \(\theta\). \includegraphics[max width=\textwidth, alt={}, center]{42de91da-d65e-40e7-8de5-f40eda927850-03_67_1571_438_328}
  2. Calculate the time after projection at which the direction of motion of the particle is \(20 ^ { \circ }\) below the horizontal.
CAIE M2 2019 November Q5
9 marks Standard +0.3
5 A particle \(P\) of mass 0.3 kg is attached to one end of a light elastic string of natural length 0.6 m and modulus of elasticity 9 N . The other end of the string is attached to a fixed point \(O\) on a smooth plane inclined at \(30 ^ { \circ }\) to the horizontal. \(O A\) is a line of greatest slope of the plane with \(A\) below the level of \(O\) and \(O A = 0.8 \mathrm {~m}\). The particle \(P\) is released from rest at \(A\).
  1. Find the initial acceleration of \(P\).
  2. Find the greatest speed of \(P\). \(6 \quad A\) and \(B\) are two fixed points on a vertical axis with \(A 0.6 \mathrm {~m}\) above \(B\). A particle \(P\) of mass 0.3 kg is attached to \(A\) by a light inextensible string of length 0.5 m . The particle \(P\) is attached to \(B\) by a light elastic string with modulus of elasticity 46 N . The particle \(P\) moves with constant angular speed \(8 \mathrm { rad } \mathrm { s } ^ { - 1 }\) in a horizontal circle with centre at the mid-point of \(A B\).
  3. Find the speed of \(P\).
  4. Calculate the tension in the string \(B P\) and hence find the natural length of this string. \includegraphics[max width=\textwidth, alt={}, center]{42de91da-d65e-40e7-8de5-f40eda927850-10_540_574_260_781} \(A B C\) is the cross-section through the centre of mass of a uniform prism which rests with \(A B\) on a rough horizontal surface. \(A B = 0.4 \mathrm {~m}\) and \(C\) is 0.9 m above the surface (see diagram). The prism is on the point of toppling about its edge through \(B\).
  5. Show that angle \(B A C = 48.4 ^ { \circ }\), correct to 3 significant figures.
    A force of magnitude 18 N acting in the plane of the cross-section and perpendicular to \(A C\) is now applied to the prism at \(C\). The prism is on the point of rotating about its edge through \(A\).
  6. Calculate the weight of the prism.
  7. Given also that the prism is on the point of slipping, calculate the coefficient of friction between the prism and the surface.
    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 M2 Specimen 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 Specimen 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 Specimen Q3
5 marks Standard +0.8
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 \(\theta\).