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CAIE M2 2017 June Q3
7 marks Standard +0.3
\includegraphics{figure_3} An object is made from a uniform solid hemisphere of radius \(0.56\) m and centre \(O\) by removing a hemisphere of radius \(0.28\) m and centre \(O\). The diagram shows a cross-section through \(O\) of the object.
  1. Calculate the distance of the centre of mass of the object from \(O\). [4] [The volume of a hemisphere is \(\frac{2}{3}\pi r^3\).]
  2. The object has weight \(24\) N. A uniform hemisphere \(H\) of radius \(0.28\) m is placed in the hollow part of the object to create a new uniform hemisphere with centre \(O\). The centre of mass of the non-uniform hemisphere is \(0.15\) m from \(O\). Calculate the weight of \(H\). [3]
CAIE M2 2017 June Q4
8 marks Moderate -0.3
A particle is projected from a point \(O\) on horizontal ground. The initial components of the velocity of the particle are \(10\,\text{m}\,\text{s}^{-1}\) horizontally and \(15\,\text{m}\,\text{s}^{-1}\) vertically. At time \(t\) s after projection, the horizontal and vertically upwards displacements of the particle from \(O\) are \(x\) m and \(y\) m respectively.
  1. Express \(x\) and \(y\) in terms of \(t\), and hence find the equation of the trajectory of the particle. [4]
  2. The horizontal ground is at the top of a vertical cliff. The point \(O\) is at a distance \(d\) m from the edge of the cliff. The particle is projected towards the edge of the cliff and does not strike the ground before it passes over the edge of the cliff. Show that \(d\) is less than \(30\). [2]
  3. Find the value of \(x\) when the particle is \(14\) m below the level of \(O\). [2]
CAIE M2 2017 June Q5
7 marks Standard +0.8
\includegraphics{figure_5} A uniform semicircular lamina of radius \(0.7\) m and weight \(14\) N has diameter \(AB\). The lamina is in a vertical plane with \(A\) freely pivoted at a fixed point. The straight edge \(AB\) rests against a small smooth peg \(P\) above the level of \(A\). The angle between \(AB\) and the horizontal is \(30°\) and \(AP = 0.9\) m (see diagram).
  1. Show that the magnitude of the force exerted by the peg on the lamina is \(7.12\) N, correct to 3 significant figures. [4]
  2. Find the angle with the horizontal of the force exerted by the pivot on the lamina at \(A\). [3]
CAIE M2 2017 June Q6
9 marks Standard +0.3
A particle \(P\) of mass \(0.15\) kg is attached to one end of a light elastic string of natural length \(0.4\) m and modulus of elasticity \(12\) N. The other end of the string is attached to a fixed point \(A\). The particle \(P\) moves in a horizontal circle which has its centre vertically below \(A\), with the string inclined at \(\theta°\) to the vertical and \(AP = 0.5\) m.
  1. Find the angular speed of \(P\) and the value of \(\theta\). [5]
  2. Calculate the difference between the elastic potential energy stored in the string and the kinetic energy of \(P\). [4]
CAIE M2 2017 June Q7
9 marks Standard +0.8
A particle \(P\) of mass \(0.5\) kg is at rest at a point \(O\) on a rough horizontal surface. At time \(t = 0\), where \(t\) is in seconds, a horizontal force acting in a fixed direction is applied to \(P\). At time \(t\) s the magnitude of the force is \(0.6t^2\) N and the velocity of \(P\) away from \(O\) is \(v\,\text{m}\,\text{s}^{-1}\). It is given that \(P\) remains at rest at \(O\) until \(t = 0.5\).
  1. Calculate the coefficient of friction between \(P\) and the surface, and show that $$\frac{\text{d}v}{\text{d}t} = 1.2t^2 - 0.3 \quad \text{for } t > 0.5.$$ [3]
  2. Express \(v\) in terms of \(t\) for \(t > 0.5\). [3]
  3. Find the displacement of \(P\) from \(O\) when \(t = 1.2\). [3]
CAIE M2 2017 June Q1
4 marks Standard +0.3
A particle is projected with speed \(20 \text{ ms}^{-1}\) at an angle of \(60°\) above the horizontal. Calculate the time after projection when the particle is descending at an angle of \(40°\) below the horizontal. [4]
CAIE M2 2017 June Q2
6 marks Standard +0.3
\includegraphics{figure_1} One end of a light inextensible string is attached to a fixed point \(A\). The other end of the string is attached to a particle \(P\) of mass \(m\) kg which hangs vertically below \(A\). The particle is also attached to one end of a light elastic string of natural length \(0.25\) m. The other end of this string is attached to a point \(B\) which is \(0.6\) m from \(P\) and on the same horizontal level as \(P\). Equilibrium is maintained by a horizontal force of magnitude \(7\) N applied to \(P\) (see Fig. 1).
  1. Calculate the modulus of elasticity of the elastic string. [2]
  2. Find the value of \(m\). [4]
CAIE M2 2017 June Q3
7 marks Standard +0.3
\includegraphics{figure_2} An object is made from a uniform solid hemisphere of radius \(0.56\) m and centre \(O\) by removing a hemisphere of radius \(0.28\) m and centre \(O\). The diagram shows a cross-section through \(O\) of the object.
  1. Calculate the distance of the centre of mass of the object from \(O\). [4] [The volume of a hemisphere is \(\frac{2}{3}\pi r^3\).]
  2. Calculate the weight of \(H\). [3]
The object has weight \(24\) N. A uniform hemisphere \(H\) of radius \(0.28\) m is placed in the hollow part of the object to create a non-uniform hemisphere with centre \(O\). The centre of mass of the non-uniform hemisphere is \(0.15\) m from \(O\).
CAIE M2 2017 June Q4
8 marks Moderate -0.3
A particle is projected from a point \(O\) on horizontal ground. The initial components of the velocity of the particle are \(10 \text{ ms}^{-1}\) horizontally and \(15 \text{ ms}^{-1}\) vertically. At time \(t\) s after projection, the horizontal and vertically upwards displacements of the particle from \(O\) are \(x\) m and \(y\) m respectively.
  1. Express \(x\) and \(y\) in terms of \(t\), and hence find the equation of the trajectory of the particle. [4]
  2. Show that \(d\) is less than \(30\). [2]
  3. Find the value of \(x\) when the particle is \(14\) m below the level of \(O\). [2]
The horizontal ground is at the top of a vertical cliff. The point \(O\) is at a distance \(d\) m from the edge of the cliff. The particle is projected towards the edge of the cliff and does not strike the ground before it passes over the edge of the cliff.
CAIE M2 2017 June Q5
7 marks Standard +0.8
\includegraphics{figure_3} A uniform semicircular lamina of radius \(0.7\) m and weight \(14\) N has diameter \(AB\). The lamina is in a vertical plane with \(A\) freely pivoted at a fixed point. The straight edge \(AB\) rests against a small smooth peg \(P\) above the level of \(A\). The angle between \(AB\) and the horizontal is \(30°\) and \(AP = 0.9\) m (see diagram).
  1. Show that the magnitude of the force exerted by the peg on the lamina is \(7.12\) N, correct to 3 significant figures. [4]
  2. Find the angle with the horizontal of the force exerted by the pivot on the lamina at \(A\). [3]
CAIE M2 2017 June Q6
9 marks Standard +0.8
A particle \(P\) of mass \(0.15\) kg is attached to one end of a light elastic string of natural length \(0.4\) m and modulus of elasticity \(12\) N. The other end of the string is attached to a fixed point \(A\). The particle \(P\) moves in a horizontal circle which has its centre vertically below \(A\), with the string inclined at \(θ°\) to the vertical and \(AP = 0.5\) m.
  1. Find the angular speed of \(P\) and the value of \(θ\). [5]
  2. Calculate the difference between the elastic potential energy stored in the string and the kinetic energy of \(P\). [4]
CAIE M2 2017 June Q7
9 marks Standard +0.3
A particle \(P\) of mass \(0.5\) kg is at rest at a point \(O\) on a rough horizontal surface. At time \(t = 0\), where \(t\) is in seconds, a horizontal force acting in a fixed direction is applied to \(P\). At time \(t\) s the magnitude of the force is \(0.6t^2\) N and the velocity of \(P\) away from \(O\) is \(v \text{ ms}^{-1}\). It is given that \(P\) remains at rest at \(O\) until \(t = 0.5\).
  1. Calculate the coefficient of friction between \(P\) and the surface, and show that $$\frac{\text{dv}}{\text{dt}} = 1.2t^2 - 0.3 \quad \text{for } t > 0.5.$$ [3]
  2. Express \(v\) in terms of \(t\) for \(t > 0.5\). [3]
  3. Find the displacement of \(P\) from \(O\) when \(t = 1.2\). [3]
CAIE M2 2018 June Q1
4 marks Moderate -0.8
A small ball \(B\) is projected from a point \(O\) on horizontal ground. The initial velocity of \(B\) has horizontal and vertically upwards components of \(18 \text{ ms}^{-1}\) and \(25 \text{ ms}^{-1}\) respectively. For the instant \(4 \text{ s}\) after projection, find the speed and direction of motion of \(B\). [4]
CAIE M2 2018 June Q2
3 marks Standard +0.3
\includegraphics{figure_2} A non-uniform rod \(AB\) of length \(0.5 \text{ m}\) and weight \(8 \text{ N}\) is freely hinged to a fixed point at \(A\). The rod makes an angle of \(30°\) with the horizontal with \(B\) above the level of \(A\). The rod is held in equilibrium by a force of magnitude \(12 \text{ N}\) acting in the vertical plane containing the rod at an angle of \(30°\) to \(AB\) applied at \(B\) (see diagram). Find the distance of the centre of mass of the rod from \(A\). [3]
CAIE M2 2018 June Q3
7 marks Standard +0.3
A particle \(P\) of mass \(0.4 \text{ kg}\) is projected horizontally along a smooth horizontal plane from a point \(O\). At time \(t \text{ s}\) after projection the velocity of \(P\) is \(v \text{ ms}^{-1}\). A force of magnitude \(0.8t \text{ N}\) directed away from \(O\) acts on \(P\) and a force of magnitude \(2e^{-t} \text{ N}\) opposes the motion of \(P\).
  1. Show that \(\frac{dv}{dt} = 2t - 5e^{-t}\). [2]
  2. Given that \(v = 8\) when \(t = 1\), express \(v\) in terms of \(t\). [3]
  3. Find the speed of projection of \(P\). [2]
CAIE M2 2018 June Q4
9 marks Moderate -0.3
A small object is projected from a point \(O\) with speed \(V \text{ ms}^{-1}\) at an angle of \(45°\) above the horizontal. At time \(t\) after projection, the horizontal and vertically upwards displacements of the object from \(O\) are \(x \text{ m}\) and \(y \text{ m}\) respectively.
  1. Express \(x\) and \(y\) in terms of \(t\), and hence find the equation of the path. [4]
The object passes through the point with coordinates \((24, 18)\).
  1. Find \(V\). [2]
  2. The object passes through two points which are \(22.5 \text{ m}\) above the level of \(O\). Find the values of \(x\) for these points. [3]
CAIE M2 2018 June Q5
8 marks Challenging +1.2
A particle \(P\) of mass \(0.7 \text{ kg}\) is attached by a light elastic string to a fixed point \(O\) on a smooth plane inclined at an angle of \(30°\) to the horizontal. The natural length of the string is \(0.5 \text{ m}\) and the modulus of elasticity is \(20 \text{ N}\). The particle \(P\) is projected up the line of greatest slope through \(O\) from a point \(A\) below the level of \(O\). The initial kinetic energy of \(P\) is \(1.8 \text{ J}\) and the initial elastic potential energy in the string is also \(1.8 \text{ J}\).
  1. Find the distance \(OA\). [2]
  2. Find the greatest speed of \(P\) in the motion. [6]
CAIE M2 2018 June Q6
9 marks Challenging +1.2
\includegraphics{figure_6} A particle \(P\) of mass \(0.2 \text{ kg}\) is attached to one end of a light inextensible string of length \(0.6 \text{ m}\). The other end of the string is attached to a particle \(Q\) of mass \(0.3 \text{ kg}\). The string passes through a small hole \(H\) in a smooth horizontal surface. A light elastic string of natural length \(0.3 \text{ m}\) and modulus of elasticity \(15 \text{ N}\) joins \(Q\) to a fixed point \(A\) which is \(0.4 \text{ m}\) vertically below \(H\). The particle \(P\) moves on the surface in a horizontal circle with centre \(H\) (see diagram).
  1. Calculate the greatest possible speed of \(P\) for which the elastic string is not extended. [4]
  2. Find the distance \(HP\) given that the angular speed of \(P\) is \(8 \text{ rad s}^{-1}\). [5]
CAIE M2 2018 June Q7
10 marks Standard +0.3
\includegraphics{figure_7} A uniform solid cone has height \(1.2 \text{ m}\) and base radius \(0.5 \text{ m}\). A uniform object is made by drilling a cylindrical hole of radius \(0.2 \text{ m}\) through the cone along the axis of symmetry (see diagram).
  1. Show that the height of the object is \(0.72 \text{ m}\) and that the volume of the cone removed by the drilling is \(0.0352\pi \text{ m}^3\). [4]
[The volume of a cone is \(\frac{1}{3}\pi r^2 h\).]
  1. Find the distance of the centre of mass of the object from its base. [6]
CAIE M2 2018 June Q1
6 marks Standard +0.3
\includegraphics{figure_1} A small ball \(B\) is projected from a point \(O\) on horizontal ground towards a point \(A\) 12 m above the ground. 0.9 s after projection \(B\) has travelled a horizontal distance of 20 m and is vertically below \(A\) (see diagram).
  1. Find the angle and the speed of projection of \(B\). [4]
  2. Calculate the distance \(AB\) when \(B\) is vertically below \(A\). [2]
CAIE M2 2018 June Q2
6 marks Standard +0.3
One end of a light elastic string is attached to a fixed point \(O\). The other end of the string is attached to a particle \(P\) of mass 0.24 kg. The string has natural length 0.6 m and modulus of elasticity 24 N. The particle is released from rest at \(O\). Find the two possible values of the distance \(OP\) for which the particle has speed 1.5 m s\(^{-1}\). [6]
CAIE M2 2018 June Q3
5 marks Standard +0.3
\includegraphics{figure_3} \(ABC\) is an object made from a uniform wire consisting of two straight portions \(AB\) and \(BC\), in which \(AB = a\), \(BC = x\) and angle \(ABC = 90°\). When the object is freely suspended from \(A\) and in equilibrium, the angle between \(AB\) and the horizontal is \(\theta\) (see diagram).
  1. Show that \(x^2 \tan \theta - 2ax - a^2 = 0\). [3]
  2. Given that \(\tan \theta = 1.25\), calculate the length of the wire in terms of \(a\). [2]
CAIE M2 2018 June Q4
7 marks Moderate -0.3
A particle \(P\) is projected from a point \(O\) on horizontal ground with initial speed 20 m s\(^{-1}\) and angle of projection 30°. At time \(t\) s after projection, the horizontal and vertically upwards displacements of \(P\) from \(O\) are \(x\) m and \(y\) m respectively.
  1. Express \(x\) and \(y\) in terms of \(t\) and hence find the equation of the trajectory of \(P\). [4]
  2. Calculate this height. [3]
\(P\) is at the same height above the ground at two points which are a horizontal distance apart of 15 m.
CAIE M2 2018 June Q5
7 marks Standard +0.8
\includegraphics{figure_5} A uniform object is made by joining a solid cone of height 0.8 m and base radius 0.6 m and a cylinder. The cylinder has length 0.4 m and radius 0.5 m. The cylinder has a cylindrical hole of length 0.4 m and radius \(x\) m drilled through it along the axis of symmetry. A plane face of the cylinder is attached to the base of the cone so that the object has an axis of symmetry perpendicular to its base and passing through the vertex of the cone. The object is placed with points on the base of the cone and the base of the cylinder in contact with a horizontal surface (see diagram). The object is on the point of toppling.
  1. Show that the centre of mass of the object is 0.15 m from the base of the cone. [3]
  2. Find \(x\). [4]
[The volume of a cone is \(\frac{1}{3}\pi r^2 h\).]
CAIE M2 2018 June Q6
9 marks Standard +0.8
\includegraphics{figure_6} A particle \(P\) of mass 0.2 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 \(A\). The particle \(P\) is also attached to one end of a second light inextensible string of length 0.6 m, the other end of which is attached to a fixed point \(B\) vertically below \(A\). The particle moves in a horizontal circle of radius 0.3 m, which has its centre at the mid-point of \(AB\), with both strings straight (see diagram).
  1. Calculate the least possible angular speed of \(P\). [4]
  2. Find the greatest possible speed of \(P\). [5]
The string \(AP\) will break if its tension exceeds 8 N. The string \(BP\) will break if its tension exceeds 5 N.