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CAIE M2 2018 November Q2
6 marks Standard +0.8
\includegraphics{figure_2} A uniform object is made by attaching the base of a solid hemisphere to the base of a solid cone so that the object has an axis of symmetry. The base of the cone has radius \(0.3\text{ m}\), and the hemisphere has radius \(0.2\text{ m}\). The object is placed on a horizontal plane with a point \(A\) on the curved surface of the hemisphere and a point \(B\) on the circumference of the cone in contact with the plane (see diagram).
  1. Given that the object is on the point of toppling about \(B\), find the distance of the centre of mass of the object from the base of the cone. [3]
  2. Given instead that the object is on the point of toppling about \(A\), calculate the height of the cone. [3]
[The volume of a cone is \(\frac{1}{3}\pi r^2 h\). The volume of a hemisphere is \(\frac{2}{3}\pi r^3\).]
CAIE M2 2018 November Q3
7 marks Standard +0.3
A particle \(P\) of mass \(0.4\text{ kg}\) is attached to a fixed point \(O\) by a light elastic string of natural length \(0.5\text{ m}\) and modulus of elasticity \(20\text{ N}\). The particle \(P\) is released from rest at \(O\).
  1. Find the greatest speed of \(P\) in the subsequent motion. [4]
  2. Find the distance below \(O\) of the point at which \(P\) comes to instantaneous rest. [3]
CAIE M2 2018 November Q4
8 marks Challenging +1.2
\includegraphics{figure_4} A particle \(P\) of mass \(0.5\text{ kg}\) is projected along a smooth horizontal surface towards a fixed point \(A\). Initially \(P\) is at a point \(O\) on the surface, and after projection, \(P\) has a displacement from \(O\) of \(x\text{ m}\) and velocity \(v\text{ m s}^{-1}\). The particle \(P\) is connected to \(A\) by a light elastic string of natural length \(0.8\text{ m}\) and modulus of elasticity \(16\text{ N}\). The distance \(OA\) is \(1.6\text{ m}\) (see diagram). The motion of \(P\) is resisted by a force of magnitude \(24x^2\text{ N}\).
  1. Show that \(v\frac{\text{d}v}{\text{d}x} = 32 - 40x - 48x^2\) while \(P\) is in motion and the string is stretched. [3]
  2. The maximum value of \(v\) is \(4.5\). Find the initial value of \(v\). [5]
CAIE M2 2018 November Q5
8 marks Standard +0.3
A particle \(P\) of mass \(0.1\text{ kg}\) is attached to one end of a light inextensible string of length \(0.5\text{ m}\). The other end of the string is attached to a fixed point \(A\). The particle \(P\) moves in a circle which has its centre \(O\) on a smooth horizontal surface \(0.3\text{ m}\) below \(A\). The tension in the string has magnitude \(T\text{ N}\) and the magnitude of the force exerted on \(P\) by the surface is \(R\text{ N}\).
  1. Given that the speed of \(P\) is \(1.5\text{ m s}^{-1}\), calculate \(T\) and \(R\). [4]
  2. Given instead that \(T = R\), calculate the angular speed of \(P\). [4]
CAIE M2 2018 November Q6
8 marks Standard +0.3
\includegraphics{figure_6} Fig. 1 shows the cross-section \(ABCDE\) through the centre of mass \(G\) of a uniform prism. The cross-section consists of a rectangle \(ABCF\) from which a triangle \(DEF\) has been removed; \(AB = 0.6\text{ m}\), \(BC = 0.7\text{ m}\) and \(DF = EF = 0.3\text{ m}\).
  1. Show that the distance of \(G\) from \(BC\) is \(0.276\text{ m}\), and find the distance of \(G\) from \(AB\). [5]
  2. The prism is placed with \(CD\) on a rough horizontal surface. A force of magnitude \(2\text{ N}\) acting in the plane of the cross-section is applied to the prism. The line of action of the force passes through \(G\) and is perpendicular to \(DE\) (see Fig. 2). The prism is on the point of toppling about the edge through \(D\). Calculate the weight of the prism. [3]
CAIE M2 2018 November Q7
9 marks Challenging +1.2
\includegraphics{figure_7} A small object is projected with speed \(24\text{ m s}^{-1}\) from a point \(O\) at the foot of a plane inclined at \(45°\) to the horizontal. The angle of projection of the object is \(15°\) above a line of greatest slope of the plane (see diagram). At time \(t\) s after projection, the horizontal and vertically upwards displacements of the object from \(O\) are \(x\) m and \(y\) m respectively.
  1. Express \(x\) and \(y\) in terms of \(t\), and hence find the value of \(t\) for the instant when the object strikes the plane. [4]
  2. Express the vertical height of the object above the plane in terms of \(t\) and hence find the greatest vertical height of the object above the plane. [5]
CAIE M2 2018 November Q4
8 marks Challenging +1.2
\includegraphics{figure_4} A particle \(P\) of mass \(0.5\text{ kg}\) is projected along a smooth horizontal surface towards a fixed point \(A\). Initially \(P\) is at a point \(O\) on the surface, and after projection, \(P\) has a displacement from \(O\) of \(x\text{ m}\) and velocity \(v\text{ m s}^{-1}\). The particle \(P\) is connected to \(A\) by a light elastic string of natural length \(0.8\text{ m}\) and modulus of elasticity \(16\text{ N}\). The distance \(OA\) is \(1.6\text{ m}\) (see diagram). The motion of \(P\) is resisted by a force of magnitude \(24x^2\text{ N}\).
  1. Show that \(v\frac{\text{d}v}{\text{d}x} = 32 - 40x - 48x^2\) while \(P\) is in motion and the string is stretched. [3] The maximum value of \(v\) is \(4.5\).
  2. Find the initial value of \(v\). [5]
CAIE M2 2018 November Q6
8 marks Standard +0.3
\includegraphics{figure_6} Fig. 1 shows the cross-section \(ABCDE\) through the centre of mass \(G\) of a uniform prism. The cross-section consists of a rectangle \(ABCF\) from which a triangle \(DEF\) has been removed; \(AB = 0.6\text{ m}\), \(BC = 0.7\text{ m}\) and \(DF = EF = 0.3\text{ m}\).
  1. Show that the distance of \(G\) from \(BC\) is \(0.276\text{ m}\), and find the distance of \(G\) from \(AB\). [5] The prism is placed with \(CD\) on a rough horizontal surface. A force of magnitude \(2\text{ N}\) acting in the plane of the cross-section is applied to the prism. The line of action of the force passes through \(G\) and is perpendicular to \(DE\) (see Fig. 2). The prism is on the point of toppling about the edge through \(D\).
  2. Calculate the weight of the prism. [3]
CAIE M2 2018 November Q7
9 marks Standard +0.8
\includegraphics{figure_7} A small object is projected with speed \(24\text{ m s}^{-1}\) from a point \(O\) at the foot of a plane inclined at \(45°\) to the horizontal. The angle of projection of the object is \(15°\) above a line of greatest slope of the plane (see diagram). At time \(t\text{ s}\) 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 value of \(t\) for the instant when the object strikes the plane. [4]
  2. Express the vertical height of the object above the plane in terms of \(t\) and hence find the greatest vertical height of the object above the plane. [5]
CAIE Further Paper 3 2020 June Q1
5 marks Standard +0.3
A particle \(P\) is projected with speed \(u\) at an angle of \(30°\) above the horizontal from a point \(O\) on a horizontal plane and moves freely under gravity. The particle reaches its greatest height at time \(T\) after projection. Find, in terms of \(u\), the speed of \(P\) at time \(\frac{2}{3}T\) after projection. [5]
CAIE Further Paper 3 2020 June Q2
5 marks Challenging +1.2
\includegraphics{figure_2} A light inextensible string of length \(a\) is threaded through a fixed smooth ring \(R\). One end of the string is attached to a particle \(A\) of mass \(3m\). The other end of the string is attached to a particle \(B\) of mass \(m\). The particle \(A\) hangs in equilibrium at a distance \(x\) vertically below the ring. The angle between \(AR\) and \(BR\) is \(\theta\) (see diagram). The particle \(B\) moves in a horizontal circle with constant angular speed \(2\sqrt{\frac{g}{a}}\). Show that \(\cos \theta = \frac{1}{3}\) and find \(x\) in terms of \(a\). [5]
CAIE Further Paper 3 2020 June Q3
7 marks Standard +0.3
One end of a light elastic spring, of natural length \(a\) and modulus of elasticity \(5mg\), is attached to a fixed point \(A\). The other end of the spring is attached to a particle \(P\) of mass \(m\). The spring hangs with \(P\) vertically below \(A\). The particle \(P\) is released from rest in the position where the extension of the spring is \(\frac{3}{5}a\).
  1. Show that the initial acceleration of \(P\) is \(\frac{3}{5}g\) upwards. [3]
  2. Find the speed of \(P\) when the spring first returns to its natural length. [4]
CAIE Further Paper 3 2020 June Q4
7 marks Challenging +1.2
\includegraphics{figure_4} A uniform square lamina \(ABCD\) has sides of length \(10\text{cm}\). The point \(E\) is on \(BC\) with \(EC = 7.5\text{cm}\), and the point \(F\) is on \(DC\) with \(CF = x\text{cm}\). The triangle \(EFC\) is removed from \(ABCD\) (see diagram). The centre of mass of the resulting shape \(ABEFD\) is a distance \(\bar{x}\text{cm}\) from \(CB\) and a distance \(\bar{y}\text{cm}\) from \(CD\).
  1. Show that \(\bar{x} = \frac{400 - x^2}{80 - 3x}\) and find a corresponding expression for \(\bar{y}\). [4]
The shape \(ABEFD\) is in equilibrium in a vertical plane with the edge \(DF\) resting on a smooth horizontal surface.
  1. Find the greatest possible value of \(x\), giving your answer in the form \(a + b\sqrt{2}\), where \(a\) and \(b\) are constants to be determined. [3]
CAIE Further Paper 3 2020 June Q5
8 marks Challenging +1.2
A particle \(P\) is moving along a straight line with acceleration \(3ku - kv\) where \(v\) is its velocity at time \(t\), \(u\) is its initial velocity and \(k\) is a constant. The velocity and acceleration of \(P\) are both in the direction of increasing displacement from the initial position.
  1. Find the time taken for \(P\) to achieve a velocity of \(2u\). [3]
  2. Find an expression for the displacement of \(P\) from its initial position when its velocity is \(2u\). [5]
CAIE Further Paper 3 2020 June Q6
8 marks Challenging +1.2
A particle \(P\) of mass \(m\) is moving with speed \(u\) on a fixed smooth horizontal surface. The particle strikes a fixed vertical barrier. At the instant of impact the direction of motion of \(P\) makes an angle \(\alpha\) with the barrier. The coefficient of restitution between \(P\) and the barrier is \(e\). As a result of the impact, the direction of motion of \(P\) is turned through \(90°\).
  1. Show that \(\tan^2 \alpha = \frac{1}{e}\). [3]
The particle \(P\) loses two-thirds of its kinetic energy in the impact.
  1. Find the value of \(\alpha\) and the value of \(e\). [5]
CAIE Further Paper 3 2020 June Q7
10 marks Challenging +1.8
A hollow cylinder of radius \(a\) is fixed with its axis horizontal. A particle \(P\), of mass \(m\), moves in part of a vertical circle of radius \(a\) and centre \(O\) on the smooth inner surface of the cylinder. The speed of \(P\) when it is at the lowest point \(A\) of its motion is \(\sqrt{\frac{7}{2}ga}\). The particle \(P\) loses contact with the surface of the cylinder when \(OP\) makes an angle \(\theta\) with the upward vertical through \(O\).
  1. Show that \(\theta = 60°\). [5]
  2. Show that in its subsequent motion \(P\) strikes the cylinder at the point \(A\). [5]
CAIE Further Paper 3 2020 June Q1
2 marks Moderate -0.5
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 \(O\) on a smooth horizontal plane. The particle \(P\) moves in horizontal circles about \(O\). The tension in the string is \(4mg\). Find, in terms of \(a\) and \(g\), the time that \(P\) takes to make one complete revolution. [2]
CAIE Further Paper 3 2020 June Q2
6 marks Standard +0.8
A particle \(Q\) of mass \(m\) kg falls from rest under gravity. The motion of \(Q\) is resisted by a force of magnitude \(mkv\) N, where \(v\) ms\(^{-1}\) is the speed of \(Q\) at time \(t\) s and \(k\) is a positive constant. Find an expression for \(v\) in terms of \(g\), \(k\) and \(t\). [6]
CAIE Further Paper 3 2020 June Q3
6 marks Standard +0.8
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 \(AB\) a diameter of the circle. \(OA\) makes an angle of \(60°\) with the downward vertical through \(O\) and \(OB\) makes an angle of \(60°\) with the upward vertical through \(O\). The speed of \(Q\) when it is at \(A\) is \(2\sqrt{ag}\). Given that \(T_A\) and \(T_B\) are the tensions in the string at \(A\) and \(B\) respectively, find the ratio \(T_A : T_B\). [6]
CAIE Further Paper 3 2020 June Q4
4 marks Standard +0.3
\includegraphics{figure_4} A uniform solid circular cone, of vertical height \(4r\) and radius \(2r\), is attached to a uniform solid cylinder, of height \(3r\) and radius \(kr\), 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 $$\frac{(99k^2 + 96)r}{18k^2 + 32}.$$ [4]
CAIE Further Paper 3 2020 June Q4
4 marks Challenging +1.2
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 \(\tan \alpha = \frac{1}{5}\).
  1. 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
4 marks Challenging +1.8
\includegraphics{figure_5} 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 \(\alpha°\) 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 \(\tan \alpha = \frac{1+e}{1-e}\). [4]
CAIE Further Paper 3 2020 June Q5
4 marks Moderate -0.5
  1. Given that \(\tan \alpha = 2\), find the speed of \(A\) after the collision. [4]
CAIE Further Paper 3 2020 June Q6
6 marks Challenging +1.2
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 \(\tan \alpha = \frac{1}{2}\tan \theta\). [6]
CAIE Further Paper 3 2020 June Q6
4 marks Challenging +1.2
  1. 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 \(g\). [4]