Questions — CAIE M2 (519 questions)

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CAIE M2 2017 March Q5
7 marks Standard +0.3
\includegraphics{figure_5} Two particles \(P\) and \(Q\) have masses \(0.4 \text{ kg}\) and \(m \text{ kg}\) respectively. \(P\) is attached to a fixed point \(A\) by a light inextensible string of length \(0.5 \text{ m}\) which is inclined at an angle of \(60°\) to the vertical. \(P\) and \(Q\) are joined to each other by a light inextensible vertical string. \(Q\) is attached to a fixed point \(B\), which is vertically below \(A\), by a light inextensible string. The string \(BQ\) is taut and horizontal. The particles rotate in horizontal circles about an axis through \(A\) and \(B\) with constant angular speed \(\omega \text{ rad s}^{-1}\) (see diagram). The tension in the string joining \(P\) and \(Q\) is \(1.5 \text{ N}\).
  1. Find the tension in the string \(AP\) and the value of \(\omega\). [4]
  2. Find \(m\) and the tension in the string \(BQ\). [3]
CAIE M2 2017 March Q6
8 marks Challenging +1.2
\(O\) and \(A\) are fixed points on a rough horizontal surface, with \(OA = 1 \text{ m}\). A particle \(P\) of mass \(0.4 \text{ kg}\) is projected horizontally with speed \(U \text{ m s}^{-1}\) from \(A\) in the direction \(OA\) and moves in a straight line. After projection, when the displacement of \(P\) from \(O\) is \(x \text{ m}\), the velocity of \(P\) is \(v \text{ m s}^{-1}\). The coefficient of friction between the surface and \(P\) is \(0.4\). A force of magnitude \(\frac{0.8}{x} \text{ N}\) acts on \(P\) in the direction \(PO\).
  1. Show that, while the particle is in motion, \(v \frac{\text{d}v}{\text{d}x} = -4 - \frac{2}{x}\). [3]
It is given that \(P\) comes to instantaneous rest between \(x = 2.0\) and \(x = 2.1\).
  1. Find the set of possible values of \(U\). [5]
CAIE M2 2017 March Q7
10 marks Standard +0.8
One end of a light elastic string of natural length \(0.6 \text{ m}\) and modulus of elasticity \(24 \text{ N}\) is attached to a fixed point \(O\). The other end of the string is attached to a particle \(P\) of mass \(0.4 \text{ kg}\) which hangs in equilibrium vertically below \(O\).
  1. Calculate the extension of the string. [2]
\(P\) is projected vertically downwards from the equilibrium position with speed \(5 \text{ m s}^{-1}\).
  1. Calculate the distance \(P\) travels before it is first at instantaneous rest. [4]
When \(P\) is first at instantaneous rest a stationary particle of mass \(0.4 \text{ kg}\) becomes attached to \(P\).
  1. Find the greatest speed of the combined particle in the subsequent motion. [4]
CAIE M2 2010 November Q1
3 marks Moderate -0.8
A horizontal circular disc rotates with constant angular speed \(9 \text{ rad s}^{-1}\) about its centre \(O\). A particle of mass \(0.05 \text{ kg}\) is placed on the disc at a distance \(0.4 \text{ m}\) from \(O\). The particle moves with the disc and no sliding takes place. Calculate the magnitude of the resultant force exerted on the particle by the disc. [3]
CAIE M2 2010 November Q2
6 marks Standard +0.3
\includegraphics{figure_2} A bow consists of a uniform curved portion \(AB\) of mass \(1.4 \text{ kg}\), and a uniform taut string of mass \(m \text{ kg}\) which joins \(A\) and \(B\). The curved portion \(AB\) is an arc of a circle centre \(O\) and radius \(0.8 \text{ m}\). Angle \(AOB\) is \(\frac{2}{3}\pi\) radians (see diagram). The centre of mass of the bow (including the string) is \(0.65 \text{ m}\) from \(O\). Calculate \(m\). [6]
CAIE M2 2010 November Q3
7 marks Standard +0.3
\includegraphics{figure_3} One end of a light inextensible string of length \(0.2 \text{ m}\) is attached to a fixed point \(A\) which is above a smooth horizontal surface. A particle \(P\) of mass \(0.6 \text{ kg}\) is attached to the other end of the string. \(P\) moves in a circle on the surface with constant speed \(v \text{ m s}^{-1}\), with the string taut and making an angle of \(30°\) to the horizontal (see diagram).
  1. Given that \(v = 1.5\), calculate the magnitude of the force that the surface exerts on \(P\). [4]
  2. Given instead that \(P\) moves with its greatest possible speed while remaining in contact with the surface, find \(v\). [3]
CAIE M2 2010 November Q4
7 marks Standard +0.3
\includegraphics{figure_4} A uniform beam \(AB\) has length \(2 \text{ m}\) and weight \(70 \text{ N}\). The beam is hinged at \(A\) to a fixed point on a vertical wall, and is held in equilibrium by a light inextensible rope. One end of the rope is attached to the wall at a point \(1.7 \text{ m}\) vertically above the hinge. The other end of the rope is attached to the beam at a point \(0.8 \text{ m}\) from \(A\). The rope is at right angles to \(AB\). The beam carries a load of weight \(220 \text{ N}\) at \(B\) (see diagram).
  1. Find the tension in the rope. [3]
  2. Find the direction of the force exerted on the beam at \(A\). [4]
CAIE M2 2010 November Q5
7 marks Standard +0.3
A particle \(P\) of mass \(0.28 \text{ kg}\) is attached to the mid-point of a light elastic string of natural length \(4 \text{ m}\). The ends of the string are attached to fixed points \(A\) and \(B\) which are at the same horizontal level and \(4.8 \text{ m}\) apart. \(P\) is released from rest at the mid-point of \(AB\). In the subsequent motion, the acceleration of \(P\) is zero when \(P\) is at a distance \(0.7 \text{ m}\) below \(AB\).
  1. Show that the modulus of elasticity of the string is \(20 \text{ N}\). [4]
  2. Calculate the maximum speed of \(P\). [3]
CAIE M2 2010 November Q6
10 marks Standard +0.3
A cyclist and his bicycle have a total mass of \(81 \text{ kg}\). The cyclist starts from rest and rides in a straight line. The cyclist exerts a constant force of \(135 \text{ N}\) and the motion is opposed by a resistance of magnitude \(9v \text{ N}\), where \(v \text{ m s}^{-1}\) is the cyclist's speed at time \(t \text{ s}\) after starting.
  1. Show that \(\frac{9}{15-v} \frac{dv}{dt} = 1\). [2]
  2. Solve this differential equation to show that \(v = 15(1-e^{-\frac{t}{9}})\). [4]
  3. Find the distance travelled by the cyclist in the first \(9 \text{ s}\) of the motion. [4]
CAIE M2 2010 November Q7
10 marks Standard +0.3
\includegraphics{figure_7} A particle \(P\) is projected from a point \(O\) with initial speed \(10 \text{ m s}^{-1}\) at an angle of \(45°\) above the horizontal. \(P\) subsequently passes through the point \(A\) which is at an angle of elevation of \(30°\) from \(O\) (see diagram). At time \(t \text{ s}\) after projection the horizontal and vertically upward displacements of \(P\) from \(O\) are \(x \text{ m}\) and \(y \text{ m}\) respectively.
  1. Write down expressions for \(x\) and \(y\) in terms of \(t\), and hence obtain the equation of the trajectory of \(P\). [3]
  2. Calculate the value of \(x\) when \(P\) is at \(A\). [3]
  3. Find the angle the trajectory makes with the horizontal when \(P\) is at \(A\). [4]
CAIE M2 2010 November Q1
6 marks Moderate -0.3
\includegraphics{figure_1} \(ABCD\) is a uniform lamina with \(AB = 1.8\) m, \(AD = DC = 0.9\) m, and \(AD\) perpendicular to \(AB\) and \(DC\) (see diagram).
  1. Find the distance of the centre of mass of the lamina from \(AB\) and the distance from \(AD\). [4]
The lamina is freely suspended at \(A\) and hangs in equilibrium.
  1. Calculate the angle between \(AB\) and the vertical. [2]
CAIE M2 2010 November Q2
7 marks Standard +0.2
A particle \(P\) is projected with speed \(26\) m s\(^{-1}\) at an angle of \(30°\) above the horizontal from a point \(O\) on a horizontal plane.
  1. For the instant when the vertical component of the velocity of \(P\) is \(5\) m s\(^{-1}\) downwards, find the direction of motion of \(P\) and the height of \(P\) above the plane. [4]
  2. \(P\) strikes the plane at the point \(A\). Calculate the time taken by \(P\) to travel from \(O\) to \(A\) and the distance \(OA\). [3]
CAIE M2 2010 November Q3
8 marks Standard +0.3
\includegraphics{figure_3} Particles \(P\) and \(Q\) have masses \(0.8\) kg and \(0.4\) kg respectively. \(P\) is attached to a fixed point \(A\) by a light inextensible string which is inclined at an angle \(\alpha°\) to the vertical. \(Q\) is attached to a fixed point \(B\), which is vertically below \(A\), by a light inextensible string of length \(0.3\) m. The string \(BQ\) is horizontal. \(P\) and \(Q\) are joined to each other by a light inextensible string which is vertical. The particles rotate in horizontal circles of radius \(0.3\) m about the axis through \(A\) and \(B\) with constant angular speed \(5\) rad s\(^{-1}\) (see diagram).
  1. By considering the motion of \(Q\), find the tensions in the strings \(PQ\) and \(BQ\). [3]
  2. Find the tension in the string \(AP\) and the value of \(\alpha\). [5]
CAIE M2 2010 November Q4
8 marks Standard +0.3
\includegraphics{figure_4} A uniform rod \(AB\) has weight \(15\) N and length \(1.2\) m. The end \(A\) of the rod is in contact with a rough plane inclined at \(30°\) to the horizontal, and the rod is perpendicular to the plane. The rod is held in equilibrium in this position by means of a horizontal force applied at \(B\), acting in the vertical plane containing the rod (see diagram).
  1. Show that the magnitude of the force applied at \(B\) is \(4.33\) N, correct to \(3\) significant figures. [3]
  2. Find the magnitude of the frictional force exerted by the plane on the rod. [2]
  3. Given that the rod is in limiting equilibrium, calculate the coefficient of friction between the rod and the plane. [3]
CAIE M2 2010 November Q5
9 marks Standard +0.3
\includegraphics{figure_5} A light elastic string has natural length \(2\) m and modulus of elasticity \(\lambda\) N. The ends of the string are attached to fixed points \(A\) and \(B\) which are at the same horizontal level and \(2.4\) m apart. A particle \(P\) of mass \(0.6\) kg is attached to the mid-point of the string and hangs in equilibrium at a point \(0.5\) m below \(AB\) (see diagram).
  1. Show that \(\lambda = 26\). [4]
\(P\) is projected vertically downwards from the equilibrium position, and comes to instantaneous rest at a point \(0.9\) m below \(AB\).
  1. Calculate the speed of projection of \(P\). [5]
CAIE M2 2010 November Q6
12 marks Challenging +1.2
\includegraphics{figure_6} A particle \(P\) of mass \(0.2\) kg is projected with velocity \(2\) m s\(^{-1}\) upwards along a line of greatest slope on a plane inclined at \(30°\) to the horizontal (see diagram). Air resistance of magnitude \(0.5v\) N opposes the motion of \(P\), where \(v\) m s\(^{-1}\) is the velocity of \(P\) at time \(t\) s after projection. The coefficient of friction between \(P\) and the plane is \(\frac{1}{2\sqrt{3}}\). The particle \(P\) reaches a position of instantaneous rest when \(t = T\).
  1. Show that, while \(P\) is moving up the plane, \(\frac{dv}{dt} = -2.5(3 + v)\). [3]
  2. Calculate \(T\). [4]
  3. Calculate the speed of \(P\) when \(t = 2T\). [5]
CAIE M2 2011 November Q1
5 marks Standard +0.3
\includegraphics{figure_1} A non-uniform rod \(AB\), of length 0.6 m and weight 9 N, has its centre of mass 0.4 m from \(A\). The end \(A\) of the rod is in contact with a rough vertical wall. The rod is held in equilibrium, perpendicular to the wall, by means of a light string attached to \(B\). The string is inclined at \(30°\) to the horizontal. The tension in the string is \(T\) N (see diagram).
  1. Calculate \(T\). [2]
  2. Find the least possible value of the coefficient of friction at \(A\). [3]
CAIE M2 2011 November Q2
7 marks Standard +0.3
\includegraphics{figure_2} A particle \(P\) is projected from a point \(O\) at an angle of \(60°\) above horizontal ground. At an instant 0.6 s after projection, the angle of elevation of \(P\) from \(O\) is \(45°\) (see diagram).
  1. Show that the speed of projection of \(P\) is 8.20 m s\(^{-1}\), correct to 3 significant figures. [4]
  2. Calculate the time after projection when the direction of motion of \(P\) is \(45°\) above the horizontal. [3]
CAIE M2 2011 November Q3
8 marks Standard +0.3
One end of a light elastic string of natural length 0.4 m and modulus of elasticity 20 N is attached to a fixed point \(O\). The other end of the string is attached to a particle \(P\) of mass 0.25 kg. \(P\) hangs in equilibrium below \(O\).
  1. Calculate the distance \(OP\). [2]
The particle \(P\) is raised, and is released from rest at \(O\).
  1. Calculate the speed of \(P\) when it passes through the equilibrium position. [3]
  2. Calculate the greatest value of the distance \(OP\) in the subsequent motion. [3]
CAIE M2 2011 November Q4
9 marks Standard +0.8
A uniform solid cylinder has radius 0.7 m and height \(h\) m. A uniform solid cone has base radius 0.7 m and height 2.4 m. The cylinder and the cone both rest in equilibrium each with a circular face in contact with a horizontal plane. The plane is now tilted so that its inclination to the horizontal, \(θ°\), is increased gradually until the cone is about to topple.
  1. Find the value of \(θ\) at which the cone is about to topple. [2]
  2. Given that the cylinder does not topple, find the greatest possible value of \(h\). [2]
The plane is returned to a horizontal position, and the cone is fixed to one end of the cylinder so that the plane faces coincide. It is given that the weight of the cylinder is three times the weight of the cone. The curved surface of the cone is placed on the horizontal plane (see diagram). \includegraphics{figure_4}
  1. Given that the solid immediately topples, find the least possible value of \(h\). [5]
CAIE M2 2011 November Q5
10 marks Standard +0.3
A ball of mass 0.05 kg is released from rest at a height \(h\) m above the ground. At time \(t\) s after its release, the downward velocity of the ball is \(v\) m s\(^{-1}\). Air resistance opposes the motion of the ball with a force of magnitude 0.01\(v\) N.
  1. Show that \(\frac{dv}{dt} = 10 - 0.2v\). Hence find \(v\) in terms of \(t\). [6]
  2. Given that the ball reaches the ground when \(t = 2\), calculate \(h\). [4]
CAIE M2 2011 November Q6
11 marks Standard +0.3
A smooth bead \(B\) of mass 0.3 kg is threaded on a light inextensible string of length 0.9 m. One end of the string is attached to a fixed point \(A\), and the other end of the string is attached to a fixed point \(C\) which is vertically below \(A\). The tension in the string is 7 N, and the bead rotates with angular speed \(ω\) rad s\(^{-1}\) in a horizontal circle about the vertical axis through \(A\) and \(C\).
  1. Given that \(B\) moves in a circle with centre \(C\) and radius 0.2 m, calculate \(ω\), and hence find the kinetic energy of \(B\). [5]
  2. Given instead that angle \(ABC = 90°\), and that \(AB\) makes an angle \(\tan^{-1}(\frac{4}{3})\) with the vertical, calculate \(T\) and \(ω\). [6]
CAIE M2 2012 November Q1
3 marks Moderate -0.8
\(ABC\) is a uniform semicircular arc with diameter \(AC = 0.5\) m. The arc rotates about a fixed axis through \(A\) and \(C\) with angular speed \(2.4\) rad s\(^{-1}\). Calculate the speed of the centre of mass of the arc. [3]
CAIE M2 2012 November Q2
7 marks Standard +0.8
\includegraphics{figure_2} A uniform rod \(AB\) has weight \(6\) N and length \(0.8\) m. The rod rests in limiting equilibrium with \(B\) in contact with a rough horizontal surface and \(AB\) inclined at \(60°\) to the horizontal. Equilibrium is maintained by a force, in the vertical plane containing \(AB\), acting at \(A\) at an angle of \(45°\) to \(AB\) (see diagram). Calculate
  1. the magnitude of the force applied at \(A\), [3]
  2. the least possible value of the coefficient of friction at \(B\). [4]
CAIE M2 2012 November Q3
7 marks Standard +0.3
A particle \(P\) of mass \(0.2\) kg is released from rest and falls vertically. At time \(t\) s after release \(P\) has speed \(v\) m s\(^{-1}\). A resisting force of magnitude \(0.8v\) N acts on \(P\).
  1. Show that the acceleration of \(P\) is \((10 - 4v)\) m s\(^{-2}\). [2]
  2. Find the value of \(v\) when \(t = 0.6\). [5]