Questions M2 (1537 questions)

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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.
CAIE M2 2018 June Q7
10 marks Standard +0.8
A particle \(P\) of mass 0.2 kg is released from rest at a point \(O\) above horizontal ground. At time \(t\) s after its release the velocity of \(P\) is 7.5 m s\(^{-1}\) downwards. A vertically downwards force of magnitude 0.6t N acts on \(P\). A vertically upwards force of magnitude \(ke^{-t}\) N, where \(k\) is a constant, also acts on \(P\).
  1. Show that \(\frac{dv}{dt} = 10 - 5ke^{-t} + 3t\). [2]
  2. Find the greatest value of \(k\) for which \(P\) does not initially move upwards. [3]
  3. Given that \(k = 1\), and that \(P\) strikes the ground when \(t = 2\), find the height of \(O\) above the ground. [5]
CAIE M2 2017 March Q1
5 marks Moderate -0.3
A small ball is projected with speed \(15 \text{ m s}^{-1}\) at an angle of \(60°\) above the horizontal. Find the distance from the point of projection of the ball at the instant when it is travelling horizontally. [5]
CAIE M2 2017 March Q2
6 marks Standard +0.3
A cylindrical container is open at the top. The curved surface and the circular base of the container are both made from the same thin uniform material. The container has radius \(0.2 \text{ m}\) and height \(0.9 \text{ m}\).
  1. Show that the centre of mass of the container is \(0.405 \text{ m}\) from the base. [3]
The container is placed with its base on a rough inclined plane. The container is in equilibrium on the point of slipping down the plane and also on the point of toppling.
  1. Find the coefficient of friction between the container and the plane. [3]
CAIE M2 2017 March Q3
7 marks Moderate -0.3
A particle \(P\) is projected with speed \(20 \text{ m s}^{-1}\) at an angle of \(60°\) below the horizontal, from a point \(O\) which is \(30 \text{ m}\) above horizontal ground.
  1. Calculate the time taken by \(P\) to reach the ground. [3]
  2. Calculate the speed and direction of motion of \(P\) immediately before it reaches the ground. [4]
CAIE M2 2017 March Q4
7 marks Standard +0.3
\includegraphics{figure_4} The diagram shows a uniform lamina \(ABCD\) with \(AB = 0.75 \text{ m}\), \(AD = 0.6 \text{ m}\) and \(BC = 0.9 \text{ m}\). Angle \(BAD =\) angle \(ABC = 90°\).
  1. Show that the distance of the centre of mass of the lamina from \(AB\) is \(0.38 \text{ m}\), and find the distance of the centre of mass from \(BC\). [5]
The lamina is freely suspended at \(B\) and hangs in equilibrium.
  1. Find the angle between \(BC\) and the vertical. [2]
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]