Questions M2 (1537 questions)

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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]
CAIE M2 2012 November Q4
6 marks Standard +0.3
\includegraphics{figure_4} A particle \(P\) is moving inside a smooth hollow cone which has its vertex downwards and its axis vertical, and whose semi-vertical angle is \(45°\). A light inextensible string parallel to the surface of the cone connects \(P\) to the vertex. \(P\) moves with constant angular speed in a horizontal circle of radius \(0.67\) m (see diagram). The tension in the string is equal to the weight of \(P\). Calculate the angular speed of \(P\). [6]
CAIE M2 2012 November Q5
7 marks Standard +0.3
A particle \(P\) is projected with speed \(30\) m s\(^{-1}\) at an angle of \(60°\) above the horizontal from a point \(O\) on horizontal ground. For the instant when the speed of \(P\) is \(17\) m s\(^{-1}\) and increasing,
  1. show that the vertical component of the velocity of \(P\) is \(8\) m s\(^{-1}\) downwards, [2]
  2. calculate the distance of \(P\) from \(O\). [5]
CAIE M2 2012 November Q6
8 marks Standard +0.3
\includegraphics{figure_6} A uniform lamina \(OABCD\) consists of a semicircle \(BCD\) with centre \(O\) and radius \(0.6\) m and an isosceles triangle \(OAB\), joined along \(OB\) (see diagram). The triangle has area \(0.36\) m\(^2\) and \(AB = AO\).
  1. Show that the centre of mass of the lamina lies on \(OB\). [4]
  2. Calculate the distance of the centre of mass of the lamina from \(O\). [4]
CAIE M2 2012 November Q7
12 marks Challenging +1.2
A light elastic string has natural length \(3\) m and modulus of elasticity \(45\) N. A particle \(P\) of weight \(6\) N is attached to the mid-point of the string. The ends of the string are attached to fixed points \(A\) and \(B\) which lie in the same vertical line with \(A\) above \(B\) and \(AB = 4\) m. The particle \(P\) is released from rest at the point \(1.5\) m vertically below \(A\).
  1. Calculate the distance \(P\) moves after its release before first coming to instantaneous rest at a point vertically above \(B\). (You may assume that at this point the part of the string joining \(P\) to \(B\) is slack.) [4]
  2. Show that the greatest speed of \(P\) occurs when it is \(2.1\) m below \(A\), and calculate this greatest speed. [5]
  3. Calculate the greatest magnitude of the acceleration of \(P\). [3]
CAIE M2 2013 November Q1
Standard +0.3
\includegraphics{figure_1}
  1. \(A\) has velocity \(\vec{x}\) and \(C\) has velocity \(\vec{v}\)
CAIE M2 2013 November Q2
Moderate -0.5
\includegraphics{figure_2}
    1. \(C\)
    2. \(C\) has velocity \(\vec{v}\)
CAIE M2 2013 November Q3
Moderate -1.0
\(A\) has velocity \(\vec{v}\), there are velocities \(\vec{x}\), \(\vec{v}\), \(\vec{v}\) around point \(O\), and velocity \(\vec{v}\)
    1. —
    2. \(v\) and \(v\)
CAIE M2 2013 November Q4
Moderate -0.3
\(A\) has velocity \(\vec{v}\)
    1. \(C\) with velocity \(\vec{v}\)
    2. \(C\)
CAIE M2 2013 November Q5
Moderate -0.5
\includegraphics{figure_5} \(A\) has velocity \(\vec{x}\), there are velocities \(\vec{x}\), \(\vec{x}\), \(\vec{v}\) and \(\vec{v}\)
    1. \(v\)
    2. \(v\) and \(v\)
CAIE M2 2013 November Q6
Moderate -0.5
\includegraphics{figure_6} \(E\) has velocity \(\vec{v}\)
    1. \(B\)
    2. \(v\)
CAIE M2 2013 November Q7
Moderate -0.5
\(A\) has velocity \(\vec{x}\)
    1. \(C\)
  2. \(A\) has velocity \(\vec{x}\)
    1. \(C\)
  3. \(C\) with velocities \(v \vec{v}\)
CAIE M2 2013 November Q1
8 marks Moderate -0.8
A particle moves in a straight line. Its displacement from a fixed point \(O\) at time \(t\) seconds is \(s\) metres, where \(s = t^3 - 9t^2 + 24t\).
  1. Find expressions for the velocity \(v\) and acceleration \(a\) of the particle at time \(t\).
  2. Find the values of \(t\) for which the particle is at rest.
  3. Find the total distance travelled by the particle in the first \(6\) seconds.
[8]
CAIE M2 2013 November Q2
6 marks Moderate -0.5
A particle moves in a straight line. At time \(t\) seconds its velocity is \(v\) ms\(^{-1}\) and its acceleration is \(a\) ms\(^{-2}\).
  1. Given that \(a = —\), express \(v\) in terms of \(t\).
  2. Given that \(v = tv\) when \(t = 0\), find \(v\) in terms of \(t\).
  3. Find the displacement from the starting point when \(t = v\).
[6]
CAIE M2 2013 November Q3
8 marks Standard +0.3
\includegraphics{figure_3} A particle moves on the inner surface of a smooth hollow cone of semi-vertical angle \(\alpha\). The axis of the cone is vertical with the vertex at the bottom. The particle moves in a horizontal circle of radius \(r\) with constant speed \(v\). Find expressions for the normal reactions on the particle from the cone surface, and show that the height of the particle above the vertex is \(\frac{v^2}{g \tan \alpha}\). [8]
CAIE M2 2013 November Q4
14 marks Standard +0.8
A particle of mass \(m\) is attached to one end of a light inextensible string of length \(l\). The other end of the string is attached to a fixed point. The particle moves in a vertical circle.
  1. Show that the speed \(v\) at the lowest point of the circle must satisfy \(v^2 \geq 5gl\) for the particle to complete the circle.
  2. Given that the particle just completes the circle, find the tensions in the string at the highest and lowest points of the circle.
  3. Given that \(v^2 = 6gl\) at the lowest point, find the tension in the string when the particle has risen through an angle \(\theta\) from the lowest point.
[14]