Questions — AQA M2 (165 questions)

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AQA M2 2014 June Q2
10 marks Standard +0.3
A particle has mass 6 kg. A single force \((24e^{-2t}\mathbf{i} - 12t^3\mathbf{j})\) newtons acts on the particle at time \(t\) seconds. No other forces act on the particle.
  1. Find the acceleration of the particle at time \(t\). [2 marks]
  2. At time \(t = 0\), the velocity of the particle is \((-7\mathbf{i} - 4\mathbf{j}) \text{ m s}^{-1}\). Find the velocity of the particle at time \(t\). [4 marks]
  3. Find the speed of the particle when \(t = 0.5\). [4 marks]
AQA M2 2014 June Q3
5 marks Moderate -0.8
Four tools are attached to a board. The board is to be modelled as a uniform lamina and the four tools as four particles. The diagram shows the lamina, the four particles \(A\), \(B\), \(C\) and \(D\), and the \(x\) and \(y\) axes. \includegraphics{figure_3} The lamina has mass 5 kg and its centre of mass is at the point \((7, 6)\). Particle \(A\) has mass 4 kg and is at the point \((11, 2)\). Particle \(B\) has mass 3 kg and is at the point \((3, 6)\). Particle \(C\) has mass 7 kg and is at the point \((5, 9)\). Particle \(D\) has mass 1 kg and is at the point \((1, 4)\). Find the coordinates of the centre of mass of the system of board and tools. [5 marks]
AQA M2 2014 June Q4
9 marks Standard +0.3
A particle, of mass 0.8 kg, is attached to one end of a light inextensible string. The other end of the string is attached to the fixed point \(O\). The particle is set in motion, so that it moves in a horizontal circle at constant speed, with the string at an angle of \(35°\) to the vertical. The centre of this circle is vertically below \(O\), as shown in the diagram. \includegraphics{figure_4} The particle moves in a horizontal circle and completes 20 revolutions each minute.
  1. Find the angular speed of the particle in radians per second. [2 marks]
  2. Find the tension in the string. [3 marks]
  3. Find the radius of the horizontal circle. [4 marks]
AQA M2 2014 June Q5
7 marks Standard +0.3
A light inextensible string, of length \(a\), has one end attached to a fixed point \(O\). A particle, of mass \(m\), is attached to the other end of the string. The particle is moving in a vertical circle with centre \(O\). The point \(Q\) is the highest point of the particle's path. When the particle is at \(P\), vertically below \(O\), the string is taut and the particle is moving with speed \(7\sqrt{ag}\), as shown in the diagram. \includegraphics{figure_5}
  1. Find, in terms of \(g\) and \(a\), the speed of the particle at the point \(Q\). [4 marks]
  2. Find, in terms of \(g\) and \(m\), the tension in the string when the particle is at \(Q\). [3 marks]
AQA M2 2014 June Q6
13 marks Standard +0.8
A puck, of mass \(m\) kg, is moving in a straight line across smooth horizontal ice. At time \(t\) seconds, the puck has speed \(v \text{ m s}^{-1}\). As the puck moves, it experiences an air resistance force of magnitude \(0.3mv^3\) newtons, until it comes to rest. No other horizontal forces act on the puck. When \(t = 0\), the speed of the puck is \(8 \text{ m s}^{-1}\). Model the puck as a particle.
  1. Show that $$v = (4 - 0.2t)^{\frac{3}{2}}$$ [6 marks]
  2. Find the value of \(t\) when the puck comes to rest. [2 marks]
  3. Find the distance travelled by the puck as its speed decreases from \(8 \text{ m s}^{-1}\) to zero. [5 marks]
AQA M2 2014 June Q7
8 marks Standard +0.3
A uniform ladder \(AB\), of length 6 metres and mass 22 kg, rests with its foot, \(A\), on rough horizontal ground. The ladder rests against the top of a smooth vertical wall at the point \(C\), where the length \(AC\) is 5 metres. The vertical plane containing the ladder is perpendicular to the wall, and the angle between the ladder and the ground is \(60°\). A man, of mass 88 kg, is standing on the ladder. The man may be modelled as a particle at the point \(D\), where the length of \(AD\) is 4 metres. The ladder is on the point of slipping. \includegraphics{figure_7}
  1. Draw a diagram to show the forces acting on the ladder. [2 marks]
  2. Find the coefficient of friction between the ladder and the horizontal ground. [6 marks]
AQA M2 2014 June Q8
15 marks Challenging +1.2
An elastic string has natural length 1.5 metres and modulus of elasticity 120 newtons. One end of the string is attached to a fixed point, \(A\), on a rough plane inclined at \(20°\) to the horizontal. The other end of the elastic string is attached to a particle of mass 4 kg. The coefficient of friction between the particle and the plane is 0.8. The three points, \(A\), \(B\) and \(C\), lie on a line of greatest slope. The point \(C\) is \(x\) metres from \(A\), as shown in the diagram. The particle is released from rest at \(C\) and moves up the plane. \includegraphics{figure_8}
  1. Show that, as the particle moves up the plane, the frictional force acting on the particle is 29.5 N, correct to three significant figures. [3 marks]
  2. The particle comes to rest for an instant at \(B\), which is 2 metres from \(A\). The particle then starts to move back towards \(A\).
    1. Find \(x\). [8 marks]
    2. Find the acceleration of the particle as it starts to move back towards \(A\). [4 marks]
AQA M2 2016 June Q1
8 marks Moderate -0.8
A stone, of mass \(0.3\) kg, is thrown with a speed of \(8 \text{ m s}^{-1}\) from a point at a height of \(5\) metres above a horizontal surface.
  1. Calculate the initial kinetic energy of the stone. [2 marks]
    1. Find the kinetic energy of the stone when it hits the surface. [3 marks]
    2. Hence find the speed of the stone when it hits the surface. [2 marks]
    3. State one modelling assumption that you have made. [1 mark]
AQA M2 2016 June Q2
13 marks Standard +0.3
A particle moves in a horizontal plane under the action of a single force, \(\mathbf{F}\) newtons. The unit vectors \(\mathbf{i}\) and \(\mathbf{j}\) are directed east and north respectively. At time \(t\) seconds, the velocity of the particle, \(\mathbf{v} \text{ m s}^{-1}\), is given by $$\mathbf{v} = (8t - t^4)\mathbf{i} + 6e^{-3t}\mathbf{j}$$
  1. Find an expression for the acceleration of the particle at time \(t\). [2 marks]
  2. The mass of the particle is \(2\) kg.
    1. Find an expression for the force \(\mathbf{F}\) acting on the particle at time \(t\). [2 marks]
    2. Find the magnitude of \(\mathbf{F}\) when \(t = 1\). [3 marks]
  3. Find the value of \(t\) when \(\mathbf{F}\) acts due south. [2 marks]
  4. When \(t = 0\), the particle is at the point with position vector \((3\mathbf{i} - 5\mathbf{j})\) metres. Find an expression for the position vector, \(\mathbf{r}\) metres, of the particle at time \(t\). [4 marks]
AQA M2 2016 June Q3
9 marks Moderate -0.3
The diagram shows a uniform lamina \(ABCDEFGHIJKL\). \includegraphics{figure_3}
  1. Explain why the centre of mass of the lamina is \(35\) cm from \(AL\). [1 mark]
  2. Find the distance of the centre of mass from \(AF\). [4 marks]
  3. The lamina is freely suspended from \(A\). Find the angle between \(AB\) and the vertical when the lamina is in equilibrium. [3 marks]
  4. Explain, briefly, how you have used the fact that the lamina is uniform. [1 mark]
AQA M2 2016 June Q4
8 marks Standard +0.3
A particle \(P\), of mass \(6\) kg, is attached to one end of a light inextensible string. The string passes through a small smooth ring, fixed at a point \(O\). A second particle \(Q\), of mass \(8\) kg, is attached to the other end of the string. The particle \(Q\) hangs at rest vertically below the ring, and the particle \(P\) moves with speed \(5 \text{ m s}^{-1}\) in a horizontal circle, as shown in the diagram. The angle between \(OP\) and the vertical is \(\theta\). \includegraphics{figure_4}
  1. Find the tension in the string. [1 mark]
  2. Find \(\theta\). [3 marks]
  3. Find the radius of the horizontal circle. [4 marks]
AQA M2 2016 June Q5
12 marks Standard +0.3
A particle of mass \(m\) is suspended from a fixed point \(O\) by a light inextensible string of length \(l\). The particle hangs in equilibrium at the point \(R\) vertically below \(O\). The particle is set into motion with a horizontal velocity \(u\) so that it moves in a complete vertical circle with centre \(O\). The point \(T\) on the circle is such that angle \(ROT\) is \(30°\), as shown in the diagram. \includegraphics{figure_5}
  1. Find, in terms of \(g\), \(l\) and \(u\), the speed of the particle at the point \(T\). [3 marks]
  2. Find, in terms of \(g\), \(l\), \(m\) and \(u\), the tension in the string when the particle is at the point \(T\). [3 marks]
  3. Find, in terms of \(g\), \(l\), \(m\) and \(u\), the tension in the string when the particle returns to the point \(R\). [2 marks]
  4. The particle makes complete revolutions. Find, in terms of \(g\) and \(l\), the minimum value of \(u\). [4 marks]
AQA M2 2016 June Q6
8 marks Standard +0.3
A stone, of mass \(m\), falls vertically downwards under gravity through still water. At time \(t\), the stone has speed \(v\) and it experiences a resistance force of magnitude \(\lambda mv\), where \(\lambda\) is a constant.
  1. Show that $$\frac{\text{d}v}{\text{d}t} = g - \lambda v$$ [2 marks]
  2. The initial speed of the stone is \(u\). Find an expression for \(v\) at time \(t\). [6 marks]
AQA M2 2016 June Q7
9 marks Standard +0.8
A uniform ladder, of weight \(W\), rests with its top against a rough vertical wall and its base on rough horizontal ground. The coefficient of friction between the wall and the ladder is \(\mu\) and the coefficient of friction between the ground and the ladder is \(2\mu\). When the ladder is on the point of slipping, the ladder is inclined at an angle of \(\theta\) to the horizontal.
  1. Draw a diagram to show the forces acting on the ladder. [2 marks]
  2. Find \(\tan \theta\) in terms of \(\mu\). [7 marks]
AQA M2 2016 June Q8
8 marks Challenging +1.8
A particle \(P\), of mass \(5\) kg is placed at the point \(A\) on a rough plane which is inclined at \(30°\) to the horizontal. The points \(Q\) and \(R\) are also on the surface of the inclined plane, with \(QR = 15\) metres. The point \(A\) is between \(Q\) and \(R\) so that \(AQ = 4\) metres and \(AR = 11\) metres. The three points \(Q\), \(A\) and \(R\) are on a line of greatest slope of the plane. \includegraphics{figure_8} The particle is attached to two light elastic strings, \(PQ\) and \(PR\). One of the strings, \(PQ\), has natural length \(4\) metres and modulus of elasticity \(160\) N, the other string, \(PR\), has natural length \(6\) metres and modulus of elasticity \(120\) N. The particle is released from rest at the point \(A\). The coefficient of friction between the particle and the plane is \(0.4\). Find the distance of the particle from \(Q\) when it is next at rest. [8 marks]