Questions M2 (1537 questions)

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Edexcel M2 2013 June Q6
12 marks Standard +0.3
\includegraphics{figure_3} A uniform rod \(AB\) has weight 30 N and length 3 m. The rod rests in equilibrium on a rough horizontal peg \(P\) with its end \(A\) on smooth horizontal ground. The rod is in a vertical plane perpendicular to the peg. The rod is inclined at 15° to the ground and the point of contact between the peg and the rod is 45 cm above the ground, as shown in Figure 3.
  1. Show that the normal reaction at \(P\) has magnitude 25 N. [4]
  2. Find the magnitude of the force on the rod at \(A\). [4]
The coefficient of friction between the rod and the peg is \(\mu\).
  1. Find the range of possible values of \(\mu\). [4]
Edexcel M2 2013 June Q7
13 marks Standard +0.3
\includegraphics{figure_4} Two smooth particles \(P\) and \(Q\) have masses \(m\) and \(2m\) respectively. The particles are moving in the same direction in the same straight line, on a smooth horizontal plane, with \(Q\) in front of \(P\). The particles are moving towards a fixed smooth vertical wall which is perpendicular to the direction of motion of the particles, as shown in Figure 4. The speed of \(P\) is \(2u\) and the speed of \(Q\) is \(3u\). The coefficient of restitution between \(Q\) and the wall is \(\frac{1}{3}\). Particle \(Q\) strikes the wall, rebounds and then collides directly with \(P\). The direction of motion of each particle is reversed by this collision. Immediately after this collision the speed of \(P\) is \(v\) and the speed of \(Q\) is \(w\).
  1. Show that \(v = 2w\). [5]
The total kinetic energy of \(P\) and \(Q\) immediately after they collide is half the total kinetic energy of \(P\) and \(Q\) immediately before they collide.
  1. Find the coefficient of restitution between \(P\) and \(Q\). [8]
AQA M2 2014 June Q1
8 marks Moderate -0.8
An eagle has caught a salmon of mass 3 kg to take to its nest. When the eagle is flying with speed \(8 \text{ m s}^{-1}\), it drops the salmon. The salmon falls a vertical distance of 13 metres back into the sea. The salmon is to be modelled as a particle. The salmon's weight is the only force that acts on it as it falls to the sea.
  1. Calculate the kinetic energy of the salmon when it is dropped by the eagle. [2 marks]
  2. Calculate the potential energy lost by the salmon as it falls to the sea. [2 marks]
    1. Find the kinetic energy of the salmon when it reaches the sea. [2 marks]
    2. Hence find the speed of the salmon when it reaches the sea. [2 marks]
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]
Edexcel M2 Q1
4 marks Moderate -0.8
A constant force acts on a particle of mass 200 grams, moving it 50 cm in a straight line on a rough horizontal surface at a constant speed. The coefficient of friction between the particle and the surface is \(\frac{1}{4}\). Calculate, in J, the work done by the force. [4 marks]
Edexcel M2 Q2
5 marks Moderate -0.3
A stone, of mass 0.9 kg, is projected vertically upwards with speed 10 ms\(^{-1}\) in a medium which exerts a constant resistance to motion. It comes to rest after rising a distance of 3.75 m. Find the magnitude of the non-gravitational resisting force acting on the stone. [5 marks]
Edexcel M2 Q3
8 marks Moderate -0.8
A particle \(P\), of mass 0.4 kg, moves in a straight line such that, at time \(t\) seconds after passing through a fixed point \(O\), its distance from \(O\) is \(x\) metres, where \(x = 3t^2 + 8t\).
  1. Show that \(P\) never returns to \(O\). [2 marks]
  2. Find the value of \(t\) when \(P\) has velocity 20 ms\(^{-1}\). [3 marks]
  3. Show that the force acting on \(P\) is constant, and find its magnitude. [3 marks]
Edexcel M2 Q4
9 marks Standard +0.3
Two smooth spheres \(A\) and \(B\), of masses \(2m\) and \(3m\) respectively, are moving on a smooth horizontal table with velocities \((3\mathbf{i} - \mathbf{j})\) ms\(^{-1}\) and \((4\mathbf{i} + \mathbf{j})\) ms\(^{-1}\), where \(\mathbf{i}\) and \(\mathbf{j}\) are perpendicular unit vectors. They collide, after which \(A\) has velocity \((5\mathbf{i} + \mathbf{j})\) ms\(^{-1}\).
  1. Find the magnitude of the impulse exerted on \(B\) by \(A\), stating the units of your answer. [4 marks]
  2. Find the speed of \(B\) immediately after the collision. [5 marks]
Edexcel M2 Q5
10 marks Standard +0.3
A small car, of mass 850 kg, moves on a straight horizontal road. Its engine is working at its maximum rate of 25 kW, and a constant resisting force of magnitude 900 N opposes the car's motion.
  1. Find the acceleration of the car when it is moving with speed 15 ms\(^{-1}\). [3 marks]
  2. Find the maximum speed of the car on the horizontal road. [3 marks]
With the engine still working at 25 kW and the non-gravitational resistance remaining at 900 N, the car now climbs a hill inclined at an angle \(\alpha\) to the horizontal, where \(\sin \alpha = \frac{1}{10}\).
  1. Find the maximum speed of the car on this hill. [4 marks]
Edexcel M2 Q6
12 marks Standard +0.3
A uniform wire \(ABCD\) is bent into the shape shown, where the sections \(AB\), \(BC\) and \(CD\) are straight and of length \(3a\), \(10a\) and \(5a\) respectively and \(AD\) is parallel to \(BC\). \includegraphics{figure_6}
  1. Show that the cosine of angle \(BCD\) is \(\frac{3}{5}\). [2 marks]
  2. Find the distances of the centre of mass of the bent wire from (i) \(AB\), (ii) \(BC\). [6 marks]
The wire is hung over a smooth peg at \(B\) and rests in equilibrium.
  1. Find, to the nearest 0.1°, the angle between \(BC\) and the vertical in this position. [4 marks]
Edexcel M2 Q7
12 marks Standard +0.3
Two particles \(P\) and \(Q\), of masses 0.3 kg and 0.2 kg respectively, are moving towards each other along a straight line. \(P\) has speed 4 ms\(^{-1}\). They collide directly. After the collision the direction of motion of both particles has been reversed, and \(Q\) has speed 2 ms\(^{-1}\). The coefficient of restitution between \(P\) and \(Q\) is \(\frac{1}{3}\). Find
  1. the speed of \(Q\) before the collision, [4 marks]
  2. the speed of \(P\) after the collision, [4 marks]
  3. the kinetic energy, in J, lost in the impact. [4 marks]