Questions — AQA M2 (165 questions)

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AQA M2 Q1
Moderate -0.3
1 A uniform beam, \(A B\), has mass 20 kg and length 7 metres. A rope is attached to the beam at \(A\). A second rope is attached to the beam at the point \(C\), which is 2 metres from \(B\). Both of the ropes are vertical. The beam is in equilibrium in a horizontal position, as shown in the diagram. \includegraphics[max width=\textwidth, alt={}, center]{88aec6ab-af83-4d5e-84b6-5fd84c16a6c9-003_298_906_756_552} Find the tensions in the two ropes.
AQA M2 Q2
Moderate -0.8
2 A particle, of mass 2 kg , is attached to one end of a light inextensible string. The other end is fixed to the point \(O\). The particle is set into motion, so that it describes a horizontal circle of radius 0.6 metres, with the string at an angle of \(30 ^ { \circ }\) to the vertical. The centre of the circle is vertically below \(O\). \includegraphics[max width=\textwidth, alt={}, center]{88aec6ab-af83-4d5e-84b6-5fd84c16a6c9-003_346_340_1580_842}
  1. Show that the tension in the string is 22.6 N , correct to three significant figures.
  2. Find the speed of the particle.
AQA M2 Q3
Moderate -0.8
3 A particle moves in a straight line and at time \(t\) has velocity \(v\), where $$v = 2 t - 12 \mathrm { e } ^ { - t } , \quad t \geqslant 0$$
    1. Find an expression for the acceleration of the particle at time \(t\).
    2. State the range of values of the acceleration of the particle.
  2. When \(t = 0\), the particle is at the origin. Find an expression for the displacement of the particle from the origin at time \(t\).
    (4 marks)
AQA M2 Q4
Standard +0.3
4 A car has a maximum speed of \(42 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) when it is moving on a horizontal road. When the speed of the car is \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\), it experiences a resistance force of magnitude \(30 v\) newtons.
  1. Show that the maximum power of the car is 52920 W .
  2. The car has mass 1200 kg . It travels, from rest, up a slope inclined at \(5 ^ { \circ }\) to the horizontal.
    1. Show that, when the car is travelling at its maximum speed \(\mathrm { V } \mathrm { m } \mathrm { s } ^ { - 1 }\) up the slope, $$V ^ { 2 } + 392 \sin 5 ^ { \circ } V - 1764 = 0$$
    2. Hence find \(V\).
AQA M2 Q5
Standard +0.3
5 A car, of mass 1600 kg , is travelling along a straight horizontal road at a speed of \(20 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) when the driving force is removed. The car then freewheels and experiences a resistance force. The resistance force has magnitude \(40 v\) newtons, where \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\) is the speed of the car after it has been freewheeling for \(t\) seconds. Find an expression for \(v\) in terms of \(t\).
AQA M2 Q6
Standard +0.3
6 A particle \(P\), of mass \(m \mathrm {~kg}\), is placed at the point \(Q\) on the top of a smooth upturned hemisphere of radius 3 metres and centre \(O\). The plane face of the hemisphere is fixed to a horizontal table. The particle is set into motion with an initial horizontal velocity of \(2 \mathrm {~ms} ^ { - 1 }\). When the particle is on the surface of the hemisphere, the angle between \(O P\) and \(O Q\) is \(\theta\) and the particle has speed \(v \mathrm {~ms} ^ { - 1 }\). \includegraphics[max width=\textwidth, alt={}, center]{88aec6ab-af83-4d5e-84b6-5fd84c16a6c9-005_419_1013_607_511}
  1. Show that \(v ^ { 2 } = 4 + 6 g ( 1 - \cos \theta )\).
  2. Find the value of \(\theta\) when the particle leaves the hemisphere.
AQA M2 Q7
Standard +0.3
7 A particle, of mass 10 kg , is attached to one end of a light elastic string of natural length 0.4 metres and modulus of elasticity 100 N . The other end of the string is fixed to the point \(O\).
  1. Find the length of the elastic string when the particle hangs in equilibrium directly below \(O\).
  2. The particle is pulled down and held at a point \(P\), which is 1 metre vertically below \(O\). Show that the elastic potential energy of the string when the particle is in this position is 45 J .
  3. The particle is released from rest at the point \(P\). In the subsequent motion, the particle has speed \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\) when it is \(x\) metres below \(\boldsymbol { O }\).
    1. Show that, while the string is taut, $$v ^ { 2 } = 39.6 x - 25 x ^ { 2 } - 14.6$$
    2. Find the value of \(x\) when the particle comes to rest for the first time after being released, given that the string is still taut.
AQA M2 Q8
Standard +0.3
8 Two small blocks, \(A\) and \(B\), of masses 0.8 kg and 1.2 kg respectively, are stuck together. A spring has natural length 0.5 metres and modulus of elasticity 49 N . One end of the spring is attached to the top of the block \(A\) and the other end of the spring is attached to a fixed point \(O\).
  1. The system hangs in equilibrium with the blocks stuck together, as shown in the diagram. \includegraphics[max width=\textwidth, alt={}, center]{88aec6ab-af83-4d5e-84b6-5fd84c16a6c9-017_385_239_669_881} Find the extension of the spring.
  2. Show that the elastic potential energy of the spring when the system is in equilibrium is 1.96 J .
  3. The system is hanging in this equilibrium position when block \(B\) falls off and block \(A\) begins to move vertically upwards. Block \(A\) next comes to rest when the spring is compressed by \(x\) metres.
    1. Show that \(x\) satisfies the equation $$x ^ { 2 } + 0.16 x - 0.008 = 0$$
    2. Find the value of \(x\).
AQA M2 2007 January Q1
8 marks Moderate -0.8
1 A child, of mass 35 kg , slides down a slide in a water park. The child, starting from rest, slides from the point \(A\) to the point \(B\), which is 10 metres vertically below the level of \(A\), as shown in the diagram. \includegraphics[max width=\textwidth, alt={}, center]{480a817d-074f-440d-829e-c8f8a9746151-2_259_595_685_705}
  1. In a simple model, all resistance forces are ignored. Use an energy method to find the speed of the child at \(B\).
  2. State one resistance force that has been ignored in answering part (a).
  3. In fact, when the child slides down the slide, she reaches \(B\) with a speed of \(12 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). Given that the slide is 20 metres long and the sum of the resistance forces has a constant magnitude of \(F\) newtons, use an energy method to find the value of \(F\).
    (4 marks)
AQA M2 2007 January Q2
6 marks Moderate -0.8
2 A hotel sign consists of a uniform rectangular lamina of weight \(W\). The sign is suspended in equilibrium in a vertical plane by two vertical light chains attached to the sign at the points \(A\) and \(B\), as shown in the diagram. The edge containing \(A\) and \(B\) is horizontal. \includegraphics[max width=\textwidth, alt={}, center]{480a817d-074f-440d-829e-c8f8a9746151-2_289_529_1859_726} The tensions in the chains attached at \(A\) and \(B\) are \(T _ { A }\) and \(T _ { B }\) respectively.
  1. Draw a diagram to show the forces acting on the sign.
  2. Find \(T _ { A }\) and \(T _ { B }\) in terms of \(W\).
  3. Explain how you have used the fact that the lamina is uniform in answering part (b).
AQA M2 2007 January Q3
6 marks Moderate -0.3
3 A light inextensible string has length \(2 a\). One end of the string is attached to a fixed point \(O\) and a particle of mass \(m\) is attached to the other end. Initially, the particle is held at the point \(A\) with the string taut and horizontal. The particle is then released from rest and moves in a circular path. Subsequently, it passes through the point \(B\), which is directly below \(O\). The points \(O , A\) and \(B\) are as shown in the diagram. \includegraphics[max width=\textwidth, alt={}, center]{480a817d-074f-440d-829e-c8f8a9746151-3_426_437_575_772}
  1. Show that the speed of the particle at \(B\) is \(2 \sqrt { a g }\).
  2. Find the tension in the string as the particle passes through \(B\). Give your answer in terms of \(m\) and \(g\).
AQA M2 2007 January Q4
9 marks Standard +0.3
4 A uniform T-shaped lamina is formed by rigidly joining two rectangles \(A B C H\) and \(D E F G\), as shown in the diagram. \includegraphics[max width=\textwidth, alt={}, center]{480a817d-074f-440d-829e-c8f8a9746151-4_748_652_456_644}
  1. Show that the centre of mass of the lamina is 26 cm from the edge \(A B\).
  2. Explain why the centre of mass of the lamina is 5 cm from the edge \(G F\).
  3. The point \(X\) is on the edge \(A B\) and is 7 cm from \(A\), as shown in the diagram below. \includegraphics[max width=\textwidth, alt={}, center]{480a817d-074f-440d-829e-c8f8a9746151-4_697_534_1576_753} The lamina is freely suspended from \(X\) and hangs in equilibrium.
    Find the angle between the edge \(A B\) and the vertical, giving your answer to the nearest degree.
    (4 marks)
AQA M2 2007 January Q5
12 marks Moderate -0.3
5 Tom is on a fairground ride.
Tom's position vector, \(\mathbf { r }\) metres, at time \(t\) seconds is given by $$\mathbf { r } = 2 \cos t \mathbf { i } + 2 \sin t \mathbf { j } + ( 10 - 0.4 t ) \mathbf { k }$$ The perpendicular unit vectors \(\mathbf { i }\) and \(\mathbf { j }\) are in the horizontal plane and the unit vector \(\mathbf { k }\) is directed vertically upwards.
    1. Find Tom's position vector when \(t = 0\).
    2. Find Tom's position vector when \(t = 2 \pi\).
    3. Write down the first two values of \(t\) for which Tom is directly below his starting point.
  2. Find an expression for Tom's velocity at time \(t\).
  3. Tom has mass 25 kg . Show that the resultant force acting on Tom during the motion has constant magnitude. State the magnitude of the resultant force.
    (5 marks)
AQA M2 2007 January Q6
11 marks Moderate -0.8
6 A particle is attached to one end of a light inextensible string. The other end of the string is attached to a fixed point \(O\). The particle is set into motion, so that it describes a horizontal circle whose centre is vertically below \(O\). The angle between the string and the vertical is \(\theta\), as shown in the diagram. \includegraphics[max width=\textwidth, alt={}, center]{480a817d-074f-440d-829e-c8f8a9746151-6_506_442_534_794}
  1. The particle completes 40 revolutions every minute. Show that the angular speed of the particle is \(\frac { 4 \pi } { 3 }\) radians per second.
  2. The radius of the circle is 0.2 metres. Find, in terms of \(\pi\), the magnitude of the acceleration of the particle.
  3. The mass of the particle is \(m \mathrm {~kg}\) and the tension in the string is \(T\) newtons.
    1. Draw a diagram showing the forces acting on the particle.
    2. Explain why \(T \cos \theta = m g\).
    3. Find the value of \(\theta\), giving your answer to the nearest degree.
AQA M2 2007 January Q7
11 marks Moderate -0.3
7 A motorcycle has a maximum power of 72 kilowatts. The motorcycle and its rider are travelling along a straight horizontal road. When they are moving at a speed of \(\mathrm { V } \mathrm { m } \mathrm { s } ^ { - 1 }\), they experience a total resistance force of magnitude \(k V\) newtons, where \(k\) is a constant.
  1. The maximum speed of the motorcycle and its rider is \(60 \mathrm {~ms} ^ { - 1 }\). Show that \(k = 20\).
  2. When the motorcycle is travelling at \(20 \mathrm {~m} \mathrm {~s} ^ { - 1 }\), the rider allows the motorcycle to freewheel so that the only horizontal force acting is the resistance force. When the motorcycle has been freewheeling for \(t\) seconds, its speed is \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\) and the magnitude of the resistance force is \(20 v\) newtons. The mass of the motorcycle and its rider is 500 kg .
    1. Show that \(\frac { \mathrm { d } v } { \mathrm {~d} t } = - \frac { v } { 25 }\).
    2. Hence find the time that it takes for the speed of the motorcycle to reduce from \(20 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) to \(10 \mathrm {~m} \mathrm {~s} ^ { - 1 }\).
      (6 marks)
AQA M2 2007 January Q8
12 marks Standard +0.3
8 Two small blocks, \(A\) and \(B\), of masses 0.8 kg and 1.2 kg respectively, are stuck together. A spring has natural length 0.5 metres and modulus of elasticity 49 N . One end of the spring is attached to the top of the block \(A\) and the other end of the spring is attached to a fixed point \(O\).
  1. The system hangs in equilibrium with the blocks stuck together, as shown in the diagram. \includegraphics[max width=\textwidth, alt={}, center]{480a817d-074f-440d-829e-c8f8a9746151-8_385_239_669_881} Find the extension of the spring.
  2. Show that the elastic potential energy of the spring when the system is in equilibrium is 1.96 J .
  3. The system is hanging in this equilibrium position when block \(B\) falls off and block \(A\) begins to move vertically upwards. Block \(A\) next comes to rest when the spring is compressed by \(x\) metres.
    1. Show that \(x\) satisfies the equation $$x ^ { 2 } + 0.16 x - 0.008 = 0$$
    2. Find the value of \(x\).
AQA M2 2009 January Q1
4 marks Moderate -0.8
1 A particle moves along a straight line. At time \(t\), it has velocity \(v\), where $$v = 4 t ^ { 3 } - 8 \sin 2 t + 5$$ When \(t = 0\), the particle is at the origin.
Find an expression for the displacement of the particle from the origin at time \(t\).
AQA M2 2009 January Q2
9 marks Moderate -0.8
2 A stone, of mass 6 kg , is thrown vertically upwards with a speed of \(12 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) from a point at a height of 4 metres above ground level.
  1. Calculate the initial kinetic energy of the stone.
    1. Show that the kinetic energy of the stone when it hits the ground is 667 J , correct to three significant figures.
    2. Hence find the speed of the stone when it hits the ground.
    3. State two modelling assumptions that you have made.
AQA M2 2009 January Q3
12 marks Moderate -0.3
3 A particle moves on a horizontal plane, in which the unit vectors \(\mathbf { i }\) and \(\mathbf { j }\) are directed east and north respectively. At time \(t\) seconds, the position vector of the particle is \(\mathbf { r }\) metres, where $$\mathbf { r } = \left( 2 \mathrm { e } ^ { \frac { 1 } { 2 } t } - 8 t + 5 \right) \mathbf { i } + \left( t ^ { 2 } - 6 t \right) \mathbf { j }$$
  1. Find an expression for the velocity of the particle at time \(t\).
    1. Find the speed of the particle when \(t = 3\).
    2. State the direction in which the particle is travelling when \(t = 3\).
  3. Find the acceleration of the particle when \(t = 3\).
  4. The mass of the particle is 7 kg . Find the magnitude of the resultant force on the particle when \(t = 3\).
AQA M2 2009 January Q4
9 marks Standard +0.3
4 A uniform rectangular lamina \(A B C D\) has a mass of 8 kg . The side \(A B\) has length 20 cm , the side \(B C\) has length 10 cm , and \(P\) is the mid-point of \(A B\). A uniform circular lamina, of mass 2 kg and radius 5 cm , is fixed to the rectangular lamina to form a sign. The centre of the circular lamina is 5 cm from each of \(A B\) and \(B C\), as shown in the diagram. \includegraphics[max width=\textwidth, alt={}, center]{851cb2a3-5bc8-4af9-b1fc-a143d37beebe-3_661_1200_589_406}
  1. Find the distance of the centre of mass of the sign from \(A D\).
  2. Write down the distance of the centre of mass of the sign from \(A B\).
  3. The sign is freely suspended from \(P\). Find the angle between \(A D\) and the vertical when the sign is in equilibrium.
  4. Explain how you have used the fact that each lamina is uniform in your solution to this question.
AQA M2 2009 January Q5
9 marks Moderate -0.5
5 A particle, of mass 6 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 \(30 ^ { \circ }\) to the vertical. The centre of this circle is vertically below \(O\). \includegraphics[max width=\textwidth, alt={}, center]{851cb2a3-5bc8-4af9-b1fc-a143d37beebe-4_586_490_541_767} The particle moves in a horizontal circle with an angular speed of 40 revolutions per minute.
  1. Show that the angular speed of the particle is \(\frac { 4 \pi } { 3 }\) radians per second.
  2. Show that the tension in the string is 67.9 N , correct to three significant figures.
  3. Find the radius of the horizontal circle.
AQA M2 2009 January Q6
7 marks Moderate -0.3
6 A train, of mass 60 tonnes, travels on a straight horizontal track. It has a maximum speed of \(40 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) when its engine is working at 800 kW .
  1. Find the magnitude of the resistance force acting on the train when the train is travelling at its maximum speed.
  2. When the train is travelling at \(40 \mathrm {~m} \mathrm {~s} ^ { - 1 }\), the power is turned off. Assume that the resistance force is constant and is equal to that found in part (a). Also assume that this resistance force is the only horizontal force acting on the train. Use an energy method to find how far the train travels when slowing from \(40 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) to \(36 \mathrm {~m} \mathrm {~s} ^ { - 1 }\).
    (4 marks)
AQA M2 2009 January Q7
7 marks Standard +0.3
7 A hollow cylinder, of internal radius 4 m , is fixed so that its axis is horizontal. The point \(O\) is on this axis. A particle, of mass 6 kg , is set in motion so that it moves on the smooth inner surface of the cylinder in a vertical circle about \(O\). Its speed at the point \(A\), which is vertically below \(O\), is \(8 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). \includegraphics[max width=\textwidth, alt={}, center]{851cb2a3-5bc8-4af9-b1fc-a143d37beebe-5_746_739_504_662} When the particle is at the point \(B\), at a height of 2 m above \(A\), find:
  1. its speed;
  2. the normal reaction between the cylinder and the particle.
AQA M2 2009 January Q8
7 marks Standard +0.3
8 A stone, of mass 0.05 kg , is moving along the smooth horizontal floor of a tank, which is filled with oil. At time \(t\), the stone has speed \(v\). As the stone moves, it experiences a resistance force of magnitude \(0.08 v ^ { 2 }\).
  1. Show that $$\frac { \mathrm { d } v } { \mathrm {~d} t } = - 1.6 v ^ { 2 }$$ (2 marks)
  2. The initial speed of the stone is \(3 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). Show that $$v = \frac { 15 } { 5 + 24 t }$$ (5 marks)
AQA M2 2009 January Q9
11 marks Standard +0.3
9 A bungee jumper, of mass 80 kg , is attached to one end of a light elastic cord, of natural length 16 metres and modulus of elasticity 784 N . The other end of the cord is attached to a horizontal platform, which is at a height of 65 metres above the ground. The bungee jumper steps off the platform at the point where the cord is attached and falls vertically. The bungee jumper can be modelled as a particle. Hooke's law can be assumed to apply throughout the motion and air resistance can be assumed to be negligible.
  1. Find the length of the cord when the acceleration of the bungee jumper is zero.
  2. The cord extends by \(x\) metres beyond its natural length before the bungee jumper first comes to rest.
    1. Show that \(x ^ { 2 } - 32 x - 512 = 0\).
    2. Find the distance above the ground at which the bungee jumper first comes to rest.