Questions — AQA M2 (163 questions)

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AQA M2 2015 June Q5
6 marks
5 An item of clothing is placed inside a washing machine. The drum of the washing machine has radius 30 cm and rotates, about a fixed horizontal axis, at a constant angular speed of 900 revolutions per minute. Model the item of clothing as a particle of mass 0.8 kg and assume that the clothing travels in a vertical circle with constant angular speed. Find the minimum magnitude of the normal reaction force exerted by the drum on the clothing and find the maximum magnitude of the normal reaction force exerted by the drum on the clothing.
[0pt] [6 marks]
\includegraphics[max width=\textwidth, alt={}]{691c50b4-50b2-4e3a-a7e0-60f8ec35ee3c-10_1883_1709_824_153}
AQA M2 2015 June Q6
9 marks
6 A van, of mass 1400 kg , is accelerating at a constant rate of \(0.2 \mathrm {~m} \mathrm {~s} ^ { - 2 }\) as it travels up a slope inclined at an angle \(\theta\) to the horizontal. The van experiences total resistance forces of 4000 N .
When the van is travelling at a speed of \(20 \mathrm {~m} \mathrm {~s} ^ { - 1 }\), the power output of the van's engine is 91.1 kW . Find \(\theta\).
[0pt] [9 marks]
AQA M2 2015 June Q7
2 marks
7 A parachutist, of mass 72 kg , is falling vertically. He opens his parachute at time \(t = 0\) when his speed is \(30 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). He then experiences an air resistance force of magnitude \(240 v\) newtons, where \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\) is his speed at time \(t\) seconds.
  1. When \(t > 0\), show that \(- \frac { 3 } { 10 } \frac { \mathrm {~d} v } { \mathrm {~d} t } = v - 2.94\).
  2. Find \(v\) in terms of \(t\).
  3. Sketch a graph to show how, for \(t \geqslant 0\), the parachutist's speed varies with time.
    [0pt] [2 marks]
AQA M2 2015 June Q8
8 Carol, a bungee jumper of mass 70 kg , is attached to one end of a light elastic cord of natural length 26 metres and modulus of elasticity 1456 N . The other end of the cord is attached to a fixed horizontal platform which is at a height of 69 metres above the ground. Carol steps off the platform at the point where the cord is attached and falls vertically. Hooke's law can be assumed to apply whilst the cord is taut. Model Carol as a particle and assume air resistance to be negligible.
When Carol has fallen \(x \mathrm {~m}\), her speed is \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\).
  1. By considering energy, show that $$5 v ^ { 2 } = 306 x - 4 x ^ { 2 } - 2704 \text { for } x \geqslant 26$$
  2. Why is the expression found in part (a) not true when \(x\) takes values less than 26?
  3. Find the maximum value of \(x\).
    1. Find the distance fallen by Carol when her speed is a maximum.
    2. Hence find Carol's maximum speed.
AQA M2 2015 June Q9
8 marks
9 A uniform rod, \(P Q\), of length \(2 a\), rests with one end, \(P\), on rough horizontal ground and a point \(T\) resting on a rough fixed prism of semicircular cross-section of radius \(a\), as shown in the diagram. The rod is in a vertical plane which is parallel to the prism's cross-section. The coefficient of friction at both \(P\) and \(T\) is \(\mu\).
\includegraphics[max width=\textwidth, alt={}, center]{691c50b4-50b2-4e3a-a7e0-60f8ec35ee3c-20_451_1093_477_475} The rod is on the point of slipping when it is inclined at an angle of \(30 ^ { \circ }\) to the horizontal. Find the value of \(\mu\).
[0pt] [8 marks]
\includegraphics[max width=\textwidth, alt={}, center]{691c50b4-50b2-4e3a-a7e0-60f8ec35ee3c-24_2488_1728_219_141}
AQA M2 2016 June Q1
1 A stone, of mass 0.3 kg , is thrown with a speed of \(8 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) from a point at a height of 5 metres above a horizontal surface.
  1. Calculate the initial kinetic energy of the stone.
    1. Find the kinetic energy of the stone when it hits the surface.
    2. Hence find the speed of the stone when it hits the surface.
    3. State one modelling assumption that you have made.
AQA M2 2016 June Q2
4 marks
2 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, \(\mathbf { v } \mathrm { m } \mathrm { s } ^ { - 1 }\), is given by $$\mathbf { v } = \left( 8 t - t ^ { 4 } \right) \mathbf { i } + 6 \mathrm { e } ^ { - 3 t } \mathbf { j }$$
  1. Find an expression for the acceleration of the particle at time \(t\).
  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. Find the magnitude of \(\mathbf { F }\) when \(t = 1\).
  3. Find the value of \(t\) when \(\mathbf { F }\) acts due south.
  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 \(t\).
    [0pt] [4 marks]
AQA M2 2016 June Q3
3 The diagram shows a uniform lamina \(A B C D E F G H I J K L\).
\includegraphics[max width=\textwidth, alt={}, center]{7c2c50e0-4976-4301-9898-61b2760a2aee-08_474_1378_351_370}
  1. Explain why the centre of mass of the lamina is 35 cm from \(A L\).
  2. Find the distance of the centre of mass from \(A F\).
  3. The lamina is freely suspended from \(A\). Find the angle between \(A B\) and the vertical when the lamina is in equilibrium.
  4. Explain, briefly, how you have used the fact that the lamina is uniform.
AQA M2 2016 June Q4
4 marks
4 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 \(5 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) in a horizontal circle, as shown in the diagram. The angle between \(O P\) and the vertical is \(\theta\).
\includegraphics[max width=\textwidth, alt={}, center]{7c2c50e0-4976-4301-9898-61b2760a2aee-10_517_433_683_790}
  1. Find the tension in the string.
  2. \(\quad\) Find \(\theta\).
  3. Find the radius of the horizontal circle.
    [0pt] [4 marks]
AQA M2 2016 June Q5
4 marks
5 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 \(R O T\) is \(30 ^ { \circ }\), as shown in the diagram.
\includegraphics[max width=\textwidth, alt={}, center]{7c2c50e0-4976-4301-9898-61b2760a2aee-12_766_736_644_651}
  1. Find, in terms of \(g , l\) and \(u\), the speed of the particle at the point \(T\).
  2. Find, in terms of \(g , l , m\) and \(u\), the tension in the string when the particle is at the point \(T\).
  3. Find, in terms of \(g , l , m\) and \(u\), the tension in the string when the particle returns to the point \(R\).
  4. The particle makes complete revolutions. Find, in terms of \(g\) and \(l\), the minimum value of \(u\).
    [0pt] [4 marks]
AQA M2 2016 June Q6
6 marks
6 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 \(\lambda m v\), where \(\lambda\) is a constant.
  1. Show that $$\frac { \mathrm { d } v } { \mathrm {~d} t } = g - \lambda v$$
  2. The initial speed of the stone is \(u\). Find an expression for \(v\) at time \(t\).
    [0pt] [6 marks] \(7 \quad\) 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.
  3. Draw a diagram to show the forces acting on the ladder.
  4. Find \(\tan \theta\) in terms of \(\mu\).
AQA M2 2016 June Q8
8 marks
8 A particle, \(P\), of mass 5 kg is placed at the point \(A\) on a rough plane which is inclined at \(30 ^ { \circ }\) to the horizontal. The points \(Q\) and \(R\) are also on the surface of the inclined plane, with \(Q R = 15\) metres. The point \(A\) is between \(Q\) and \(R\) so that \(A Q = 4\) metres and \(A R = 11\) metres. The three points \(Q , A\) and \(R\) are on a line of greatest slope of the plane.
\includegraphics[max width=\textwidth, alt={}, center]{7c2c50e0-4976-4301-9898-61b2760a2aee-20_391_882_676_587} The particle is attached to two light elastic strings, \(P Q\) and \(P R\).
One of the strings, \(P Q\), has natural length 4 metres and modulus of elasticity 160 N , the other string, \(P R\), 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.
[0pt] [8 marks]
\includegraphics[max width=\textwidth, alt={}]{7c2c50e0-4976-4301-9898-61b2760a2aee-23_2488_1709_219_153}
\section*{DO NOT WRITE ON THIS PAGE ANSWER IN THE SPACES PROVIDED}
AQA M2 2012 January Q7
  1. Show that \(v ^ { 2 } = u ^ { 2 } - 4 a g\).
  2. The ratio of the tensions in the string when the bead is at the two points \(A\) and \(B\) is \(2 : 5\).
    1. Find \(u\) in terms of \(g\) and \(a\).
    2. Find the ratio \(u : v\).
AQA M2 2010 June Q7
  1. Draw a diagram to show the forces acting on the rod.
  2. Find the magnitude of the normal reaction force between the rod and the ground.
    1. Find the normal reaction acting on the rod at \(C\).
    2. Find the friction force acting on the rod at \(C\).
  4. In this position, the rod is on the point of slipping. Calculate the coefficient of friction between the rod and the peg.
    \includegraphics[max width=\textwidth, alt={}]{3ffa0a2b-aa7d-46eb-b92b-3e3ee59f235c-15_2484_1709_223_153}
AQA M2 Q1
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
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
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
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
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
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
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
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
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
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
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 \(g\).