Questions M2 (1391 questions)

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AQA M2 2007 January 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]{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
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
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
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
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
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
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 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
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
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.
AQA M2 2010 January Q1
1 An inextensible rope is attached to a sledge which is at rest on a horizontal surface. A constant force of magnitude 40 newtons at an angle of \(30 ^ { \circ }\) to the horizontal is applied to the sledge, as shown in the diagram.
\includegraphics[max width=\textwidth, alt={}, center]{06b431ca-d3a8-46d6-b9f8-bac08d3fd51e-2_312_1086_664_466} Calculate the work done by the force as the sledge is moved 5 metres along the surface.
AQA M2 2010 January Q2
2 A piece of modern art is modelled as a uniform lamina and three particles. The diagram shows the lamina, the three particles \(A , B\) and \(C\), and the \(x\) - and \(y\)-axes.
\includegraphics[max width=\textwidth, alt={}, center]{06b431ca-d3a8-46d6-b9f8-bac08d3fd51e-2_875_1004_1414_502} The lamina, which is fixed in the \(x - y\) plane, has mass 10 kg and its centre of mass is at the point (12, 9). The three particles are attached to the lamina.
Particle \(A\) has mass 3 kg and is at the point (15, 6).
Particle \(B\) has mass 1 kg and is at the point ( 7,14 ).
Particle \(C\) has mass 6 kg and is at the point ( 8,7 ).
Find the coordinates of the centre of mass of the piece of modern art.
AQA M2 2010 January Q3
3 A uniform plank, of length 8 metres, has mass 30 kg . The plank is supported in equilibrium in a horizontal position by two smooth supports at the points \(A\) and \(B\), as shown in the diagram. A block, of mass 20 kg , is placed on the plank at point \(A\).
\includegraphics[max width=\textwidth, alt={}, center]{06b431ca-d3a8-46d6-b9f8-bac08d3fd51e-3_193_1216_477_404}
  1. Draw a diagram to show the forces acting on the plank.
  2. Show that the magnitude of the force exerted on the plank by the support at \(B\) is \(19.2 g\) newtons.
  3. Find the magnitude of the force exerted on the plank by the support at \(A\).
  4. Explain how you have used the fact that the plank is uniform in your solution.
AQA M2 2010 January Q4
4 A particle moves so that at time \(t\) seconds its velocity \(\mathbf { v } \mathrm { m } \mathrm { s } ^ { - 1 }\) is given by $$\mathbf { v } = \left( 4 t ^ { 3 } - 12 t + 3 \right) \mathbf { i } + 5 \mathbf { j } + 8 t \mathbf { k }$$
  1. When \(t = 0\), the position vector of the particle is \(( - 5 \mathbf { i } + 6 \mathbf { k } )\) metres. Find the position vector of the particle at time \(t\).
  2. Find the acceleration of the particle at time \(t\).
  3. Find the magnitude of the acceleration of the particle at time \(t\). Do not simplify your answer.
  4. Hence find the time at which the magnitude of the acceleration is a minimum.
  5. The particle is moving under the action of a single variable force \(\mathbf { F }\) newtons. The mass of the particle is 7 kg . Find the minimum magnitude of \(\mathbf { F }\).
AQA M2 2010 January Q5
5 A golf ball, of mass \(m \mathrm {~kg}\), is moving in a straight line across smooth horizontal ground. At time \(t\) seconds, the golf ball has speed \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\). As the golf ball moves, it experiences a resistance force of magnitude \(0.2 m v ^ { \frac { 1 } { 2 } }\) newtons until it comes to rest. No other horizontal force acts on the golf ball. Model the golf ball as a particle.
  1. Show that $$\frac { \mathrm { d } v } { \mathrm {~d} t } = - 0.2 v ^ { \frac { 1 } { 2 } }$$
  2. When \(t = 0\), the speed of the golf ball is \(16 \mathrm {~ms} ^ { - 1 }\). Show that \(v = ( 4 - 0.1 t ) ^ { 2 }\).
  3. Find the value of \(t\) when \(v = 1\).
  4. Find the distance travelled by the golf ball as its speed decreases from \(16 \mathrm {~ms} ^ { - 1 }\) to \(1 \mathrm {~ms} ^ { - 1 }\).
AQA M2 2010 January Q6
6 A particle, of mass 4 kg , is attached to one end of a light inextensible string of length 1.2 metres. The other end of the string is attached to a fixed point \(O\). The particle moves in a horizontal circle at a constant speed. The angle between the string and the vertical is \(\theta\).
\includegraphics[max width=\textwidth, alt={}, center]{06b431ca-d3a8-46d6-b9f8-bac08d3fd51e-4_529_554_1580_737}
  1. Find the radius of the horizontal circle in terms of \(\theta\).
  2. The angular speed of the particle is 5 radians per second. Find \(\theta\).
AQA M2 2010 January Q7
7 A smooth hemisphere, of radius \(a\) and centre \(O\), is fixed with its plane face on a horizontal surface. A particle, of mass \(m\), can move freely on the surface of the hemisphere. The particle is placed at the point \(A\), the highest point of the hemisphere, and is set in motion along the surface with speed \(u\).
  1. While the particle is in contact with the hemisphere at a point \(P , O P\) makes an angle \(\theta\) with the upward vertical.
    \includegraphics[max width=\textwidth, alt={}, center]{06b431ca-d3a8-46d6-b9f8-bac08d3fd51e-5_366_1246_715_395} Show that the speed of the particle at \(P\) is $$\left( u ^ { 2 } + 2 g a [ 1 - \cos \theta ] \right) ^ { \frac { 1 } { 2 } }$$
  2. The particle leaves the surface of the hemisphere when \(\theta = \alpha\). Find \(\cos \alpha\) in terms of \(a , u\) and \(g\).
AQA M2 2010 January Q8
8 A bungee jumper, of mass 49 kg , is attached to one end of a light elastic cord of natural length 22 metres and modulus of elasticity 1078 newtons. The other end of the cord is attached to a horizontal platform, which is at a height of 60 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. Assume that Hooke's Law applies whilst the cord is taut and that air resistance is negligible throughout the motion. When the bungee jumper has fallen \(x\) metres, his speed is \(v \mathrm {~ms} ^ { - 1 }\).
  1. By considering energy, show that, when \(x\) is greater than 22, $$5 v ^ { 2 } = 318 x - 5 x ^ { 2 } - 2420$$
  2. Explain why \(x\) must be greater than 22 for the equation in part (a) to be valid. ( 1 mark)
  3. Find the maximum value of \(x\).
    1. Show that the speed of the bungee jumper is a maximum when \(x = 31.8\).
    2. Hence find the maximum speed of the bungee jumper.
AQA M2 2008 June Q1
1 A particle moves in a straight line and at time \(t\) seconds has velocity \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\), where $$v = 6 t ^ { 2 } + 4 t - 7 , \quad t \geqslant 0$$
  1. Find an expression for the acceleration of the particle at time \(t\).
  2. The mass of the particle is 3 kg . Find the resultant force on the particle when \(t = 4\).
  3. When \(t = 0\), the displacement of the particle from the origin is 5 metres. Find an expression for the displacement of the particle from the origin at time \(t\).
AQA M2 2008 June Q2
2 A uniform plank, of length 6 metres, has mass 40 kg . The plank is held in equilibrium in a horizontal position by two vertical ropes attached to the plank at \(A\) and \(B\), as shown in the diagram.
\includegraphics[max width=\textwidth, alt={}, center]{03994596-21ad-4201-8d64-ba2d7b7e0a77-2_323_1162_1464_440}
  1. Draw a diagram to show the forces acting on the plank.
  2. Show that the tension in the rope attached to the plank at \(B\) is \(21 g \mathrm {~N}\).
  3. Find the tension in the rope that is attached to the plank at \(A\).
  4. State where in your solution you have used the fact that the plank is uniform.
AQA M2 2008 June Q3
3 Three particles are attached to a light rectangular lamina \(O A B C\), which is fixed in a horizontal plane. Take \(O A\) and \(O C\) as the \(x\) - and \(y\)-axes, as shown. Particle \(P\) has mass 1 kg and is attached at the point \(( 25,10 )\).
Particle \(Q\) has mass 4 kg and is attached at the point ( 12,7 ).
Particle \(R\) has mass 5 kg and is attached at the point \(( 4,18 )\).
\includegraphics[max width=\textwidth, alt={}, center]{03994596-21ad-4201-8d64-ba2d7b7e0a77-3_782_1033_703_482} Find the coordinates of the centre of mass of the three particles.
AQA M2 2008 June Q4
4 A van, of mass 1500 kg , has a maximum speed of \(50 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) on a straight horizontal road. When the van travels at a speed of \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\), it experiences a resistance force of magnitude \(40 v\) newtons.
  1. Show that the maximum power of the van is 100000 watts.
  2. The van is travelling along a straight horizontal road. Find the maximum possible acceleration of the van when its speed is \(25 \mathrm {~m} \mathrm {~s} ^ { - 1 }\).
  3. The van starts to climb a hill which is inclined at \(6 ^ { \circ }\) to the horizontal. Find the maximum possible constant speed of the van as it travels in a straight line up the hill.
    (6 marks)
AQA M2 2008 June Q5
5 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 particle's position vector, \(\mathbf { r }\) metres, is given by $$\mathbf { r } = 8 \left( \cos \frac { 1 } { 4 } t \right) \mathbf { i } - 8 \left( \sin \frac { 1 } { 4 } t \right) \mathbf { j }$$
  1. Find an expression for the velocity of the particle at time \(t\).
  2. Show that the speed of the particle is a constant.
  3. Prove that the particle is moving in a circle.
  4. Find the angular speed of the particle.
  5. Find an expression for the acceleration of the particle at time \(t\).
  6. State the magnitude of the acceleration of the particle.
AQA M2 2008 June Q6
6 A car, of mass \(m\), is moving along a straight smooth horizontal road. At time \(t\), the car has speed \(v\). As the car moves, it experiences a resistance force of magnitude \(0.05 m v\). No other horizontal force acts on the car.
  1. Show that $$\frac { \mathrm { d } v } { \mathrm {~d} t } = - 0.05 v$$
  2. When \(t = 0\), the speed of the car is \(20 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). Show that \(v = 20 \mathrm { e } ^ { - 0.05 t }\).
  3. Find the time taken for the speed of the car to reduce to \(10 \mathrm {~ms} ^ { - 1 }\).
AQA M2 2008 June Q7
7 A small bead, of mass \(m\), is suspended from a fixed point \(O\) by a light inextensible string, of length \(a\). The bead is then set into circular motion with the string taut at \(B\), where \(B\) is vertically below \(O\), with a horizontal speed \(u\).
\includegraphics[max width=\textwidth, alt={}, center]{03994596-21ad-4201-8d64-ba2d7b7e0a77-5_451_458_461_760}
  1. Given that the string does not become slack, show that the least value of \(u\) required for the bead to make complete revolutions about \(O\) is \(\sqrt { 5 a g }\).
  2. In the case where \(u = \sqrt { 5 a g }\), find, in terms of \(g\) and \(m\), the tension in the string when the bead is at the point \(C\), which is at the same horizontal level as \(O\), as shown in the diagram.
  3. State one modelling assumption that you have made in your solution.