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AQA M2 2011 June Q8
10 marks Standard +0.3
8 A smooth wire is fixed in a vertical plane so that it forms a circle of radius \(a\) metres and centre \(O\). A bead, \(B\), of mass 0.3 kg , is threaded on the wire and is set in motion with a speed \(u \mathrm {~m} \mathrm {~s} ^ { - 1 }\) at the lowest point of its circular path, as shown in the diagram. \includegraphics[max width=\textwidth, alt={}, center]{31ba38f7-38a8-4e4e-96a3-19e819fabfb0-6_364_378_466_845}
  1. Show that, if the bead is going to make complete revolutions around the wire, $$u > 2 \sqrt { a g }$$
  2. At time \(t\) seconds, the angle between \(O B\) and the horizontal is \(\theta\), as shown in the diagram. \includegraphics[max width=\textwidth, alt={}, center]{31ba38f7-38a8-4e4e-96a3-19e819fabfb0-6_330_328_1231_858} It is given that \(u = \sqrt { \frac { 9 } { 2 } a g }\).
    1. Find the reaction of the bead on the wire, giving your answer in terms of \(g\) and \(\theta\).
    2. Find \(\theta\) when this reaction is zero.
AQA M2 2011 June Q9
14 marks Standard +0.3
9 At a theme park, a light elastic rope is used to bring a carriage to rest at the end of a ride. The carriage has mass 200 kg and is travelling at \(8 \mathrm {~ms} ^ { - 1 }\) when the elastic rope is attached to the carriage as it passes over a point \(O\). The other end of the elastic rope is fixed to the point \(O\). The carriage then moves along a horizontal surface until it is brought to rest. The elastic rope is then detached so that the carriage remains at rest. The elastic rope has natural length 6 m and modulus of elasticity 1800 N . The rope, once taut, remains horizontal throughout the motion.
  1. Calculate the elastic potential energy of the rope when the carriage is 10 m from \(O\).
    (3 marks)
  2. A student's simple model assumes that there are no resistance forces acting on the carriage so that it is brought to rest by the elastic rope alone. Find the distance of the carriage from \(O\) when it is brought to rest.
  3. The student improves the model by also including a constant resistance force of 800 N which acts while the carriage is in motion. Find the distance of the carriage from \(O\) when it is brought to rest.
    (8 marks)
AQA M2 2012 June Q1
8 marks Moderate -0.8
1 Alan, of mass 76 kg , performed a ski jump. He took off at the point \(A\) at the end of the ski run with a speed of \(28 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) and landed at the point \(B\). The level of the point \(B\) is 31 metres vertically below the level of the point \(A\), as shown in the diagram. Assume that his weight is the only force that acted on Alan during the jump. \includegraphics[max width=\textwidth, alt={}, center]{088327c1-acd3-486d-b76f-1fe2560ffaff-2_581_914_664_571}
  1. Calculate the kinetic energy of Alan when he was at the point \(A\).
  2. Calculate the potential energy lost by Alan during the jump as he moved from the point \(A\) to the point \(B\).
    1. Find the kinetic energy of Alan when he reached the point \(B\).
    2. Hence find the speed of Alan when he reached the point \(B\).
AQA M2 2012 June Q2
9 marks Moderate -0.3
2 A particle moves in a straight line. At time \(t\) seconds, it has velocity \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\), where $$v = 6 t ^ { 2 } - 2 \mathrm { e } ^ { - 4 t } + 8$$ and \(t \geqslant 0\).
    1. Find an expression for the acceleration of the particle at time \(t\).
    2. Find the acceleration of the particle when \(t = 0.5\).
  2. The particle has mass 4 kg . Find the magnitude of the force acting on the particle when \(t = 0.5\).
  3. 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 2012 June Q3
11 marks Standard +0.3
3 A uniform rectangular lamina \(A B C D\), of mass 1.6 kg , has side \(A B\) of length 12 cm and side \(B C\) of length 8 cm . To create a logo, a uniform circular lamina, of mass 0.4 kg , is attached. The centre of the circular lamina is at the point \(C\), as shown in the diagram. \includegraphics[max width=\textwidth, alt={}, center]{088327c1-acd3-486d-b76f-1fe2560ffaff-3_520_780_593_630}
  1. Find the distance of the centre of mass of the logo:
    1. from the line \(A B\);
    2. from the line \(A D\).
  2. The logo is suspended in equilibrium, with \(A B\) horizontal, by two vertical strings. One string is attached at the point \(A\) and the other string is attached at the point \(B\). Find the tension in each of the two strings.
AQA M2 2012 June Q4
9 marks Moderate -0.3
4 A particle moves on a horizontal plane, in which the unit vectors \(\mathbf { i }\) and \(\mathbf { j }\) are perpendicular. At time \(t\), the particle's position vector, \(\mathbf { r }\), is given by $$\mathbf { r } = 4 \cos 3 t \mathbf { i } - 4 \sin 3 t \mathbf { j }$$
  1. Prove that the particle is moving on a circle, which has its centre at the origin.
  2. Find an expression for the velocity of the particle at time \(t\).
  3. Find an expression for the acceleration of the particle at time \(t\).
  4. The acceleration of the particle can be written as $$\mathbf { a } = k \mathbf { r }$$ where \(k\) is a constant. Find the value of \(k\).
  5. State the direction of the acceleration of the particle.
AQA M2 2012 June Q5
8 marks Moderate -0.3
5 Two particles, \(A\) and \(B\), are connected by a light inextensible string which passes through a hole in a smooth horizontal table. The edges of the hole are also smooth. Particle \(A\), of mass 1.4 kg , moves, on the table, with constant speed in a circle of radius 0.3 m around the hole. Particle \(B\), of mass 2.1 kg , hangs in equilibrium under the table, as shown in the diagram. \includegraphics[max width=\textwidth, alt={}, center]{088327c1-acd3-486d-b76f-1fe2560ffaff-4_684_1022_1176_504}
  1. Find the angular speed of particle \(A\).
  2. Find the speed of particle \(A\).
  3. Find the time taken for particle \(A\) to complete one full circle around the hole.
AQA M2 2012 June Q6
7 marks Standard +0.3
6 Simon, a small child of mass 22 kg , is on a swing. He is swinging freely through an angle of \(18 ^ { \circ }\) on both sides of the vertical. Model Simon as a particle, \(P\), of mass 22 kg , attached to a fixed point, \(Q\), by a light inextensible rope of length 2.4 m . \includegraphics[max width=\textwidth, alt={}, center]{088327c1-acd3-486d-b76f-1fe2560ffaff-5_700_310_466_849}
  1. Find Simon's maximum speed as he swings.
  2. Calculate the tension in the rope when Simon's speed is a maximum.
AQA M2 2012 June Q7
7 marks Standard +0.3
7 A stone, of mass 5 kg , is projected vertically downwards, in a viscous liquid, with an initial speed of \(7 \mathrm {~ms} ^ { - 1 }\). At time \(t\) seconds after it is projected, the stone has speed \(v \mathrm {~ms} ^ { - 1 }\) and it experiences a resistance force of magnitude \(9.8 v\) newtons.
  1. When \(t \geqslant 0\), show that $$\frac { \mathrm { d } v } { \mathrm {~d} t } = - 1.96 ( v - 5 )$$ (2 marks)
  2. Find \(v\) in terms of \(t\).
AQA M2 2012 June Q8
16 marks Standard +0.3
8 Zoë carries out an experiment with a block, which she places on the horizontal surface of an ice rink. She attaches one end of a light elastic string to a fixed point, \(A\), on a vertical wall at the edge of the ice rink at the height of the surface of the ice rink. The block, of mass 0.4 kg , is attached to the other end of the string. The string has natural length 5 m and modulus of elasticity 120 N . The block is modelled as a particle which is placed on the surface of the ice rink at a point \(B\), where \(A B\) is perpendicular to the wall and of length 5.5 m . \includegraphics[max width=\textwidth, alt={}, center]{088327c1-acd3-486d-b76f-1fe2560ffaff-6_499_1429_813_333} The block is set into motion at the point \(B\) with speed \(9 \mathrm {~ms} ^ { - 1 }\) directly towards the point \(A\). The string remains horizontal throughout the motion.
  1. Initially, Zoë assumes that the surface of the ice rink is smooth. Using this assumption, find the speed of the block when it reaches the point \(A\).
  2. Zoë now assumes that friction acts on the block. The coefficient of friction between the block and the surface of the ice rink is \(\mu\).
    1. Find, in terms of \(g\) and \(\mu\), the speed of the block when it reaches the point \(A\).
    2. The block rebounds from the wall in the direction of the point \(B\). The speed of the block immediately after the rebound is half of the speed with which it hit the wall. Find \(\mu\) if the block comes to rest just as it reaches the point \(B\).
AQA M2 2013 June Q1
6 marks Easy -1.2
1 A particle, of mass 3 kg , moves along a straight line. At time \(t\) seconds, the displacement, \(s\) metres, of the particle from the origin is given by $$s = 8 t ^ { 3 } + 15$$
  1. Find the velocity of the particle at time \(t\).
  2. Find the magnitude of the resultant force acting on the particle when \(t = 2\).
AQA M2 2013 June Q2
8 marks Moderate -0.8
2 Carol, a circus performer, is on a swing. She jumps off the swing and lands in a safety net. When Carol leaves the swing, she has a speed of \(7 \mathrm {~ms} ^ { - 1 }\) and she is at a height of 8 metres above the safety net. Carol is to be modelled as a particle of mass 52 kg being acted upon only by gravity.
  1. Find the kinetic energy of Carol when she leaves the swing.
  2. Show that the kinetic energy of Carol when she hits the net is 5350 J , correct to three significant figures.
  3. Find the speed of Carol when she hits the net.
AQA M2 2013 June Q3
8 marks Standard +0.3
3 A particle, of mass 10 kg , moves on a smooth horizontal plane. At time \(t\) seconds, the acceleration of the particle is given by $$\left\{ \left( 40 t + 3 t ^ { 2 } \right) \mathbf { i } + 20 \mathrm { e } ^ { - 4 t } \mathbf { j } \right\} \mathrm { m } \mathrm {~s} ^ { - 2 }$$ where the vectors \(\mathbf { i }\) and \(\mathbf { j }\) are perpendicular unit vectors.
  1. At time \(t = 1\), the velocity of the particle is \(\left( 6 \mathbf { i } - 5 \mathrm { e } ^ { - 4 } \mathbf { j } \right) \mathrm { m } \mathrm { s } ^ { - 1 }\). Find the velocity of the particle at time \(t\).
  2. Calculate the initial speed of the particle.
AQA M2 2013 June Q4
12 marks Moderate -0.3
4 A uniform plank \(A B\), of length 6 m , has mass 25 kg . It is supported in equilibrium in a horizontal position by two vertical inextensible ropes. One of the ropes is attached to the plank at the point \(P\) and the other rope is attached to the plank at the point \(Q\), where \(A P = 1 \mathrm {~m}\) and \(Q B = 0.8 \mathrm {~m}\), as shown in the diagram. \includegraphics[max width=\textwidth, alt={}, center]{5b1c9e8d-459a-474c-bd29-6dadff40de14-2_227_1187_2252_424}
    1. Find the tension in each rope.
    2. State how you have used the fact that the plank is uniform in your solution. (1 mark)
  2. A particle of mass \(m \mathrm {~kg}\) is attached to the plank at point \(B\), and the tension in each rope is now the same. Find \(m\).
AQA M2 2013 June Q5
4 marks Standard +0.3
5 Tom is travelling on a train which is moving at a constant speed of \(15 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) on a horizontal track. Tom has placed his mobile phone on a rough horizontal table. The coefficient of friction between the phone and the table is 0.2 . The train moves round a bend of constant radius. The phone does not slide as the train travels round the bend. Model the phone as a particle moving round part of a circle, with centre \(O\) and radius \(r\) metres. Find the least possible value of \(r\).
AQA M2 2013 June Q6
8 marks Standard +0.3
6 A car accelerates from rest along a straight horizontal road. The car's engine produces a constant horizontal force of magnitude 4000 N .
At time \(t\) seconds, the speed of the car is \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\), and a resistance force of magnitude \(40 v\) newtons acts upon the car. The mass of the car is 1600 kg .
  1. Show that \(\frac { \mathrm { d } v } { \mathrm {~d} t } = \frac { 100 - v } { 40 }\).
  2. Find the velocity of the car at time \(t\).
AQA M2 2013 June Q7
6 marks Moderate -0.3
7 A train, of mass 22 tonnes, moves along a straight horizontal track. A constant resistance force of 5000 N acts on the train. The power output of the engine of the train is 240 kW . Find the acceleration of the train when its speed is \(20 \mathrm {~ms} ^ { - 1 }\).
AQA M2 2013 June Q8
9 marks Standard +0.3
8 A bead, of mass \(m\), moves on a smooth circular ring, of radius \(a\) and centre \(O\), which is fixed in a vertical plane. At \(P\), the highest point on the ring, the speed of the bead is \(2 u\); at \(Q\), the lowest point on the ring, the speed of the bead is \(5 u\).
  1. Show that \(u = \sqrt { \frac { 4 a g } { 21 } }\).
    (4 marks)
  2. \(\quad S\) is a point on the ring so that angle \(P O S\) is \(60 ^ { \circ }\), as shown in the diagram. \includegraphics[max width=\textwidth, alt={}, center]{5b1c9e8d-459a-474c-bd29-6dadff40de14-4_600_540_657_760} Find, in terms of \(m\) and \(g\), the magnitude of the reaction of the ring on the bead when the bead is at \(S\).
AQA M2 2013 June Q9
14 marks Challenging +1.2
9 Two particles, \(A\) and \(B\), are connected by a light elastic string that passes through a hole at a point \(O\) in a rough horizontal table. The edges of the hole are smooth. Particle \(A\) has a mass of 8 kg and particle \(B\) has a mass of 3 kg . The elastic string has natural length 3 metres and modulus of elasticity 60 newtons.
Initially, particle \(A\) is held 3.5 metres from the point \(O\) on the surface of the table and particle \(B\) is held at a point 2 metres vertically below \(O\). The coefficient of friction between the table and particle \(A\) is 0.4 .
The two particles are released from rest.
    1. Show that initially particle \(A\) moves towards the hole in the table.
    2. Show that initially particle \(B\) also moves towards the hole in the table.
  2. Calculate the initial elastic potential energy in the string.
  3. Particle \(A\) comes permanently to rest when it has moved 0.46 metres, at which time particle \(B\) is still moving upwards. Calculate the distance that particle \(B\) has moved when it is at rest for the first time.
AQA M2 2014 June Q1
8 marks Moderate -0.8
1 An eagle has caught a salmon of mass 3 kg to take to its nest. When the eagle is flying with speed \(8 \mathrm {~m} \mathrm {~s} ^ { - 1 }\), 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. Calculate the potential energy lost by the salmon as it falls to the sea.
    1. Find the kinetic energy of the salmon when it reaches the sea.
    2. Hence find the speed of the salmon when it reaches the sea.
      \includegraphics[max width=\textwidth, alt={}]{8ca9234e-1a01-4e35-8b1b-17c6039cf8d7-02_1291_1709_1416_153}
      \(2 \quad\) A particle has mass 6 kg . A single force \(\left( 24 \mathrm { e } ^ { - 2 t } \mathbf { i } - 12 t ^ { 3 } \mathbf { j } \right)\) newtons acts on the particle at time \(t\) seconds. No other forces act on the particle.
  4. Find the acceleration of the particle at time \(t\).
  5. At time \(t = 0\), the velocity of the particle is \(( - 7 \mathbf { i } - 4 \mathbf { j } ) \mathrm { ms } ^ { - 1 }\). Find the velocity of the particle at time \(t\).
  6. Find the speed of the particle when \(t = 0.5\).
AQA M2 2014 June Q3
5 marks Moderate -0.8
3 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[max width=\textwidth, alt={}, center]{8ca9234e-1a01-4e35-8b1b-17c6039cf8d7-06_597_960_550_532} 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.
[0pt] [5 marks]
AQA M2 2014 June Q4
9 marks Moderate -0.8
4 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 \(35 ^ { \circ }\) to the vertical. The centre of this circle is vertically below \(O\), as shown in the diagram. \includegraphics[max width=\textwidth, alt={}, center]{8ca9234e-1a01-4e35-8b1b-17c6039cf8d7-08_808_600_520_721} 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. Find the tension in the string.
  3. Find the radius of the horizontal circle.
AQA M2 2014 June Q5
7 marks Standard +0.3
5 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 \(7 \sqrt { a g }\), as shown in the diagram. \includegraphics[max width=\textwidth, alt={}, center]{8ca9234e-1a01-4e35-8b1b-17c6039cf8d7-10_887_812_525_628}
  1. Find, in terms of \(g\) and \(a\), the speed of the particle at the point \(Q\).
  2. Find, in terms of \(g\) and \(m\), the tension in the string when the particle is at \(Q\).
AQA M2 2014 June Q6
13 marks Standard +0.3
6 A puck, of mass \(m \mathrm {~kg}\), is moving in a straight line across smooth horizontal ice. At time \(t\) seconds, the puck has speed \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\). As the puck moves, it experiences an air resistance force of magnitude \(0.3 m v ^ { \frac { 1 } { 3 } }\) newtons, until it comes to rest. No other horizontal forces act on the puck. When \(t = 0\), the speed of the puck is \(8 \mathrm {~m} \mathrm {~s} ^ { - 1 }\).
Model the puck as a particle.
  1. Show that $$v = ( 4 - 0.2 t ) ^ { \frac { 3 } { 2 } }$$
  2. Find the value of \(t\) when the puck comes to rest.
  3. Find the distance travelled by the puck as its speed decreases from \(8 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) to zero.
    \includegraphics[max width=\textwidth, alt={}]{8ca9234e-1a01-4e35-8b1b-17c6039cf8d7-12_1479_1709_1228_153}
AQA M2 2014 June Q7
8 marks Standard +0.3
7 A uniform ladder \(A B\), 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 \(A C\) is 5 metres. The vertical plane containing the ladder is perpendicular to the wall, and the angle between the ladder and the ground is \(60 ^ { \circ }\). 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 \(A D\) is 4 metres. The ladder is on the point of slipping. \includegraphics[max width=\textwidth, alt={}, center]{8ca9234e-1a01-4e35-8b1b-17c6039cf8d7-14_864_808_758_616}
  1. Draw a diagram to show the forces acting on the ladder.
  2. Find the coefficient of friction between the ladder and the horizontal ground.
    [0pt] [6 marks]
    \includegraphics[max width=\textwidth, alt={}]{8ca9234e-1a01-4e35-8b1b-17c6039cf8d7-15_2484_1707_221_153}