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AQA M2 2008 January Q6
10 marks Standard +0.3
6 A light elastic string has one end attached to a point \(A\) fixed on a smooth plane inclined at \(30 ^ { \circ }\) to the horizontal. The other end of the string is attached to a particle of mass 6 kg . The elastic string has natural length 4 metres and modulus of elasticity 300 newtons. The particle is pulled down the plane in the direction of the line of greatest slope through \(A\). The particle is released from rest when it is 5.5 metres from \(A\).
\includegraphics[max width=\textwidth, alt={}, center]{1bc18163-b20e-4dc6-bd35-496efec8dc73-4_314_713_1900_660}
  1. Calculate the elastic potential energy of the string when the particle is 5.5 metres from the point \(A\).
  2. Show that the speed of the particle when the string becomes slack is \(3.66 \mathrm {~m} \mathrm {~s} ^ { - 1 }\), correct to three significant figures.
  3. Show that the particle will not reach point \(A\) in the subsequent motion.
AQA M2 2008 January Q7
8 marks Standard +0.3
7 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. The particle is moving in a vertical circle, centre \(O\). When the particle is at \(B\), vertically above \(O\), the string is taut and the particle is moving with speed \(3 \sqrt { a g }\).
\includegraphics[max width=\textwidth, alt={}, center]{1bc18163-b20e-4dc6-bd35-496efec8dc73-5_422_399_497_778}
  1. Find, in terms of \(g\) and \(a\), the speed of the particle at the lowest point, \(A\), of its path.
  2. Find, in terms of \(g\) and \(m\), the tension in the string when the particle is at \(A\).
AQA M2 2008 January Q8
10 marks Standard +0.3
8 A car of mass 600 kg is driven along a straight horizontal road. The resistance to motion of the car is \(k v ^ { 2 }\) newtons, where \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\) is the velocity of the car at time \(t\) seconds and \(k\) is a constant.
  1. When the engine of the car has power 8 kW , show that the equation of motion of the car is $$600 \frac { \mathrm {~d} v } { \mathrm {~d} t } - \frac { 8000 } { v } + k v ^ { 2 } = 0$$
  2. When the velocity of the car is \(20 \mathrm {~m} \mathrm {~s} ^ { - 1 }\), the engine is turned off.
    1. Show that the equation of motion of the car now becomes $$600 \frac { \mathrm {~d} v } { \mathrm {~d} t } = - k v ^ { 2 }$$
    2. Find, in terms of \(k\), the time taken for the velocity of the car to drop to \(10 \mathrm {~ms} ^ { - 1 }\).
AQA M2 2011 January Q1
10 marks Moderate -0.3
1 The velocity of a particle at time \(t\) seconds is \(\mathbf { v } \mathrm { m } \mathrm { s } ^ { - 1 }\), where $$\mathbf { v } = \left( 4 + 3 t ^ { 2 } \right) \mathbf { i } + ( 12 - 8 t ) \mathbf { j }$$
  1. When \(t = 0\), the particle is at the point with position vector \(( 5 \mathbf { i } - 7 \mathbf { j } ) \mathrm { m }\). Find the position vector, \(\mathbf { r }\) metres, of the particle at time \(t\).
  2. Find the acceleration of the particle at time \(t\).
  3. The particle has mass 2 kg . Find the magnitude of the force acting on the particle when \(t = 1\).
AQA M2 2011 January Q2
5 marks Moderate -0.8
2 A particle is placed on a smooth plane which is inclined at an angle of \(20 ^ { \circ }\) to the horizontal. The particle, of mass 4 kg , is released from rest at a point \(A\) and travels down the plane, passing through a point \(B\). The distance \(A B\) is 5 m .
\includegraphics[max width=\textwidth, alt={}, center]{9d039ec3-fd0a-40ae-9afe-7627439081df-04_371_693_500_680}
  1. Find the potential energy lost as the particle moves from point \(A\) to point \(B\).
  2. Hence write down the kinetic energy of the particle when it reaches point \(B\).
  3. Hence find the speed of the particle when it reaches point \(B\).
AQA M2 2011 January Q3
4 marks Moderate -0.8
3 A pump is being used to empty a flooded basement.
In one minute, 400 litres of water are pumped out of the basement.
The water is raised 8 metres and is ejected through a pipe at a speed of \(2 \mathrm {~ms} ^ { - 1 }\).
The mass of 400 litres of water is 400 kg .
  1. Calculate the gain in potential energy of the 400 litres of water.
  2. Calculate the gain in kinetic energy of the 400 litres of water.
  3. Hence calculate the power of the pump, giving your answer in watts.
AQA M2 2011 January Q4
14 marks Standard +0.3
4 A uniform rectangular lamina \(A B C D\) has a mass of 5 kg . The side \(A B\) has length 60 cm and the side \(B C\) has length 30 cm . The points \(P , Q , R\) and \(S\) are the mid-points of the sides, as shown in the diagram below. A uniform triangular lamina \(S R D\), of mass 4 kg , is fixed to the rectangular lamina to form a shop sign. The centre of mass of the triangular lamina \(S R D\) is 10 cm from the side \(A D\) and 5 cm from the side \(D C\).
\includegraphics[max width=\textwidth, alt={}, center]{9d039ec3-fd0a-40ae-9afe-7627439081df-08_613_1086_660_518}
  1. Find the distance of the centre of mass of the shop sign from \(A D\).
  2. Find the distance of the centre of mass of the shop sign from \(A B\).
  3. The shop sign is freely suspended from \(P\). Find the angle between \(A B\) and the horizontal when the shop sign is in equilibrium.
  4. To ensure that the side \(A B\) is horizontal when the shop sign is freely suspended from point \(P\), a particle of mass \(m \mathrm {~kg}\) is attached to the shop sign at point \(B\). Calculate \(m\).
  5. Explain how you have used the fact that the rectangular lamina \(A B C D\) is uniform in your solution to this question.
    (1 mark)
    \includegraphics[max width=\textwidth, alt={}]{9d039ec3-fd0a-40ae-9afe-7627439081df-10_2486_1714_221_153}
    \includegraphics[max width=\textwidth, alt={}]{9d039ec3-fd0a-40ae-9afe-7627439081df-11_2486_1714_221_153}
AQA M2 2011 January Q5
10 marks Moderate -0.8
5
  1. A shiny coin is on a rough horizontal turntable at a distance 0.8 m from its centre. The turntable rotates at a constant angular speed. The coefficient of friction between the shiny coin and the turntable is 0.3 . Find the maximum angular speed, in radians per second, at which the turntable can rotate if the shiny coin is not going to slide.
  2. The turntable is stopped and the shiny coin is removed. An old coin is placed on the turntable at a distance 0.15 m from its centre. The turntable is made to rotate at a constant angular speed of 45 revolutions per minute.
    1. Find the angular speed of the turntable in radians per second.
    2. The old coin remains in the same position on the turntable. Find the least value of the coefficient of friction between the old coin and the turntable needed to prevent the old coin from sliding.
AQA M2 2011 January Q6
9 marks Standard +0.3
6 A light inextensible string, of length \(a\), has one end attached to a fixed point \(O\). A small bead, of mass \(m\), is attached to the other end of the string. The bead is moving in a vertical circle, centre \(O\). When the bead is at \(B\), vertically below \(O\), the string is taut and the bead is moving with speed \(5 v\).
\includegraphics[max width=\textwidth, alt={}, center]{9d039ec3-fd0a-40ae-9afe-7627439081df-14_536_554_502_774}
  1. The speed of the bead at the highest point of its path is \(3 v\). Find \(v\) in terms of \(a\) and \(g\).
  2. Find the ratio of the greatest tension to the least tension in the string, as the bead travels around its circular path.
    \includegraphics[max width=\textwidth, alt={}]{9d039ec3-fd0a-40ae-9afe-7627439081df-14_1261_1709_1446_153}
AQA M2 2011 January Q7
15 marks Standard +0.3
7
  1. An elastic string has natural length \(l\) and modulus of elasticity \(\lambda\). The string is stretched from length \(l\) to length \(l + e\). Show, by integration, that the work done in stretching the string is \(\frac { \lambda e ^ { 2 } } { 2 l }\).
  2. A block, of mass 4 kg , is attached to one end of a light elastic string. The string has natural length 2 m and modulus of elasticity 196 N . The other end of the string is attached to a fixed point \(O\).
    1. A second block, of mass 3 kg , is attached to the 4 kg block and the system hangs in equilibrium, as shown in the diagram.
      \includegraphics[max width=\textwidth, alt={}, center]{9d039ec3-fd0a-40ae-9afe-7627439081df-16_374_291_890_877} Find the extension in the string.
    2. The block of mass 3 kg becomes detached from the 4 kg block and falls to the ground. The 4 kg block now begins to move vertically upwards. Find the extension of the string when the 4 kg block is next at rest.
    3. Find the extension of the string when the speed of the 4 kg block is a maximum.
      (3 marks)
      \includegraphics[max width=\textwidth, alt={}]{9d039ec3-fd0a-40ae-9afe-7627439081df-18_2486_1714_221_153}
      \includegraphics[max width=\textwidth, alt={}]{9d039ec3-fd0a-40ae-9afe-7627439081df-19_2486_1714_221_153}
AQA M2 2011 January Q8
8 marks Moderate -0.3
8 Vicky has mass 65 kg and is skydiving. She steps out of a helicopter and falls vertically. She then waits a short period of time before opening her parachute. The parachute opens at time \(t = 0\) when her speed is \(19.6 \mathrm {~ms} ^ { - 1 }\), and she then experiences an air resistance force of magnitude \(260 v\) newtons, where \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\) is her speed at time \(t\) seconds.
  1. When \(t > 0\) :
    1. show that the resultant downward force acting on Vicky is 65(9.8-4v) newtons
    2. show that \(\frac { \mathrm { d } v } { \mathrm {~d} t } = - 4 ( v - 2.45 )\).
  2. By showing that \(\int \frac { 1 } { v - 2.45 } \mathrm {~d} v = - \int 4 \mathrm {~d} t\), find \(v\) in terms of \(t\).
    \includegraphics[max width=\textwidth, alt={}]{9d039ec3-fd0a-40ae-9afe-7627439081df-21_2349_1707_221_153}
    DO NOT WRITE ON THIS PAGE ANSWER IN THE SPACES PROVIDED DO NOT WRITE ON THIS PAGE ANSWER IN THE SPACES PROVIDED
AQA M2 2012 January Q1
8 marks Moderate -0.8
1 A plane is dropping packets of aid as it flies over a flooded village. The speed of a packet when it leaves the plane is \(60 \mathrm {~ms} ^ { - 1 }\). The packet has mass 25 kg . The packet falls a vertical distance of 34 metres to reach the ground.
  1. Calculate the kinetic energy of the packet when it leaves the plane.
  2. Calculate the potential energy lost by the packet as it falls to the ground.
  3. Assume that the effect of air resistance on the packet as it falls can be neglected.
    1. Find the kinetic energy of the packet when it reaches the ground.
    2. Hence find the speed of the packet when it reaches the ground.
AQA M2 2012 January Q2
10 marks Standard +0.3
2 A particle, of mass 50 kg , moves on a smooth horizontal plane. A single horizontal force $$\left[ \left( 300 t - 60 t ^ { 2 } \right) \mathbf { i } + 100 \mathrm { e } ^ { - 2 t } \mathbf { j } \right] \text { newtons }$$ acts on the particle at time \(t\) seconds.
The vectors \(\mathbf { i }\) and \(\mathbf { j }\) are perpendicular unit vectors.
  1. Find the acceleration of the particle at time \(t\).
  2. When \(t = 0\), the velocity of the particle is \(( 7 \mathbf { i } - 4 \mathbf { j } ) \mathrm { ms } ^ { - 1 }\). Find the velocity of the particle at time \(t\).
  3. Calculate the speed of the particle when \(t = 1\).
AQA M2 2012 January Q3
10 marks Standard +0.3
3 A uniform ladder \(P Q\), of length 8 metres and mass 28 kg , rests in equilibrium with its foot, \(P\), on a rough horizontal floor and its top, \(Q\), leaning against a smooth vertical wall. The vertical plane containing the ladder is perpendicular to the wall and the angle between the ladder and the floor is \(69 ^ { \circ }\). A man, of mass 72 kg , is standing at the point \(C\) on the ladder so that the distance \(P C\) is 6 metres. The man may be modelled as a particle at \(C\).
\includegraphics[max width=\textwidth, alt={}, center]{06c3e260-8167-4616-97d4-0f360a376a0f-3_679_679_685_678}
  1. Draw a diagram to show the forces acting on the ladder.
  2. With the man standing at the point \(C\), the ladder is on the point of slipping.
    1. Show that the magnitude of the reaction between the ladder and the vertical wall is 256 N , correct to three significant figures.
    2. Find the coefficient of friction between the ladder and the horizontal floor.
AQA M2 2012 January Q4
6 marks Moderate -0.3
4 A car travels along a straight horizontal road. When its speed is \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\), the car experiences a resistance force of magnitude \(25 v\) newtons.
  1. The car has a maximum constant speed of \(42 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) on this road. Show that the power being used to propel the car at this speed is 44100 watts.
  2. The car has mass 1500 kg . Find the acceleration of the car when it is travelling at \(15 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) on this road under a power of 44100 watts.
AQA M2 2012 January Q5
6 marks Moderate -0.8
5 A parcel is placed on a flat rough horizontal surface in a van. The van is travelling along a horizontal road. It travels around a bend of radius 34 m at a constant speed. The coefficient of friction between the parcel and the horizontal surface in the van is 0.85 . Model the parcel as a particle travelling around part of a circle of radius 34 m and centre \(O\), as shown in the diagram.
\includegraphics[max width=\textwidth, alt={}, center]{06c3e260-8167-4616-97d4-0f360a376a0f-4_348_700_687_667} Find the greatest speed at which the van can travel around the bend without causing the parcel to slide.
AQA M2 2012 January Q6
10 marks Standard +0.3
6 Alice places a toy, of mass 0.4 kg , on a slope. The toy is set in motion with an initial velocity of \(1 \mathrm {~ms} ^ { - 1 }\) down the slope. The resultant force acting on the toy is \(( 2 - 4 v )\) newtons, where \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\) is the toy's velocity at time \(t\) seconds after it is set in motion.
  1. Show that \(\frac { \mathrm { d } v } { \mathrm {~d} t } = - 10 ( v - 0.5 )\).
  2. By using \(\int \frac { 1 } { v - 0.5 } \mathrm {~d} v = - \int 10 \mathrm {~d} t\), find \(v\) in terms of \(t\).
  3. Find the time taken for the toy's velocity to reduce to \(0.55 \mathrm {~m} \mathrm {~s} ^ { - 1 }\).
    \(7 \quad\) A small bead, of mass \(m\), is suspended from a fixed point \(O\) by a light inextensible string of length \(a\). With the string taut, the bead is at the point \(B\), vertically below \(O\), when it is set into vertical circular motion with an initial horizontal velocity \(u\), as shown in the diagram.
    \includegraphics[max width=\textwidth, alt={}, center]{06c3e260-8167-4616-97d4-0f360a376a0f-5_616_613_520_733} The string does not become slack in the subsequent motion. The velocity of the bead at the point \(A\), where \(A\) is vertically above \(O\), is \(v\).
AQA M2 2012 January Q8
14 marks Standard +0.3
8 An elastic string has one end attached to a point \(O\) fixed on a rough horizontal surface. The other end of the string is attached to a particle of mass 2 kg . The elastic string has natural length 0.8 metres and modulus of elasticity 32 newtons. The particle is pulled so that it is at the point \(A\), on the surface, 3 metres from the point \(O\).
  1. Calculate the elastic potential energy when the particle is at the point \(A\).
  2. The particle is released from rest at the point \(A\) and moves in a straight line towards \(O\). The particle is next at rest at the point \(B\). The distance \(A B\) is 5 metres.
    \includegraphics[max width=\textwidth, alt={}, center]{06c3e260-8167-4616-97d4-0f360a376a0f-6_179_1055_877_497} Find the frictional force acting on the particle as it moves along the surface.
  3. Show that the particle does not remain at rest at the point \(B\).
  4. The particle next comes to rest at a point \(C\) with the string slack. Find the distance \(B C\).
  5. Hence, or otherwise, find the total distance travelled by the particle after it is released from the point \(A\).
AQA M2 2013 January Q1
8 marks Moderate -0.8
1 Tim is playing cricket. He hits a ball at a point \(A\). The speed of the ball immediately after being hit is \(11 \mathrm {~ms} ^ { - 1 }\). The ball strikes a tree at a point \(B\). The height of \(B\) is 5 metres above the height of \(A\).
The ball is to be modelled as a particle of mass 0.16 kg being acted upon only by gravity.
  1. Calculate the initial kinetic energy of the ball.
  2. Calculate the potential energy gained by the ball as it moves from the point \(A\) to the point \(B\).
    1. Find the kinetic energy of the ball immediately before it strikes the tree.
    2. Hence find the speed of the ball immediately before it strikes the tree.
AQA M2 2013 January Q2
11 marks Standard +0.3
2 A particle moves in a horizontal plane. The vectors \(\mathbf { i }\) and \(\mathbf { j }\) are perpendicular unit vectors in the horizontal plane. At time \(t\) seconds, the velocity of the particle, \(\mathbf { v } \mathrm { m } \mathrm { s } ^ { - 1 }\), is given by $$\mathbf { v } = 12 \cos \left( \frac { \pi } { 3 } t \right) \mathbf { i } - 9 t ^ { 2 } \mathbf { j }$$
  1. Find an expression for the acceleration of the particle at time \(t\).
  2. The particle, which has mass 4 kg , moves under the action of a single force, \(\mathbf { F }\) newtons.
    1. Find an expression for the force \(\mathbf { F }\) in terms of \(t\).
    2. Find the magnitude of \(\mathbf { F }\) when \(t = 3\).
  3. When \(t = 3\), the particle is at the point with position vector \(( 4 \mathbf { i } - 2 \mathbf { j } ) \mathrm { m }\). Find the position vector, \(\mathbf { r }\) metres, of the particle at time \(t\).
AQA M2 2013 January Q3
5 marks Moderate -0.3
3 A van, of mass 1500 kg , travels at a constant speed of \(22 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) up a slope inclined at an angle \(\theta\) to the horizontal, where \(\sin \theta = \frac { 1 } { 25 }\). The van experiences a resistance force of 8000 N .
Find the power output of the van's engine, giving your answer in kilowatts.
AQA M2 2013 January Q4
8 marks Moderate -0.3
4 The diagram shows a uniform lamina which is in the shape of two identical rectangles \(A X G H\) and \(Y B C D\) and a square \(X Y E F\), arranged as shown. The length of \(A X\) is 10 cm , the length of \(X Y\) is 10 cm and the length of \(A H\) is 30 cm .
\includegraphics[max width=\textwidth, alt={}, center]{85514b55-3f13-4746-a3ef-747239b64cca-3_1183_1278_513_374}
  1. Explain why the centre of mass of the lamina is 15 cm from \(A H\).
  2. Find the distance of the centre of mass of the lamina from \(A B\).
  3. The lamina is freely suspended from the point \(H\). Find, to the nearest degree, the angle between \(H G\) and the horizontal when the lamina is in equilibrium.
AQA M2 2013 January Q5
7 marks Standard +0.8
5 A particle, of mass 12 kg , is moving along a straight horizontal line. At time \(t\) seconds, the particle has speed \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\). As the particle moves, it experiences a resistance force of magnitude \(4 v ^ { \frac { 1 } { 3 } }\). No other horizontal force acts on the particle. The initial speed of the particle is \(8 \mathrm {~m} \mathrm {~s} ^ { - 1 }\).
  1. Show that $$v = \left( 4 - \frac { 2 } { 9 } t \right) ^ { \frac { 3 } { 2 } }$$
  2. Find the value of \(t\) when the particle comes to rest.
AQA M2 2013 January Q6
8 marks Moderate -0.8
6 A light inextensible string has one end attached to a particle, \(P\), of mass 2 kg . The other end of the string is attached to the fixed point \(A\). The point \(A\) is vertically above the point \(B\). The particle moves at a constant speed in a horizontal circle of radius 0.8 m and centre \(B\). The tension in the string is 34 N . The string is inclined at an angle \(\theta\) to the vertical, as shown in the diagram.
\includegraphics[max width=\textwidth, alt={}, center]{85514b55-3f13-4746-a3ef-747239b64cca-4_760_816_1436_596}
  1. Find the angle \(\theta\).
  2. Find the speed of the particle.
  3. Find the time taken for the particle to make one complete revolution.
AQA M2 2013 January Q7
7 marks Standard +0.3
7 A small ball, of mass 3 kg , is suspended from a fixed point \(O\) by a light inextensible string of length 1.2 m . Initially, the string is taut and the ball is at the point \(P\), vertically below \(O\). The ball is then set into motion with an initial horizontal velocity of \(4 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). The ball moves in a vertical circle, centre \(O\). The point \(A\), on the circle, is such that angle \(A O P\) is \(25 ^ { \circ }\), as shown in the diagram.
\includegraphics[max width=\textwidth, alt={}, center]{85514b55-3f13-4746-a3ef-747239b64cca-5_663_702_660_701}
  1. Find the speed of the ball at the point \(A\).
  2. Find the tension in the string when the ball is at the point \(A\).