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AQA M2 2013 January Q8
13 marks Standard +0.3
8
  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 particle, of mass 5 kg , is attached to one end of a light elastic string. The other end of the string is attached to a fixed point \(O\). The string has natural length 1.6 m and modulus of elasticity 392 N .
    1. Find the extension of the string when the particle hangs in equilibrium.
    2. The particle is pulled down to a point \(A\), which is 2.2 m below the point \(O\). Calculate the elastic potential energy in the string.
    3. The particle is released when it is at rest at the point \(A\). Calculate the distance of the particle from the point \(A\) when its speed first reaches \(0.8 \mathrm {~m} \mathrm {~s} ^ { - 1 }\).
AQA M2 2013 January Q9
8 marks Challenging +1.2
9 A smooth hollow hemisphere, of radius \(a\) and centre \(O\), is fixed so that its rim is in a horizontal plane. A smooth uniform \(\operatorname { rod } A B\), of mass \(m\), is in equilibrium, with one end \(A\) resting on the inside of the hemisphere and the point \(C\) on the rod being in contact with the rim of the hemisphere. The rod, of length \(l\), is inclined at an angle \(\theta\) to the horizontal, as shown in the diagram.
\includegraphics[max width=\textwidth, alt={}, center]{85514b55-3f13-4746-a3ef-747239b64cca-6_453_828_559_591}
  1. Explain why the reaction between the rod and the hemisphere at point \(A\) acts through \(O\).
  2. Draw a diagram to show the forces acting on the rod.
  3. Show that \(l = \frac { 4 a \cos 2 \theta } { \cos \theta }\).
AQA M2 2006 June Q1
12 marks Moderate -0.8
1 A particle moves in a horizontal plane, in which the unit vectors \(\mathbf { i }\) and \(\mathbf { j }\) are directed east and north respectively. At time \(t\) seconds, its position vector, \(\mathbf { r }\) metres, is given by $$\mathbf { r } = \left( 2 t ^ { 3 } - t ^ { 2 } + 6 \right) \mathbf { i } + \left( 8 - 4 t ^ { 3 } + t \right) \mathbf { j }$$
  1. Find an expression for the velocity of the particle at time \(t\).
    1. Find the velocity of the particle when \(t = \frac { 1 } { 3 }\).
    2. State the direction in which the particle is travelling at this time.
  3. Find the acceleration of the particle when \(t = 4\).
  4. The mass of the particle is 6 kg . Find the magnitude of the resultant force on the particle when \(t = 4\).
AQA M2 2006 June Q2
11 marks Moderate -0.3
2 A ball of mass 0.6 kg is thrown vertically upwards from ground level with an initial speed of \(14 \mathrm {~m} \mathrm {~s} ^ { - 1 }\).
  1. Calculate the initial kinetic energy of the ball.
  2. Assuming that no resistance forces act on the ball, use an energy method to find the maximum height reached by the ball.
  3. An experiment is conducted to confirm the maximum height for the ball calculated in part (b). In this experiment the ball rises to a height of only 8 metres.
    1. Find the work done against the air resistance force that acts on the ball as it moves.
    2. Assuming that the air resistance force is constant, find its magnitude.
  4. Explain why it is not realistic to model the air resistance as a constant force.
AQA M2 2006 June Q3
12 marks Moderate -0.3
3 The diagram shows a uniform rod, \(A B\), of mass 10 kg and length 5 metres. The rod is held in equilibrium in a horizontal position, by a support at \(C\) and a light vertical rope attached to \(A\), where \(A C\) is 2 metres.
\includegraphics[max width=\textwidth, alt={}, center]{c02cf013-365b-44e2-8c16-aa8209cbe250-3_237_680_479_648}
  1. Draw and label a diagram to show the forces acting on the rod.
  2. Show that the tension in the rope is 24.5 N .
  3. A package of mass \(m \mathrm {~kg}\) is suspended from \(B\). The tension in the rope has to be doubled to maintain equilibrium.
    1. Find \(m\).
    2. Find the magnitude of the force exerted on the rod by the support.
  4. Explain how you have used the fact that the rod is uniform in your solution.
AQA M2 2006 June Q4
11 marks Standard +0.3
4 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 \(P\) vertically below \(O\). The particle is then set into motion with a horizontal velocity \(U\) so that it moves in a complete vertical circle with centre \(O\). The point \(Q\) on the circle is such that \(\angle P O Q = 60 ^ { \circ }\), as shown in the diagram.
\includegraphics[max width=\textwidth, alt={}, center]{c02cf013-365b-44e2-8c16-aa8209cbe250-3_566_540_1797_751}
  1. Find, in terms of \(g , l\) and \(U\), the speed of the particle at \(Q\).
  2. Find, in terms of \(g , l , m\) and \(U\), the tension in the string when the particle is at \(Q\).
  3. Find, in terms of \(g , l , m\) and \(U\), the tension in the string when the particle returns to \(P\).
    (2 marks)
AQA M2 2006 June Q5
14 marks Standard +0.3
5 The graph shows a model for the resultant horizontal force on a car, which varies as it accelerates from rest for 20 seconds. The mass of the car is 1200 kg .
\includegraphics[max width=\textwidth, alt={}, center]{c02cf013-365b-44e2-8c16-aa8209cbe250-4_373_1203_445_390}
  1. The acceleration of the car at time \(t\) seconds is \(a \mathrm {~m} \mathrm {~s} ^ { - 2 }\). Show that $$a = \frac { 2 } { 3 } + \frac { t } { 20 } , \text { for } 0 \leqslant t \leqslant 20$$
  2. Find an expression for the velocity of the car at time \(t\).
  3. Find the distance travelled by the car in the 20 seconds.
  4. An alternative model assumes that the resultant force increases uniformly from 900 to 2100 newtons during the 20 seconds. Which term in your expression for the velocity would change as a result of this modification? Explain why.
AQA M2 2006 June Q6
7 marks Moderate -0.8
6 A car of mass 1200 kg travels round a roundabout on a horizontal, circular path at a constant speed of \(14 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). The radius of the circle is 50 metres. Assume that there is no resistance to the motion of the car and that the car can be modelled as a particle.
  1. A friction force, directed towards the centre of the roundabout, acts on the car as it moves. Show that the magnitude of this friction force is 4704 N .
  2. The coefficient of friction between the car and the road is \(\mu\). Show that \(\mu \geqslant 0.4\).
AQA M2 2006 June Q7
8 marks Standard +0.3
7 A particle of mass 20 kg moves along a straight horizontal line. At time \(t\) seconds the velocity of the particle is \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\). A resistance force of magnitude \(10 \sqrt { v }\) newtons acts on the particle while it is moving. At time \(t = 0\) the velocity of the particle is \(25 \mathrm {~ms} ^ { - 1 }\).
  1. Show that, at time \(t\) $$v = \left( \frac { 20 - t } { 4 } \right) ^ { 2 }$$
  2. State the value of \(t\) when the particle comes to rest.
AQA M2 2007 June Q1
10 marks Moderate -0.8
1 A hot air balloon moves vertically upwards with a constant velocity. When the balloon is at a height of 30 metres above ground level, a box of mass 5 kg is released from the balloon. After the box is released, it initially moves vertically upwards with speed \(10 \mathrm {~ms} ^ { - 1 }\).
  1. Find the initial kinetic energy of the box.
  2. Show that the kinetic energy of the box when it hits the ground is 1720 J .
  3. Hence find the speed of the box when it hits the ground.
  4. State two modelling assumptions which you have made.
AQA M2 2007 June Q2
9 marks Standard +0.3
2 A uniform lamina is in the shape of a rectangle \(A B C D\) and a square \(E F G H\), as shown in the diagram. The length \(A B\) is 20 cm , the length \(B C\) is 30 cm , the length \(D E\) is 5 cm and the length \(E F\) is 10 cm . The point \(P\) is the midpoint of \(A B\) and the point \(Q\) is the midpoint of \(H G\).
\includegraphics[max width=\textwidth, alt={}, center]{676e753d-1b80-413c-a4b9-21861db8dde5-2_615_1221_1585_429}
  1. Explain why the centre of mass of the lamina lies on \(P Q\).
  2. Find the distance of the centre of mass of the lamina from \(A B\).
  3. The lamina is freely suspended from \(A\). Find, to the nearest degree, the angle between \(A D\) and the vertical when the lamina is in equilibrium.
AQA M2 2007 June Q3
11 marks Moderate -0.3
3 A particle has mass 800 kg . A single force of \(( 2400 \mathbf { i } - 4800 t \mathbf { j } )\) newtons acts on the particle at time \(t\) seconds. No other forces act on the particle.
  1. Find the acceleration of the particle at time \(t\).
  2. At time \(t = 0\), the velocity of the particle is \(( 6 \mathbf { i } + 30 \mathbf { j } ) \mathrm { m } \mathrm { s } ^ { - 1 }\). The velocity of the particle at time \(t\) is \(\mathbf { v } \mathrm { m } \mathrm { s } ^ { - 1 }\). Show that $$\mathbf { v } = ( 6 + 3 t ) \mathbf { i } + \left( 30 - 3 t ^ { 2 } \right) \mathbf { j }$$
  3. Initially, the particle is at the point with position vector \(( 2 \mathbf { i } + 5 \mathbf { j } ) \mathrm { m }\). Find the position vector, \(\mathbf { r }\) metres, of the particle at time \(t\).
AQA M2 2007 June Q4
9 marks Standard +0.3
4 A uniform plank is 10 m long and has mass 15 kg . It is placed on horizontal ground at the edge of a vertical river bank, so that 2 m of the plank is projecting over the edge, as shown in the diagram below.
\includegraphics[max width=\textwidth, alt={}, center]{676e753d-1b80-413c-a4b9-21861db8dde5-3_250_1285_1361_388}
  1. A woman of mass 50 kg stands on the part of the plank which projects over the river. Find the greatest distance from the river bank at which she can safely stand.
  2. The woman wishes to stand safely at the end of the plank which projects over the river. Find the minimum mass which she should place on the other end of the plank so that she can do this.
  3. State how you have used the fact that the plank is uniform in your solution.
  4. State one other modelling assumption which you have made.
AQA M2 2007 June Q5
9 marks Standard +0.3
5 A bead of mass \(m\) moves on a smooth circular ring of radius \(a\) which is fixed in a vertical plane, as shown in the diagram. Its speed at \(A\), the highest point of its path, is \(v\) and its speed at \(B\), the lowest point of its path, is \(7 v\).
\includegraphics[max width=\textwidth, alt={}, center]{676e753d-1b80-413c-a4b9-21861db8dde5-4_419_317_484_842}
  1. Show that \(v = \sqrt { \frac { a g } { 12 } }\).
  2. Find the reaction of the ring on the bead, in terms of \(m\) and \(g\), when the bead is at \(A\).
AQA M2 2007 June Q6
12 marks Standard +0.3
6 An elastic string has one end attached to a point \(O\), fixed on a horizontal table. The other end of the string is attached to a particle of mass 5 kilograms. The elastic string has natural length 2 metres and modulus of elasticity 200 newtons. The particle is pulled so that it is 2.5 metres from the point \(O\) and it is then released from rest on the table.
  1. Calculate the elastic potential energy when the particle is 2.5 m from the point \(O\).
  2. If the table is smooth, show that the speed of the particle when the string becomes slack is \(\sqrt { 5 } \mathrm {~m} \mathrm {~s} ^ { - 1 }\).
  3. The table is, in fact, rough and the coefficient of friction between the particle and the table is 0.4 . Find the speed of the particle when the string becomes slack.
AQA M2 2007 June Q7
6 marks Moderate -0.5
7 A stone of mass \(m\) is moving along the smooth horizontal floor of a tank which is filled with a viscous liquid. At time \(t\), the stone has speed \(v\). As the stone moves, it experiences a resistance force of magnitude \(\lambda m v\), where \(\lambda\) is a constant.
  1. Show that $$\frac { \mathrm { d } v } { \mathrm {~d} t } = - \lambda v$$
  2. The initial speed of the stone is \(U\). Show that $$v = U \mathrm { e } ^ { - \lambda t }$$
AQA M2 2007 June Q8
9 marks Standard +0.3
8 A particle, \(P\), of mass 3 kg is attached to one end of a light inextensible string. The string passes through a smooth fixed ring, \(O\), and a second particle, \(Q\), of mass 5 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 \(4 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) 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]{676e753d-1b80-413c-a4b9-21861db8dde5-5_474_476_1425_774}
  1. Explain why the tension in the string is 49 N .
  2. Find \(\theta\).
  3. Find the radius of the horizontal circle.
AQA M2 2010 June Q1
3 marks Easy -1.2
1 A particle moves along a straight line through the origin. At time \(t\), the displacement, \(s\), of the particle from the origin is given by $$s = 5 t ^ { 2 } + 3 \cos 4 t$$ Find the velocity of the particle at time \(t\).
\includegraphics[max width=\textwidth, alt={}]{3ffa0a2b-aa7d-46eb-b92b-3e3ee59f235c-03_2484_1709_223_153}
AQA M2 2010 June Q2
9 marks Moderate -0.8
2 John is at the top of a cliff, looking out over the sea. He throws a rock, of mass 3 kg , horizontally with a velocity of \(4 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). The rock falls a vertical distance of 51 metres to reach the surface of the sea.
  1. Calculate the kinetic energy of the rock when it is thrown.
  2. Calculate the potential energy lost by the rock when it reaches the surface of the sea.
    1. Find the kinetic energy of the rock when it reaches the surface of the sea.
    2. Hence find the speed of the rock when it reaches the surface of the sea.
  4. State one modelling assumption which has been made.
    \includegraphics[max width=\textwidth, alt={}]{3ffa0a2b-aa7d-46eb-b92b-3e3ee59f235c-05_2484_1709_223_153}
AQA M2 2010 June Q3
4 marks Moderate -0.3
3 A uniform circular lamina, of radius 4 cm and mass 0.4 kg , has a centre \(O\), and \(A B\) is a diameter. To create a medal, a smaller uniform circular lamina, of radius 2 cm and mass 0.1 kg , is attached so that the centre of the smaller lamina is at the point \(A\), as shown in the diagram.
\includegraphics[max width=\textwidth, alt={}, center]{3ffa0a2b-aa7d-46eb-b92b-3e3ee59f235c-06_671_878_513_598}
  1. Explain why the centre of mass of the medal is on the line \(A B\).
  2. Find the distance of the centre of mass of the medal from the point \(B\).
    \includegraphics[max width=\textwidth, alt={}, center]{3ffa0a2b-aa7d-46eb-b92b-3e3ee59f235c-06_1259_1705_1448_155}
    \includegraphics[max width=\textwidth, alt={}]{3ffa0a2b-aa7d-46eb-b92b-3e3ee59f235c-07_2484_1709_223_153}
AQA M2 2010 June Q4
12 marks Standard +0.3
4 A particle has mass 200 kg and moves on a smooth horizontal plane. A single horizontal force, \(\left( 400 \cos \left( \frac { \pi } { 2 } t \right) \mathbf { i } + 600 t ^ { 2 } \mathbf { j } \right)\) newtons, acts on the particle at time \(t\) seconds. The unit vectors \(\mathbf { i }\) and \(\mathbf { j }\) are directed east and north respectively.
  1. Find the acceleration of the particle at time \(t\).
  2. When \(t = 4\), the velocity of the particle is \(( - 3 \mathbf { i } + 56 \mathbf { j } ) \mathrm { m } \mathrm { s } ^ { - 1 }\). Find the velocity of the particle at time \(t\).
  3. Find \(t\) when the particle is moving due west.
  4. Find the speed of the particle when it is moving due west.
    \includegraphics[max width=\textwidth, alt={}]{3ffa0a2b-aa7d-46eb-b92b-3e3ee59f235c-09_2484_1709_223_153}
AQA M2 2010 June Q5
7 marks Standard +0.8
5 A particle is moving along a straight line. At time \(t\), the velocity of the particle is \(v\). The acceleration of the particle throughout the motion is \(- \frac { \lambda } { v ^ { \frac { 1 } { 4 } } }\), where \(\lambda\) is a positive constant. The velocity of the particle is \(u\) when \(t = 0\). Find \(v\) in terms of \(u , \lambda\) and \(t\).
(7 marks)
\includegraphics[max width=\textwidth, alt={}]{3ffa0a2b-aa7d-46eb-b92b-3e3ee59f235c-10_2078_1719_632_150}
\includegraphics[max width=\textwidth, alt={}]{3ffa0a2b-aa7d-46eb-b92b-3e3ee59f235c-11_2484_1709_223_153}
AQA M2 2010 June Q6
13 marks Standard +0.3
6 When a car, of mass 1200 kg , travels at a speed of \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\), it experiences a resistance force of magnitude \(30 v\) newtons. The car has a maximum constant speed of \(48 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) on a straight horizontal road.
  1. Show that the maximum power of the car is 69120 watts.
  2. The car is travelling along a straight horizontal road. Find the maximum possible acceleration of the car when it is travelling at a speed of \(40 \mathrm {~m} \mathrm {~s} ^ { - 1 }\).
  3. The car starts to descend a hill on a straight road which is inclined at an angle of \(3 ^ { \circ }\) to the horizontal. Find the maximum possible constant speed of the car as it travels on this road down the hill. \includegraphics[max width=\textwidth, alt={}, center]{3ffa0a2b-aa7d-46eb-b92b-3e3ee59f235c-13_2484_1709_223_153}
    \(7 \quad\) A uniform rod \(A B\), of length 4 m and mass 6 kg , rests in equilibrium with one end, \(A\), on smooth horizontal ground. The rod rests on a rough horizontal peg at the point \(C\), where \(A C\) is 3 m . The rod is inclined at an angle of \(20 ^ { \circ }\) to the horizontal.
    \includegraphics[max width=\textwidth, alt={}, center]{3ffa0a2b-aa7d-46eb-b92b-3e3ee59f235c-14_422_984_447_529}
AQA M2 2010 June Q8
7 marks Standard +0.3
8 A particle is attached to one end of a light inextensible string of length 3 metres. The other end of the string is attached to a fixed point \(O\). The particle is set into motion horizontally at point \(P\) with speed \(v\), so that it describes part of a vertical circle whose centre is \(O\). The point \(P\) is vertically below \(O\).
\includegraphics[max width=\textwidth, alt={}, center]{3ffa0a2b-aa7d-46eb-b92b-3e3ee59f235c-16_510_334_493_861} The particle first comes momentarily to rest at the point \(Q\), where \(O Q\) makes an angle of \(15 ^ { \circ }\) to the vertical.
  1. Find the value of \(v\).
  2. When the particle is at rest at the point \(Q\), the tension in the string is 22 newtons. Find the mass of the particle.
    \includegraphics[max width=\textwidth, alt={}]{3ffa0a2b-aa7d-46eb-b92b-3e3ee59f235c-17_2484_1709_223_153}
AQA M2 2010 June Q9
8 marks Standard +0.3
9 A particle, of mass 8 kg , is attached to one end of a length of elastic string. The particle is placed on a smooth horizontal surface. The other end of the elastic string is attached to a point \(O\) fixed on the horizontal surface. The elastic string has natural length 1.2 m and modulus of elasticity 192 N .
\includegraphics[max width=\textwidth, alt={}, center]{3ffa0a2b-aa7d-46eb-b92b-3e3ee59f235c-18_165_789_571_630} The particle is set in motion on the horizontal surface so that it moves in a circle, centre \(O\), with constant speed \(3 \mathrm {~ms} ^ { - 1 }\). Find the radius of the circle. \includegraphics[max width=\textwidth, alt={}, center]{3ffa0a2b-aa7d-46eb-b92b-3e3ee59f235c-19_2349_1691_221_153}
\includegraphics[max width=\textwidth, alt={}, center]{3ffa0a2b-aa7d-46eb-b92b-3e3ee59f235c-20_2505_1730_212_139}