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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}
AQA M2 2014 June Q8
15 marks Standard +0.8
8 An elastic string has natural length 1.5 metres and modulus of elasticity 120 newtons. One end of the string is attached to a fixed point, \(A\), on a rough plane inclined at \(20 ^ { \circ }\) to the horizontal. The other end of the elastic string is attached to a particle of mass 4 kg . The coefficient of friction between the particle and the plane is 0.8 . The three points, \(A , B\) and \(C\), lie on a line of greatest slope.
The point \(C\) is \(x\) metres from \(A\), as shown in the diagram. The particle is released from rest at \(C\) and moves up the plane. \includegraphics[max width=\textwidth, alt={}, center]{8ca9234e-1a01-4e35-8b1b-17c6039cf8d7-16_250_615_703_717}
  1. Show that, as the particle moves up the plane, the frictional force acting on the particle is 29.5 N , correct to three significant figures.
  2. The particle comes to rest for an instant at \(B\), which is 2 metres from \(A\). The particle then starts to move back towards \(A\).
    1. Find \(x\).
    2. Find the acceleration of the particle as it starts to move back towards \(A\).
      \includegraphics[max width=\textwidth, alt={}]{8ca9234e-1a01-4e35-8b1b-17c6039cf8d7-17_2484_1707_221_153}
AQA M2 2015 June Q1
10 marks Standard +0.3
1 A particle, of mass 4 kg , moves in a horizontal plane under the action of a single force, \(\mathbf { F }\) newtons. The unit vectors \(\mathbf { i }\) and \(\mathbf { j }\) are in the horizontal plane, perpendicular to each other. At time \(t\) seconds, the velocity of the particle, \(\mathbf { v } \mathrm { m } \mathrm { s } ^ { - 1 }\), is given by $$\mathbf { v } = 4 \cos 2 t \mathbf { i } + 3 \sin t \mathbf { j }$$
    1. Find an expression for the force, \(\mathbf { F }\), acting on the particle at time \(t\) seconds.
    2. Find the magnitude of \(\mathbf { F }\) when \(t = \pi\).
  2. When \(t = 0\), the particle is at the point with position vector \(( 2 \mathbf { i } - 14 \mathbf { j } )\) metres. Find the position vector, \(\mathbf { r }\) metres, of the particle at time \(t\) seconds.
    [0pt] [5 marks]
    \includegraphics[max width=\textwidth, alt={}]{691c50b4-50b2-4e3a-a7e0-60f8ec35ee3c-02_1346_1717_1361_150}
AQA M2 2015 June Q2
4 marks Moderate -0.8
2 A uniform rod \(A B\), of mass 4 kg and length 6 metres, has three masses attached to it. A 3 kg mass is attached at the end \(A\) and a 5 kg mass is attached at the end \(B\). An 8 kg mass is attached at a point \(C\) on the rod. Find the distance \(A C\) if the centre of mass of the system is 4.3 m from point \(A\).
[0pt] [4 marks]
AQA M2 2015 June Q3
9 marks Standard +0.3
3 A diagram shows a children's slide, \(P Q R\). \includegraphics[max width=\textwidth, alt={}, center]{691c50b4-50b2-4e3a-a7e0-60f8ec35ee3c-06_352_640_338_699} Simon, a child of mass 32 kg , uses the slide, starting from rest at \(P\). The curved section of the slide, \(P Q\), is one sixth of a circle of radius 4 metres so that the child is travelling horizontally at point \(Q\). The centre of this circle is at point \(O\), which is vertically above point \(Q\). The section \(Q R\) is horizontal and of length 5 metres. Assume that air resistance may be ignored.
  1. Assume that the two sections of the slide, \(P Q\) and \(Q R\), are both smooth.
    1. Find the kinetic energy of Simon when he reaches the point \(R\).
    2. Hence find the speed of Simon when he reaches the point \(R\).
  2. In fact, the section \(Q R\) is rough. Assume that the section \(P Q\) is smooth.
    Find the coefficient of friction between Simon and the section \(Q R\) if Simon comes to rest at the point \(R\).
    [0pt] [4 marks]
    \includegraphics[max width=\textwidth, alt={}]{691c50b4-50b2-4e3a-a7e0-60f8ec35ee3c-06_923_1707_1784_153}
AQA M2 2015 June Q4
10 marks Standard +0.3
4 A particle, \(P\), of mass 5 kg is attached to two light inextensible strings, \(A P\) and \(B P\). The other ends of the strings are attached to the fixed points \(A\) and \(B\). The point \(A\) is vertically above the point \(B\). The particle moves at a constant speed, \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\), in a horizontal circle of radius 0.6 metres with centre \(B\). The string \(A P\) is inclined at \(20 ^ { \circ }\) to the vertical, as shown in the diagram. Both strings are taut when the particle is moving. \includegraphics[max width=\textwidth, alt={}, center]{691c50b4-50b2-4e3a-a7e0-60f8ec35ee3c-08_835_568_568_719}
  1. Find the tension in the string \(A P\).
  2. The speed of the particle is \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\). Show that the tension, \(T _ { B P }\), in the string \(B P\) is given by $$T _ { B P } = \frac { 25 } { 3 } v ^ { 2 } - 5 g \tan 20 ^ { \circ }$$
  3. Find \(v\) when the tensions in the two strings are equal.
AQA M2 2015 June Q5
6 marks Standard +0.3
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 Standard +0.3
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
9 marks Standard +0.3
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
10 marks Standard +0.3
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 Challenging +1.8
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
8 marks Moderate -0.8
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
13 marks Moderate -0.3
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
9 marks Moderate -0.3
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
8 marks Standard +0.3
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
12 marks Standard +0.3
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
8 marks Standard +0.3
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 Challenging +1.8
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 M3 Q5
Moderate -0.3
5 A football is kicked from a point \(O\) on a horizontal football ground with a velocity of \(20 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) at an angle of elevation of \(30 ^ { \circ }\). During the motion, the horizontal and upward vertical displacements of the football from \(O\) are \(x\) metres and \(y\) metres respectively.
  1. Show that \(x\) and \(y\) satisfy the equation $$y = x \tan 30 ^ { \circ } - \frac { g x ^ { 2 } } { 800 \cos ^ { 2 } 30 ^ { \circ } }$$
  2. On its downward flight the ball hits the horizontal crossbar of the goal at a point which is 2.5 m above the ground. Using the equation given in part (a), find the horizontal distance from \(O\) to the goal.
    (4 marks) \includegraphics[max width=\textwidth, alt={}, center]{fc5bfc4b-68bb-4a23-874b-87e9558dc990-04_330_1411_1902_303}
  3. State two modelling assumptions that you have made.