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AQA Further Paper 3 Mechanics 2019 June Q3
3 marks Standard +0.3
3 A disc, of mass \(m\) and radius \(r\), rotates about an axis through its centre, perpendicular to the plane face of the disc. The angular speed of the disc is \(\omega\).
A possible model for the kinetic energy \(E\) of the disc is $$E = k m ^ { a } r ^ { b } \omega ^ { c }$$ where \(a , b\) and \(c\) are constants and \(k\) is a dimensionless constant.
Find the values of \(a , b\) and \(c\).
AQA Further Paper 3 Mechanics 2019 June Q4
8 marks Moderate -0.3
4 In this question use \(g = 10 \mathrm {~m} \mathrm {~s} ^ { - 2 }\) An inelastic string has length 1.2 metres.
One end of the string is attached to a fixed point \(O\).
A sphere, of mass 500 grams, is attached to the other end of the string.
The sphere is held, with the string taut and at an angle of \(20 ^ { \circ }\) to the vertical, touching the chin of a student, as shown in the diagram below. \includegraphics[max width=\textwidth, alt={}, center]{f2470caa-0f73-4ec1-b08f-525c02ed2e67-04_739_511_676_762} The sphere is released from rest.
Assume that the student stays perfectly still once the sphere has been released.
4
  1. Calculate the maximum speed of the sphere.
    4
AQA Further Paper 3 Mechanics 2019 June Q5
11 marks Standard +0.8
5 The triangular region shown below is rotated through \(360 ^ { \circ }\) around the \(x\)-axis, to form a solid cone. \includegraphics[max width=\textwidth, alt={}, center]{f2470caa-0f73-4ec1-b08f-525c02ed2e67-06_328_755_415_644} The coordinates of the vertices of the triangle are \(( 0,0 ) , ( 8,0 )\) and \(( 0,4 )\).
All units are in centimetres. 5
  1. State an assumption that you should make about the cone in order to find the position of its centre of mass. 5
  2. Using integration, prove that the centre of mass of the cone is 2 cm from its plane face.
    5
  3. The cone is placed with its plane face on a rough board. One end of the board is lifted so that the angle between the board and the horizontal is gradually increased. Eventually the cone topples without sliding. 5 (c) (i) Find the angle between the board and the horizontal when the cone topples, giving your answer to the nearest degree. 5 (c) (ii) Find the range of possible values for the coefficient of friction between the cone and the board.
AQA Further Paper 3 Mechanics 2019 June Q6
6 marks Standard +0.3
6 A ball moving on a smooth horizontal surface collides with a fixed vertical wall. Before the collision, the ball moves with speed \(7 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) and at an angle of \(40 ^ { \circ }\) to the wall. After the collision, the ball moves with speed \(5 \mathrm {~ms} ^ { - 1 }\) and at an angle of \(26 ^ { \circ }\) to the wall. Model the ball as a particle.
6
  1. Find the coefficient of restitution between the ball and the wall, giving your answer correct to two significant figures.
    6
  2. Determine whether or not the wall is smooth. Fully justify your answer.
AQA Further Paper 3 Mechanics 2019 June Q7
9 marks Challenging +1.8
7 A particle of mass 2.5 kilograms is attached to one end of a light, inextensible string of length 75 cm . The other end of this string is attached to a point \(A\). The particle is also attached to one end of an elastic string of natural length 30 cm and modulus of elasticity \(\lambda \mathrm { N }\). The other end of this string is attached to a point \(B\), which is 60 cm vertically below \(A\). The particle is set in motion so that it describes a horizontal circle with centre \(B\). The angular speed of the particle is \(8 \mathrm { rad } \mathrm { s } { } ^ { - 1 }\) Find \(\lambda\), giving your answer in terms of \(g\).
AQA Further Paper 3 Mechanics 2019 June Q8
11 marks Challenging +1.8
8 In this question use \(g = 9.8 \mathrm {~m} \mathrm {~s} ^ { - 2 }\) A 'reverse' bungee jump consists of two identical elastic ropes. One end of each elastic rope is attached to either side of the top of a gorge. The other ends are both attached to Hannah, who has mass 84 kg
Hannah is modelled as a particle, as shown in the diagram below. \includegraphics[max width=\textwidth, alt={}, center]{f2470caa-0f73-4ec1-b08f-525c02ed2e67-12_467_844_678_598} The depth of the gorge is 50 metres and the width of the gorge is 40 metres.
Each elastic rope has natural length 30 metres and modulus of elasticity 3150 N
Hannah is released from rest at the centre of the bottom of the gorge.
8
  1. Show that the speed of Hannah when the ropes become slack is \(30 \mathrm {~ms} ^ { - 1 }\) correct to two significant figures.
    8
  2. Determine whether Hannah is moving up or down when the ropes become taut again. [5 marks] \includegraphics[max width=\textwidth, alt={}, center]{f2470caa-0f73-4ec1-b08f-525c02ed2e67-14_2492_1721_217_150} Additional page, if required.
    Write the question numbers in the left-hand margin. Question number Additional page, if required.
    Write the question numbers in the left-hand margin. Question number Additional page, if required.
    Write the question numbers in the left-hand margin. Question number Additional page, if required.
    Write the question numbers in the left-hand margin.
AQA Further Paper 3 Mechanics 2020 June Q1
1 marks Moderate -0.8
1 A rigid rod, \(A B\), has mass 2 kg and length 4 metres.
Two particles of masses 5 kg and 3 kg are fixed to \(A\) and \(B\) respectively to create a composite body, as shown in the diagram. \includegraphics[max width=\textwidth, alt={}, center]{b0d0c552-71cb-4e5a-b545-de8a9052def0-02_120_730_769_653} Find the distance of the centre of mass of the composite body from \(B\). Circle your answer.
1.5 metres
1.6 metres
2.4 metres
2.5 metres
AQA Further Paper 3 Mechanics 2020 June Q2
1 marks Standard +0.3
2 The tension, \(T\) newtons, in a spring is given by \(T = 20 e\), where \(e\) metres is the extension of the spring. Calculate the work done when the extension is increased from 0.2 metres to 0.4 metres. Circle your answer.
[0pt] [1 mark]
0.4 J 0.9 J 1.2 J 1.6 J
AQA Further Paper 3 Mechanics 2020 June Q3
2 marks Easy -1.8
3 The speed, \(v\), of a particle moving in a horizontal circle is given by the formula \(v = r \omega\) where: \(v =\) speed \(r =\) radius \(\omega =\) angular speed.
Show that the dimensions of angular speed are \(T ^ { - 1 }\) [0pt] [2 marks]
AQA Further Paper 3 Mechanics 2020 June Q4
8 marks Standard +0.3
4 A car has mass 1000 kg and travels on a straight horizontal road. The maximum speed of the car on this road is \(48 \mathrm {~ms} ^ { - 1 }\) In a simple model, it is assumed that the car experiences a resistance force that is proportional to its speed. When the car travels at \(20 \mathrm {~m} \mathrm {~s} ^ { - 1 }\), the magnitude of the resistance force is 600 newtons. 4
  1. Show that the maximum power of the car is 69120 W
    4
  2. Find the maximum acceleration of the car when it is travelling at \(25 \mathrm {~ms} ^ { - 1 }\) 4
  3. Find the maximum acceleration of the car when it is travelling at \(3 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) 4
  4. Comment on the validity of the model in the context of your answers to parts (b) and (c).
AQA Further Paper 3 Mechanics 2020 June Q5
17 marks Standard +0.8
5 A ball, of mass 0.3 kg , is moving on a smooth horizontal surface. The ball collides with a smooth fixed vertical wall and rebounds.
Before the ball hits the wall, the ball is moving at \(4 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) at an angle of \(30 ^ { \circ }\) to the wall as shown in the diagram. \includegraphics[max width=\textwidth, alt={}, center]{b0d0c552-71cb-4e5a-b545-de8a9052def0-06_634_268_584_886} The magnitude of the force, \(F\) newtons, exerted on the ball by the wall at time \(t\) seconds is modelled by $$F = k t ^ { 2 } ( 0.1 - t ) ^ { 2 } \quad \text { for } \quad 0 \leq t \leq 0.1$$ where \(k\) is a constant. The ball is in contact with the wall for 0.1 seconds.
\includegraphics[max width=\textwidth, alt={}]{b0d0c552-71cb-4e5a-b545-de8a9052def0-07_2484_1709_219_153}
5 (b) Explain why \(1800000 < k \leq 3600000\) Fully justify your answer.
5 (c) Given that \(k = 2400000\) Find the speed of the ball after the collision with the wall.
[0pt] [4 marks]
AQA Further Paper 3 Mechanics 2020 June Q6
9 marks Standard +0.3
6 A particle moves with constant speed on a circular path of radius 2 metres. The centre of the circle has position vector \(2 \mathbf { j }\) metres.
At time \(t = 0\), the particle is at the origin and is moving in the positive \(\mathbf { i }\) direction.
The particle returns to the origin every 4 seconds.
The unit vectors \(\mathbf { i }\) and \(\mathbf { j }\) are perpendicular.
6
  1. Calculate the angular speed of the particle.
    6
  2. Write down an expression for the position vector of the particle at time \(t\) seconds.
    6
  3. Find an expression for the acceleration of the particle at time \(t\) seconds.
    6
  4. State the magnitude of the acceleration of the particle.
    6
  5. State the time when the acceleration is first directed towards the origin.
AQA Further Paper 3 Mechanics 2020 June Q7
8 marks Standard +0.3
7 In this question use \(g = 9.8 \mathrm {~m} \mathrm {~s} ^ { - 2 }\) A box, of mass 8 kg , is on a rough horizontal surface.
A string attached to the box is used to pull it along the surface.
The string is inclined at an angle of \(40 ^ { \circ }\) above the horizontal.
The tension in the string is 50 newtons.
As the box moves a distance of \(x\) metres, its speed increases from \(2 \mathrm {~ms} ^ { - 1 }\) to \(5 \mathrm {~ms} ^ { - 1 }\) The coefficient of friction between the box and the surface is 0.4
7
  1. By using an energy method, find \(x\).
    7
  2. Describe how the model could be refined to obtain a more realistic value of \(x\) and use an energy argument to explain whether this would increase or decrease the value of \(x\).
AQA Further Paper 3 Mechanics 2020 June Q8
8 marks Challenging +1.8
8 A ladder has length 4 metres and mass 20 kg The ladder rests in equilibrium with one end on a horizontal surface and the ladder resting on the top of a vertical wall. In this position the ladder is on the point of slipping.
The top of the wall is 1.5 metres above the horizontal surface.
The angle between the ladder and the horizontal surface is \(\alpha\), as shown in the diagram. \includegraphics[max width=\textwidth, alt={}, center]{b0d0c552-71cb-4e5a-b545-de8a9052def0-14_362_863_804_593} The coefficient of friction between the ladder and the wall is 0.5
The coefficient of friction between the ladder and the ground is also 0.5
Show that $$\cos \alpha \sin ^ { 2 } \alpha = \frac { 3 } { 10 }$$ stating clearly any assumptions you make. \includegraphics[max width=\textwidth, alt={}, center]{b0d0c552-71cb-4e5a-b545-de8a9052def0-16_2490_1735_219_139}
AQA Further Paper 3 Mechanics 2022 June Q1
1 marks Easy -1.2
1 The graph shows how a force, \(F\) newtons, varies during a 5 second period of time. \includegraphics[max width=\textwidth, alt={}, center]{0afe3ff2-0af5-4aeb-98c5-1346fa803388-02_575_1182_680_429} Calculate the magnitude of the impulse of the force.
Circle your answer.
[0pt] [1 mark]
17.5 N s
25 Ns
35 Ns
70 Ns
AQA Further Paper 3 Mechanics 2022 June Q2
1 marks Easy -1.8
2 A car of mass 1200 kg is travelling at a constant speed of \(18 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) on a straight horizontal road. The car experiences a total resistive force of 240 newtons.
Calculate the power of the car's engine.
Circle your answer.
[0pt] [1 mark]
900 W
4320 W
16000 W
21600 W
AQA Further Paper 3 Mechanics 2022 June Q3
1 marks Easy -1.8
3 Three particles are attached to a light rod, \(A B\), of length 40 cm The particles are attached at \(A , B\) and the midpoint of the rod.
The particle at \(A\) has mass 5 kg
The particle at \(B\) has mass 1 kg
The particle at the midpoint has mass 4 kg
Find the distance of the centre of mass of this system from the midpoint of the rod.
Circle your answer.
[0pt] [1 mark] \(4 \mathrm {~cm} \quad 8 \mathrm {~cm} \quad 12 \mathrm {~cm} \quad 28 \mathrm {~cm}\) Turn over for the next question
AQA Further Paper 3 Mechanics 2022 June Q4
5 marks Standard +0.3
4
  1. State the dimensions of force. 4
  2. The velocity of an object in a circular orbit can be calculated using the formula $$v = G ^ { a } m ^ { b } r ^ { c }$$ where: \(G =\) Universal constant of gravitation in \(\mathrm { Nm } ^ { 2 } \mathrm {~kg} ^ { - 2 }\) \(m =\) Mass of the Earth in kg \(r =\) Radius of the orbit in metres
    Use dimensional analysis to find the values of \(a , b\) and \(c\) [0pt] [4 marks]
AQA Further Paper 3 Mechanics 2022 June Q5
4 marks Standard +0.3
5 A train of mass 10000 kg is travelling at \(0.3 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) when it collides with a buffer. The buffer brings the train to rest. As the buffer brings the train to rest it compresses by 0.2 metres.
When the buffer is compressed by a distance of \(x\) metres it exerts a force of magnitude \(F\) newtons, where $$F = A x + 9000 x ^ { 2 }$$ where \(A\) is a constant. 5
  1. Find, in terms of \(A\), the work done in compressing the buffer by 0.2 metres.
    5
  2. Find the value of \(A\)
AQA Further Paper 3 Mechanics 2022 June Q6
7 marks Standard +0.3
6 A particle, of mass 5 kg , moves on a circular path so that at time \(t\) seconds it has position vector \(\mathbf { r }\) metres, where $$\mathbf { r } = ( 2 \sin 3 t ) \mathbf { i } + ( 2 \cos 3 t ) \mathbf { j }$$ 6
  1. Prove that the velocity of the particle is perpendicular to its position vector.
    6
  2. Prove that the magnitude of the resultant force on the particle is constant.
AQA Further Paper 3 Mechanics 2022 June Q7
11 marks Standard +0.8
7 Two snooker balls, one white and one red, have equal mass. The balls are on a horizontal table \(A B C D\) The white ball is struck so that it moves at a speed of \(2 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) parallel to \(A B\) The white ball hits a stationary red ball.
After the collision, the white ball moves at a speed of \(0.8 \mathrm {~ms} ^ { - 1 }\) and at an angle of \(30 ^ { \circ }\) to \(A B\) After the collision, the red ball moves at a speed \(v \mathrm {~ms} ^ { - 1 }\) and at an angle \(\theta\) to \(C D\) When the collision takes place, the white ball is the same distance from \(A B\) as the distance the red ball is from CD The diagram below shows the table and the velocities of the balls after the collision. \includegraphics[max width=\textwidth, alt={}, center]{0afe3ff2-0af5-4aeb-98c5-1346fa803388-08_595_1370_1121_335} Not to scale After the collision, the white ball hits \(A B\) and the red ball hits \(C D\) Model the balls as particles that do not experience any air resistance.
7
  1. Explain why the two balls hit the sides of the table at the same time.
    7
  2. Show that \(\theta = 17.0 ^ { \circ }\) correct to one decimal place.
    7
  3. \(\quad\) Find \(v\) 7
  4. Determine which ball travels the greater distance after the collision and before hitting the side of the table. Fully justify your answer.
    7
  5. State one possible refinement to the model that you have used. \(8 \quad\) In this question use \(g\) as \(9.8 \mathrm {~ms} ^ { - 2 }\) A rope is used to pull a crate, of mass 60 kg , along a rough horizontal surface.
    The coefficient of friction between the crate and the surface is 0.4 The crate is at rest when the rope starts to pull on it.
    The tension in the rope is 240 N and the rope makes an angle of \(30 ^ { \circ }\) to the horizontal.
    When the crate has moved 5 metres, the rope becomes detached from the crate.
AQA Further Paper 3 Mechanics 2022 June Q8
8 marks Standard +0.8
8
  1. Use an energy method to find the maximum speed of the crate.
    8
  2. Use an energy method to find the total distance travelled by the crate.
    8
  3. A student claims that in reality the crate is unlikely to travel more than 5.3 metres in total. Comment on the validity of this claim. \includegraphics[max width=\textwidth, alt={}, center]{0afe3ff2-0af5-4aeb-98c5-1346fa803388-12_2488_1732_219_139}
AQA Further Paper 3 Mechanics 2022 June Q9
14 marks Challenging +1.2
9 Two blocks have square cross sections. One block has mass 9 kg and its cross section has sides of length 20 cm
The other block has mass 1 kg and its cross section has sides of length 4 cm
The blocks are fixed together to form the composite body shown in Figure 1. \begin{figure}[h]
\captionsetup{labelformat=empty} \caption{Figure 1} \includegraphics[alt={},max width=\textwidth]{0afe3ff2-0af5-4aeb-98c5-1346fa803388-13_570_492_717_776}
\end{figure} 9
  1. Find the distance of the centre of mass of the composite body from \(A F\) [0pt] [2 marks]
    Question 9 continues on the next page 9
  2. A uniform rod has mass 12 kg and length 1 metre. One end of the rod rests against a smooth vertical wall.
    The other end of the rod rests on the composite body at point \(B\) The composite body is on a horizontal surface.
    The coefficient of friction between the composite body and the horizontal surface is 0.3 The angle between the rod and \(A B\) is \(60 ^ { \circ }\) A particle of mass \(m \mathrm {~kg}\) is fixed to the rod at a distance of 75 cm from \(B\) The rod, particle and composite body are shown in Figure 2. \begin{figure}[h]
    \captionsetup{labelformat=empty} \caption{Figure 2} \includegraphics[alt={},max width=\textwidth]{0afe3ff2-0af5-4aeb-98c5-1346fa803388-14_939_1020_1133_511}
    \end{figure} 9 (b) (i) Write down the magnitude of the vertical reaction force acting on the rod at \(B\) in terms of \(m\) and \(g\) [0pt] [1 mark] 9 (b) (ii) Show that the magnitude of the horizontal reaction force acting on the rod at \(B\) is $$\frac { g ( 6 + 0.75 m ) } { \sqrt { 3 } }$$ 9 (b) (iii) Find the maximum value of \(m\) for which the composite body does not slide or topple. Fully justify your answer.
AQA Further Paper 3 Mechanics 2023 June Q1
1 marks Easy -1.8
1 State the dimensions of power.
Circle your answer. \(M L ^ { 2 } T ^ { - 3 }\) \(M L ^ { 3 } T ^ { - 3 }\) \(M L ^ { 3 } T ^ { - 2 }\) \(M L ^ { 2 } T ^ { - 2 }\)
AQA Further Paper 3 Mechanics 2023 June Q2
1 marks Moderate -0.5
2 The force \(( 3 \mathbf { i } + 4 \mathbf { j } ) \mathrm { N }\) acts at the point with coordinates \(( 0,2 )\) The unit vectors \(\mathbf { i }\) and \(\mathbf { j }\) are directed along the \(x\)-axis and the \(y\)-axis respectively.
Calculate the magnitude of the moment of this force about the origin.
Circle your answer.
[0pt] [1 mark]
6 Nm
8 Nm
10 Nm
14 Nm