Questions — AQA Further Paper 3 Mechanics (58 questions)

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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
AQA Further Paper 3 Mechanics 2023 June Q3
1 marks Standard +0.3
3 A uniform disc has mass 6 kg and diameter 8 cm A uniform rectangular lamina, \(A B C D\), has mass 4 kg , width 8 cm and length 20 cm
The disc is fixed to the lamina to form a composite body as shown in the diagram below. \includegraphics[max width=\textwidth, alt={}, center]{cd0d239b-ab92-4d17-9cb8-45722e2894cb-03_448_881_587_577} The sides \(A B , A D\) and \(C D\) are tangents to the disc.
Calculate the distance of the centre of mass of the composite body from \(A D\) Circle your answer.
4 cm
5.6 cm
6.4 cm
8.8 cm
AQA Further Paper 3 Mechanics 2023 June Q4
2 marks Easy -1.2
4 A car of mass 1400 kg drives around a horizontal circular bend of radius 60 metres.
The car has a constant speed of \(12 \mathrm {~ms} ^ { - 1 }\) on the bend.
Calculate the magnitude of the resultant force acting on the car.
[0pt] [2 marks] \(5 \quad\) A region bounded by the curve with equation \(y = 4 - x ^ { 2 }\), the \(x\)-axis and the \(y\)-axis is shown below. \includegraphics[max width=\textwidth, alt={}, center]{cd0d239b-ab92-4d17-9cb8-45722e2894cb-04_641_380_408_831} The region is rotated through \(360 ^ { \circ }\) around the \(x\)-axis to create a uniform solid.
AQA Further Paper 3 Mechanics 2023 June Q6
12 marks Standard +0.3
6 Nm
8 Nm
10 Nm
14 Nm 3 A uniform disc has mass 6 kg and diameter 8 cm A uniform rectangular lamina, \(A B C D\), has mass 4 kg , width 8 cm and length 20 cm
The disc is fixed to the lamina to form a composite body as shown in the diagram below. \includegraphics[max width=\textwidth, alt={}, center]{cd0d239b-ab92-4d17-9cb8-45722e2894cb-03_448_881_587_577} The sides \(A B , A D\) and \(C D\) are tangents to the disc.
Calculate the distance of the centre of mass of the composite body from \(A D\) Circle your answer.
4 cm
5.6 cm
6.4 cm
8.8 cm 4 A car of mass 1400 kg drives around a horizontal circular bend of radius 60 metres.
The car has a constant speed of \(12 \mathrm {~ms} ^ { - 1 }\) on the bend.
Calculate the magnitude of the resultant force acting on the car.
[0pt] [2 marks] \(5 \quad\) A region bounded by the curve with equation \(y = 4 - x ^ { 2 }\), the \(x\)-axis and the \(y\)-axis is shown below. \includegraphics[max width=\textwidth, alt={}, center]{cd0d239b-ab92-4d17-9cb8-45722e2894cb-04_641_380_408_831} The region is rotated through \(360 ^ { \circ }\) around the \(x\)-axis to create a uniform solid.
5
  1. Show that the distance of the centre of mass of the solid from the circular face is \(\frac { 5 } { 8 }\) [0pt] [5 marks]
    5
  2. The solid is suspended in equilibrium from a point on the edge of the circular face.
    Find the angle between the circular face and the horizontal, giving your answer to the nearest degree.
    6 In this question use \(g = 10 \mathrm {~m} \mathrm {~s} ^ { - 2 }\) A sphere of mass 0.8 kg is attached to one end of a string of length 2 metres.
    The other end of the string is attached to a fixed point \(O\) The sphere is released from rest with the string taut and at an angle of \(30 ^ { \circ }\) to the vertical, as shown in the diagram below. \includegraphics[max width=\textwidth, alt={}, center]{cd0d239b-ab92-4d17-9cb8-45722e2894cb-06_464_218_676_909} 6
    1. Find the speed of the sphere when it is directly below \(O\) 6
    2. State one assumption that you made about the string.
      6
    3. As the sphere moves, the string makes an angle \(\theta\) with the downward vertical. By finding an expression for the tension in the string in terms of \(\theta\), show that the tension is a maximum when the sphere is directly below \(O\) 6
    4. A physics student conducts an experiment and uses a device to measure the tension in the string when the sphere is directly below \(O\) They find that the tension is 9.5 newtons.
      Explain why this result is reasonable, showing any calculations that you make.
AQA Further Paper 3 Mechanics 2023 June Q14
12 marks Moderate -0.3
14 Nm 3 A uniform disc has mass 6 kg and diameter 8 cm A uniform rectangular lamina, \(A B C D\), has mass 4 kg , width 8 cm and length 20 cm
The disc is fixed to the lamina to form a composite body as shown in the diagram below. \includegraphics[max width=\textwidth, alt={}, center]{cd0d239b-ab92-4d17-9cb8-45722e2894cb-03_448_881_587_577} The sides \(A B , A D\) and \(C D\) are tangents to the disc.
Calculate the distance of the centre of mass of the composite body from \(A D\) Circle your answer.
4 cm
5.6 cm
6.4 cm
8.8 cm 4 A car of mass 1400 kg drives around a horizontal circular bend of radius 60 metres.
The car has a constant speed of \(12 \mathrm {~ms} ^ { - 1 }\) on the bend.
Calculate the magnitude of the resultant force acting on the car.
[0pt] [2 marks] \(5 \quad\) A region bounded by the curve with equation \(y = 4 - x ^ { 2 }\), the \(x\)-axis and the \(y\)-axis is shown below. \includegraphics[max width=\textwidth, alt={}, center]{cd0d239b-ab92-4d17-9cb8-45722e2894cb-04_641_380_408_831} The region is rotated through \(360 ^ { \circ }\) around the \(x\)-axis to create a uniform solid.
5
  1. Show that the distance of the centre of mass of the solid from the circular face is \(\frac { 5 } { 8 }\) [0pt] [5 marks]
    5
  2. The solid is suspended in equilibrium from a point on the edge of the circular face.
    Find the angle between the circular face and the horizontal, giving your answer to the nearest degree.
    6 In this question use \(g = 10 \mathrm {~m} \mathrm {~s} ^ { - 2 }\) A sphere of mass 0.8 kg is attached to one end of a string of length 2 metres.
    The other end of the string is attached to a fixed point \(O\) The sphere is released from rest with the string taut and at an angle of \(30 ^ { \circ }\) to the vertical, as shown in the diagram below. \includegraphics[max width=\textwidth, alt={}, center]{cd0d239b-ab92-4d17-9cb8-45722e2894cb-06_464_218_676_909} 6
    1. Find the speed of the sphere when it is directly below \(O\) 6
    2. State one assumption that you made about the string.
      6
    3. As the sphere moves, the string makes an angle \(\theta\) with the downward vertical. By finding an expression for the tension in the string in terms of \(\theta\), show that the tension is a maximum when the sphere is directly below \(O\) 6
    4. A physics student conducts an experiment and uses a device to measure the tension in the string when the sphere is directly below \(O\) They find that the tension is 9.5 newtons.
      Explain why this result is reasonable, showing any calculations that you make.
      7 Two particles, \(A\) and \(B\), are moving on a smooth horizontal surface. A straight line has been marked on the surface and the particles are on opposite sides of the line. Particle \(A\) has mass 2 kg and moves with velocity \(4 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) at an angle of \(30 ^ { \circ }\) to the line. Particle \(B\) has mass 3 kg and moves with velocity \(6 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) at an angle of \(45 ^ { \circ }\) to the line. The particles and their velocities are shown in the diagram below. \includegraphics[max width=\textwidth, alt={}, center]{cd0d239b-ab92-4d17-9cb8-45722e2894cb-08_451_739_858_653} The particles collide when they reach the line and then move together as a single combined particle. 7
    5. Show that the magnitude of the impulse on particle \(A\) during the collision is 7.55 Ns correct to three significant figures.
      7
    6. State the magnitude of the impulse on \(B\) during the collision, giving a reason for your answer. 7
    7. Find the size of the angle between the straight line and the impulse acting on \(B\), giving your answer to the nearest degree. 7
    8. During the collision, one particle crosses the straight line.
      State which particle crosses the line, giving a reason for your answer.
      [0pt] [1 mark] 8 In this question use \(g = 9.8 \mathrm {~m} \mathrm {~s} ^ { - 2 }\) A block has mass 10 kg and is at rest 1 metre from a fixed point \(O\) on a horizontal surface. One end of an elastic string is attached to the block and the other end of the elastic string is attached to the point \(O\) The elastic string has modulus of elasticity 40 newtons and natural length 2 metres.
      The coefficient of friction between the block and the surface is 0.6 A force is applied to the block so that it starts to move towards a vertical wall.
      The block moves on a line that is perpendicular to the wall.
      The force has magnitude 100 newtons and acts at an angle of \(30 ^ { \circ }\) to the horizontal.
      The situation is shown in the diagram below. \includegraphics[max width=\textwidth, alt={}, center]{cd0d239b-ab92-4d17-9cb8-45722e2894cb-10_239_1339_1176_354} 8
    9. Show that the distance that the block has moved, when the forces acting on it are in equilibrium, is 3.9 metres correct to two significant figures.
      [0pt] [4 marks]
      8
    10. State one limitation of the model that you have used. 8
    11. Find the maximum speed of the block.
      8
    12. The vertical wall is 8.7 metres from \(O\) Determine whether the block reaches the wall, showing any calculations that you make. \includegraphics[max width=\textwidth, alt={}, center]{cd0d239b-ab92-4d17-9cb8-45722e2894cb-13_2492_1721_217_150}
AQA Further Paper 3 Mechanics 2021 June Q1
1 marks Easy -1.2
A spring of natural length 50 cm and modulus of elasticity \(\lambda\) newtons has an elastic potential energy of 4 J when compressed by 5 cm. Find the value of \(\lambda\) Circle your answer. [1 mark] 8 16 800 1600
AQA Further Paper 3 Mechanics 2021 June Q2
1 marks Easy -1.8
A force of magnitude 7 N acts at each end of a rod of length 20 cm, forming a couple. The forces act at right angles to the rod, as shown in the diagram below. \includegraphics{figure_2} Find the magnitude of the resultant moment of the couple. Circle your answer. [1 mark] 1.4 N m 2.8 N m 140 N m 280 N m
AQA Further Paper 3 Mechanics 2021 June Q3
3 marks Moderate -0.5
A ball has mass 0.4 kg and is hit by a wooden bat. The speed of the ball just before it is hit by the bat is \(6 \text{ m s}^{-1}\) The velocity of the ball immediately after being hit by the bat is perpendicular to its initial velocity. The speed of the ball just after it is hit by the bat is \(8 \text{ m s}^{-1}\) Show that the impulse on the ball has magnitude 4 N s [3 marks]
AQA Further Paper 3 Mechanics 2021 June Q4
4 marks Standard +0.3
A spring has stiffness \(k\)
  1. Determine the dimensions of \(k\) [1 mark]
  2. One end of the spring is attached to a fixed point. A particle of mass \(m\) kg is attached to the other end of the spring. The particle is set into vertical motion and moves up and down, taking \(t\) seconds to complete one oscillation. A possible model for \(t\) is $$t = pm^a g^b k^c$$ where \(p\) is a dimensionless constant and \(g \text{ m s}^{-2}\) is the acceleration due to gravity. Find the values of \(a\), \(b\) and \(c\) for this model to be dimensionally consistent. [3 marks]
AQA Further Paper 3 Mechanics 2021 June Q5
7 marks Standard +0.8
A uniform lamina has the shape of the region enclosed by the curve \(y = x^2 + 1\) and the lines \(x = 0\), \(x = 4\) and \(y = 0\) The diagram below shows the lamina. \includegraphics{figure_5}
  1. Find the coordinates of the centre of mass of the lamina, giving your answer in exact form. [4 marks]
  2. The lamina is suspended from the point where the curve intersects the line \(x = 4\) and hangs in equilibrium. Find the angle between the vertical and the longest straight edge of the lamina, giving your answer correct to the nearest degree. [3 marks]
AQA Further Paper 3 Mechanics 2021 June Q6
4 marks Standard +0.3
A ball of mass \(m\) kg is held at rest at a height \(h\) metres above a horizontal surface. The ball is released and bounces on the surface. The coefficient of restitution between the ball and the surface is \(e\) Prove that the kinetic energy lost during the first bounce is given by $$mgh(1 - e^2)$$ [4 marks]
AQA Further Paper 3 Mechanics 2021 June Q7
9 marks Challenging +1.8
A light string has length 1.5 metres. A small sphere is attached to one end of the string. The other end of the string is attached to a fixed point O A thin horizontal bar is positioned 0.9 metres directly below O The bar is perpendicular to the plane in which the sphere moves. The sphere is released from rest with the string taut and at an angle \(\alpha\) to the downward vertical through O The string becomes slack when the angle between the two sections of the string is 60° Ben draws the diagram below to show the initial position of the sphere, the bar and the path of the sphere. \includegraphics{figure_7}
  1. State two reasons why Ben's diagram is not a good representation of the situation. [2 marks]
  2. Using your answer to part (a), sketch an improved diagram. [1 mark]
  3. Find \(\alpha\), giving your answer to the nearest degree. [6 marks]
AQA Further Paper 3 Mechanics 2021 June Q8
11 marks Challenging +1.2
In this question use \(g = 9.8 \text{ m s}^{-2}\) A lift is used to raise a crate of mass 250 kg The lift exerts an upward force of magnitude \(P\) newtons on the crate. When the crate is at a height of \(x\) metres above its initial position $$P = k(x + 1)(12 - x) + 2450$$ where \(k\) is a constant. The crate is initially at rest, at the point where \(x = 0\)
  1. Show that the work done by the upward force as the crate rises to a height of 12 metres is given by $$29400 + 360k$$ [3 marks]
  2. The speed of the crate is \(3 \text{ m s}^{-1}\) when it has risen to a height of 12 metres. Find the speed of the crate when it has risen to a height of 15 metres. [5 marks]
  3. Find the height of the crate when its speed becomes zero. [2 marks]
  4. Air resistance has been ignored. Explain why this is reasonable in this context. [1 mark]
AQA Further Paper 3 Mechanics 2021 June Q9
10 marks Challenging +1.8
In this question use \(g = 9.81 \text{ m s}^{-2}\) A conical pendulum is made from an elastic string and a sphere of mass 0.2 kg The string has natural length 1.6 metres and modulus of elasticity 200 N The sphere describes a horizontal circle of radius 0.5 metres at a speed of \(v \text{ m s}^{-1}\) The angle between the elastic string and the vertical is \(\alpha\)
  1. Show that $$62.5 - 200 \sin \alpha = 1.962 \tan \alpha$$ [5 marks]
  2. Use your calculator to find \(\alpha\) [1 mark]
  3. Find the value of \(v\) [4 marks]
AQA Further Paper 3 Mechanics 2024 June Q1
1 marks Easy -1.8
A particle moves in a circular path so that at time \(t\) seconds its position vector, \(\mathbf{r}\) metres, is given by $$\mathbf{r} = 4\sin(2t)\mathbf{i} + 4\cos(2t)\mathbf{j}$$ Find the velocity of the particle, in m s\(^{-1}\), when \(t = 0\) Circle your answer. [1 mark] \(8\mathbf{i}\) \quad \(-8\mathbf{j}\) \quad \(8\mathbf{j}\) \quad \(8\mathbf{i} - 8\mathbf{j}\)