Questions Further Paper 3 Mechanics (58 questions)

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AQA Further Paper 3 Mechanics Specimen Q1
1 marks Easy -1.8
1 A ball of mass 0.2 kg is travelling horizontally at \(7 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) when it hits a vertical wall.
It rebounds horizontally at \(5 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) Find the magnitude of the impulse exerted on the ball by the wall.
Circle your answer.
[0pt] [1 mark]
0.4 N s
1.4 N s
AQA Further Paper 3 Mechanics Specimen Q2
1 marks Easy -1.2
2 Ns
2.4 N s 2 In this question
\(a\)represents acceleration,
\(T\)represents time,
\(l\)represents length,
\(m\)represents mass,
\(v\)represents velocity,
\(F\)represents force.
One of these formulae is dimensionally consistent.
Circle your answer.
[0pt] [1 mark] $$T = 2 \pi \sqrt { \frac { a } { l } } \quad v ^ { 2 } = \frac { 2 a l } { T } \quad F l = m v ^ { 2 } \quad F T = m \sqrt { a }$$ Turn over for the next question
AQA Further Paper 3 Mechanics Specimen Q3
6 marks Standard +0.3
3 In this question use \(g = 9.8 \mathrm {~m} \mathrm {~s} ^ { - 2 }\).
A composite body consists of a uniform rod, \(A B\), and a particle.
The rod has length 4 metres and mass 22.5 kilograms.
The particle, \(P\), has mass 20 kilograms and is placed on the rod at a distance of 0.3 metres from \(B\), as shown in the diagram. \includegraphics[max width=\textwidth, alt={}, center]{4fdb2637-6368-422c-99da-85b80efe31c5-04_163_1323_767_402} 3
  1. Find the distance of the centre of mass of the body from \(A\). 3
  2. The body rests in equilibrium in a horizontal position on two supports, \(C\) and \(D\).
    The support \(C\) is 0.5 metres from \(A\) and the support \(D\) is 1 metre from \(B\). Find the magnitudes of the forces exerted on the body by the supports.
    [0pt] [4 marks]
AQA Further Paper 3 Mechanics Specimen Q4
6 marks Moderate -0.3
4 Two discs, \(A\) and \(B\), have equal radii and masses 0.8 kg and 0.4 kg respectively. The discs are placed on a horizontal surface. The discs are set in motion when they are 3 metres apart, so that they move directly towards each other, each travelling at a speed of \(6 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). The discs collide directly with each other. After the collision \(A\) moves in the opposite direction with a speed of \(1.2 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) The coefficient of restitution between the two discs is \(e\). 4
  1. Assuming that the surface is smooth, show that \(e = 0.8\) 4
  2. Describe one way in which the model you have used could be refined. Turn over for the next question
AQA Further Paper 3 Mechanics Specimen Q5
6 marks Moderate -0.5
5 In this question use \(\boldsymbol { g } = 9.8 \mathbf { m ~ s } ^ { \mathbf { - 2 } }\).
A conical pendulum consists of a string of length 60 cm and a particle of mass 400 g . The string is at an angle of \(30 ^ { \circ }\) to the vertical, as shown in the diagram. \includegraphics[max width=\textwidth, alt={}, center]{4fdb2637-6368-422c-99da-85b80efe31c5-08_501_606_644_854} 5
  1. Show that the tension in the string is 4.5 N . 5
  2. Find the angular speed of the particle.
    [0pt] [3 marks]
    5
  3. State two assumptions that you have made about the string.
AQA Further Paper 3 Mechanics Specimen Q6
7 marks Challenging +1.8
6 A uniform solid is formed by rotating the region enclosed by the positive \(x\)-axis, the line \(x = 2\) and the curve \(y = \frac { 1 } { 2 } x ^ { 2 }\) through \(360 ^ { \circ }\) around the \(x\)-axis. 6
  1. Find the centre of mass of this solid.
    6
  2. The solid is placed with its plane face on a rough inclined plane and does not slide. The angle between the inclined plane and the horizontal is gradually increased. When the angle between the inclined plane and the horizontal is \(\alpha\), the solid is on the point of toppling. Find \(\alpha\), giving your answer to the nearest \(0.1 ^ { \circ }\)
AQA Further Paper 3 Mechanics Specimen Q7
5 marks Standard +0.3
7 In this question use \(g = 9.8 \mathrm {~m} \mathrm {~s} ^ { - 2 }\).
When a car, of mass 1200 kg , travels at a speed of \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\) it experiences a total resistive force which can be modelled as being of magnitude \(36 v\) newtons.
The maximum power of the car is 90 kilowatts.
The car starts to descend a hill, inclined at \(5.2 ^ { \circ }\) to the horizontal, along a straight road.
Find the maximum speed of the car down this hill.
[0pt] [5 marks]
AQA Further Paper 3 Mechanics Specimen Q8
8 marks Challenging +1.8
8 The diagram shows part of a water park slide, \(A B C\).
The slide is in the shape of two circular arcs, \(A B\) and \(B C\), each of radius \(r\).
The point \(A\) is at a height of \(\frac { r } { 4 }\) above \(B\).
The circular \(\operatorname { arc } B C\) has centre \(O\) and \(B\) is vertically above \(O\).
These points are joined as shown in the diagram. \includegraphics[max width=\textwidth, alt={}, center]{4fdb2637-6368-422c-99da-85b80efe31c5-12_590_1173_756_443} A child starts from rest at \(A\), moves along the slide past the point \(B\) and then loses contact with the slide at a point \(D\). The angle between the vertical, \(O B\), and \(O D\) is \(\theta\) Assume that the slide is smooth. 8
  1. Show that the speed \(v\) of the child at \(D\) is given by \(v = \sqrt { \frac { g r } { 2 } ( 5 - 4 \cos \theta ) }\), where \(g\) is the acceleration due to gravity. 8
  2. Find \(\theta\), giving your answer to the nearest degree.
    8
  3. A refined model takes into account air resistance. Explain how taking air resistance into account would affect your answer to part (b).
    [0pt] [2 marks]
    8
  4. In reality the slide is not smooth. It has a surface with the same coefficient of friction between the slide and the child for its entire length. Explain why the frictional force experienced by the child is not constant.
    [0pt] [1 mark]
AQA Further Paper 3 Mechanics Specimen Q9
10 marks Challenging +1.2
9 In this question use \(g = 9.8 \mathrm {~m} \mathrm {~s} ^ { - 2 }\).
A light elastic string has one end attached to a fixed point, \(A\), on a rough plane inclined at \(30 ^ { \circ }\) to the horizontal. The other end of the string is attached to a particle, \(P\), of mass 2 kg .
The elastic string has natural length 1.3 metres and modulus of elasticity 65 N .
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 2 metres from \(A\), as shown in the diagram. \includegraphics[max width=\textwidth, alt={}, center]{4fdb2637-6368-422c-99da-85b80efe31c5-14_549_744_861_785} The coefficient of friction between the particle and the plane is 0.6
After the particle is released it moves up the plane.
The particle comes to rest at a point \(B\), which is a distance, \(d\) metres, from \(A\). 9
  1. Show that the value of \(d\) is 1.4.
    [0pt] [7 marks] 9
  2. Determine what happens after \(P\) reaches the point \(B\). Fully justify your answer.
    [0pt] [3 marks]
AQA Further Paper 3 Mechanics 2019 June Q1
1 marks Moderate -0.8
1 A spring has natural length 0.4 metres and modulus of elasticity 55 N
Calculate the elastic potential energy stored in the spring when the extension of the spring is 0.08 metres. Circle your answer. \(0.176 \mathrm {~J} \quad 0.44 \mathrm {~J} \quad 0.88 \mathrm {~J} \quad 1.76 \mathrm {~J}\)
AQA Further Paper 3 Mechanics 2019 June Q2
1 marks Easy -1.8
2 A particle has an angular speed of 72 revolutions per minute.
Find the angular speed in radians per second.
Circle your answer.
[0pt] [1 mark] \(\frac { 6 \pi } { 5 } \quad \frac { 12 \pi } { 5 } \quad 12 \pi \quad 24 \pi\)
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}