AQA Further Paper 3 Mechanics (Further Paper 3 Mechanics) 2022 June

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

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

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

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]

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 \(A\)

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.

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.

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}

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] \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
3. 1. 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
4. (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
5. (iii) Find the maximum value of \(m\) for which the composite body does not slide or topple. Fully justify your answer.