AQA Further Paper 3 Mechanics (Further Paper 3 Mechanics) 2024 June

2 As a particle moves along a straight horizontal line, it is subjected to a force \(F\) newtons that acts in the direction of motion of the particle. At time \(t\) seconds, \(F = \frac { t } { 5 }\)
Calculate the magnitude of the impulse on the particle between \(t = 0\) and \(t = 3\)
Circle your answer.
[0pt] [1 mark]
\(0.3 \mathrm {~N} \mathrm {~s} \quad 0.6 \mathrm {~N} \mathrm {~s} \quad 0.9 \mathrm {~N} \mathrm {~s} \quad 1.8 \mathrm {~N} \mathrm {~s}\) A conical pendulum consists of a light string and a particle of mass \(m \mathrm {~kg}\) The conical pendulum completes horizontal circles with radius \(r\) metres and angular speed \(\omega\) radians per second. The string makes an angle \(\theta\) with the downward vertical. The tension in the string is \(T\) newtons. The conical pendulum and the forces acting on the particle are shown in the diagram.
\includegraphics[max width=\textwidth, alt={}, center]{86817115-46a1-4702-8a33-8f9b05d69bb9-03_616_593_689_703} Which one of the following statements is correct?
Tick ( ✓ ) one box.
\(T \cos \theta = m r \omega ^ { 2 }\)
\includegraphics[max width=\textwidth, alt={}, center]{86817115-46a1-4702-8a33-8f9b05d69bb9-03_108_108_1567_900}
\(T \sin \theta = m r \omega ^ { 2 }\)
\includegraphics[max width=\textwidth, alt={}, center]{86817115-46a1-4702-8a33-8f9b05d69bb9-03_109_108_1726_900}
\(T \cos \theta = \frac { m \omega ^ { 2 } } { r }\)
\includegraphics[max width=\textwidth, alt={}, center]{86817115-46a1-4702-8a33-8f9b05d69bb9-03_109_108_1886_900}
\(T \sin \theta = \frac { m \omega ^ { 2 } } { r }\) □

4 A particle of mass 3 kg is attached to one end of a light inextensible string. The other end of the string is attached to a fixed point on a smooth horizontal surface.
The particle is set into motion so that it moves with a constant speed \(4 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) in a circular path with radius 0.8 metres on the horizontal surface. 4

1. Find the acceleration of the particle.
   4
2. Find the tension in the string.
   4
3. Show that the angular speed of the particle is 48 revolutions per minute correct to two significant figures.

5 When a sphere of radius \(r\) metres is falling at \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\) it experiences an air resistance force \(F\) newtons. The force is to be modelled as $$F = k r ^ { \alpha } { } _ { V } { } ^ { \beta }$$ where \(k\) is a constant with units \(\mathrm { kg } \mathrm { m } ^ { - 2 }\) 5

1. State the dimensions of \(F\)
   5
2. Use dimensional analysis to find the value of \(\alpha\) and the value of \(\beta\)

6 In this question use \(\boldsymbol { g } = 9.8 \mathbf { m ~ s } ^ { - 2 }\) A light elastic string has natural length 3 metres and modulus of elasticity 18 newtons.
One end of the elastic string is attached to a particle of mass 0.25 kg
The other end of the elastic string is attached to a fixed point \(O\)
The particle is released from rest at a point \(A\), which is 4.5 metres vertically below \(O\) 6

1. Calculate the elastic potential energy of the string when the particle is at \(A\)
   6
2. The point \(B\) is 3 metres vertically below \(O\) Calculate the gravitational potential energy gained by the particle as it moves from \(A\) to \(B\)
   6
3. Find the speed of the particle at \(B\)
   6
4. The point \(C\) is 3.6 metres vertically below \(O\)
   Explain, showing any calculations that you make, why the speed of the particle is increasing the first time that the particle is at \(C\)

7 A sphere, of mass 0.2 kg , moving on a smooth horizontal surface, collides with a fixed wall. Before the collision the sphere moves with speed \(5 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) at an angle of \(60 ^ { \circ }\) to the wall. After the collision the sphere moves with speed \(\nu \mathrm { m } \mathrm { s } ^ { - 1 }\) at an angle of \(\theta ^ { \circ }\) to the wall. The velocities are shown in the diagram below.
\includegraphics[max width=\textwidth, alt={}, center]{86817115-46a1-4702-8a33-8f9b05d69bb9-08_303_762_735_625} The coefficient of restitution between the wall and the sphere is 0.7 7

1. Assume that the wall is smooth. 7
2. 1. Find the value of \(v\) Give your answer to two significant figures.
      7
3. (ii) Find the value of \(\theta\) Give your answer to the nearest whole number.
   7
4. (iii) Find the magnitude of the impulse exerted on the sphere by the wall.
   Give your answer to two significant figures.
   7
5. In reality the wall is not smooth.
   Explain how this would cause a change in the magnitude of the impulse calculated in part (a)(iii).

8 The finite region enclosed by the line \(y = k x\), the \(x\)-axis and the line \(x = 5\) is rotated through \(360 ^ { \circ }\) around the \(x\) axis to form a solid cone. 8

1. 1. Use integration to show that the position of the centre of mass of the cone is independent of \(k\)
      8
2. (ii) State the distance between the base of the cone and its centre of mass.
   8
3. State one assumption that you have made about the cone. 8
4. The plane face of the cone is placed on a rough inclined plane.
   The coefficient of friction between the cone and the plane is 0.8
   The angle between the plane and the horizontal is gradually increased from \(0 ^ { \circ }\) Find the range of values of \(k\) for which the cone slides before it topples.
   [0pt] [4 marks]

9 A small sphere, of mass \(m\), is attached to one end of a light inextensible string of length \(a\) The other end of the string is attached to a fixed point \(O\) The sphere is at rest in equilibrium directly below \(O\) when it is struck, giving it a horizontal impulse of magnitude \(m U\) After the impulse, the sphere follows a circular path in a vertical plane containing the point \(O\) until the string becomes slack at the point \(C\) At \(C\) the string makes an angle of \(30 ^ { \circ }\) with the upward vertical through \(O\), as shown in the diagram below.
\includegraphics[max width=\textwidth, alt={}, center]{86817115-46a1-4702-8a33-8f9b05d69bb9-12_583_331_875_901} 9

1. Show that $$U ^ { 2 } = \frac { a g } { 2 } ( 4 + 3 \sqrt { 3 } )$$ where \(g\) is the acceleration due to gravity.
   9
2. With reference to any modelling assumptions that you have made, explain why giving your answer as an inequality would be more appropriate, and state this inequality.
   \includegraphics[max width=\textwidth, alt={}, center]{86817115-46a1-4702-8a33-8f9b05d69bb9-14_2491_1755_173_123} \begin{center} \begin{tabular}{|l|l|} \hline Question number & Additional page, if required. Write the question numbers in the left-hand margin.
   \hline & \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\)
   \hline & \begin{tabular}{l}