AQA Further Paper 3 Mechanics (Further Paper 3 Mechanics) 2019 June

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.
\(0.176 \mathrm {~J} \quad 0.44 \mathrm {~J} \quad 0.88 \mathrm {~J} \quad 1.76 \mathrm {~J}\)

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] \(\frac { 6 \pi } { 5 } \quad \frac { 12 \pi } { 5 } \quad 12 \pi \quad 24 \pi\)

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 \(c\).

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

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
4. 1. Find the angle between the board and the horizontal when the cone topples, giving your answer to the nearest degree. 5
5. (ii) Find the range of possible values for the coefficient of friction between the cone and the board.

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.

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 \(g\).

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.