AQA A Level Further Maths Paper 3M June 2019 question paper 7367/3M

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 J
0.44 J
0.88 J
1.76 J

A particle has an angular speed of 72 revolutions per minute.

Find the angular speed in radians per second.

Circle your answer.

In this question use = 10 m s²

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 to the vertical, touching the chin of a student, as shown in the diagram below.

The sphere is released from rest.

Assume that the student stays perfectly still once the sphere has been released.

Calculate the maximum speed of the sphere.

Find the maximum tension in the string.

State, with a reason, whether or not the sphere touches the student’s chin again after it has been released.

The triangular region shown below is rotated through 360 around the x-axis, to form a solid cone.

The coordinates of the vertices of the triangle are (0, 0), (8, 0) and (0, 4).

All units are in centimetres.

State an assumption that you should make about the cone in order to find the position of its centre of mass.

Using integration, prove that the centre of mass of the cone is 2 cm from its plane face.

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.

Find the angle between the board and the horizontal when the cone topples, giving your answer to the nearest degree.

Find the range of possible values for the coefficient of friction between the cone and the board.

A ball moving on a smooth horizontal surface collides with a fixed vertical wall.

wall.

wall.

Model the ball as a particle.

Find the coefficient of restitution between the ball and the wall, giving your answer correct to two significant figures.

Determine whether or not the wall is smooth.

Fully justify your answer.

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 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 rad s¹

Find , giving your answer in terms of g.

In this question use = 9.8 m s²

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.

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.

Show that the speed of Hannah when the ropes become slack is 30 m s¹ correct to two significant figures.

Determine whether Hannah is moving up or down when the ropes become taut again.