AQA A Level Further Maths Paper 3M June 2024 question paper 7367/3M

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.

Calculate the magnitude of the impulse on the particle between t = 0 and t = 3

Circle your answer.

0.3 N s
0.6 N s
0.9 N s
1.8 N s

A conical pendulum consists of a light string and a particle of mass m kg

The conical pendulum completes horizontal circles with radius r metres and angular speed ω radians per second. The string makes an angle θ 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.

Which one of the following statements is correct?

Tick (✓) one box.

T cosθ = mrω²

T sinθ = mrω²

Find the acceleration of the particle.

Find the tension in the string.

Show that the angular speed of the particle is 48 revolutions per minute correct to two significant figures.

State the dimensions of F

Use dimensional analysis to find the value of α and the value of β

Calculate the elastic potential energy of the string when the particle is at A

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

Find the speed of the particle at B

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

Assume that the wall is smooth.

Find the value of v

Give your answer to two significant figures.

Find the value of θ

Give your answer to the nearest whole number.

Find the magnitude of the impulse exerted on the sphere by the wall.

Give your answer to two significant figures.

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).

The finite region enclosed by the line y = kx, the x-axis and the line x = 5 is rotated through 360 ° around the x axis to form a solid cone.

Use integration to show that the position of the centre of mass of the cone is independent of k

State the distance between the base of the cone and its centre of mass.

State one assumption that you have made about the cone.

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°

Find the range of values of k for which the cone slides before it topples.

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 mU

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° with the upward vertical through O, as shown in the diagram below.

Show that

where g is the acceleration due to gravity.

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.