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AQA A Level Further Maths

Topic questions Mechanics Simple harmonic motion

Simple harmonic motion

7 questions · 74 marks · about 1.5 hours

Simple harmonic motion questions

TopicSimple harmonic motion

AQA 7367 Simple harmonic motion 7 questions

Paper 1 June 2024, Question 18 12 marks

Figure for question 18
(a)

Find the extension of each string when the system is in equilibrium.

[3]
Reveal mark scheme Mark scheme for question 18(a)
(b)
Figure for question 18
(b)(i)

The ball is pulled a distance of 0.6 metres downwards from its equilibrium position towards C, and released from rest.

Show that during the subsequent motion the particle satisfies the differential equation

Question 18, shown as an image

where x metres is the displacement of the particle below the equilibrium position at time t seconds after the particle is released.

[3]
Reveal mark scheme Mark scheme for question 18(b)(i)
(b)(ii)

Find x in terms of t

[5]
Reveal mark scheme Mark scheme for question 18(b)(ii)
(c)

State one limitation of the model used in part (b)

[1]
Reveal mark scheme Mark scheme for question 18(c)

Paper 2 June 2023, Question 16 16 marks

A bungee jumper of mass m kg is attached to an elastic rope. The other end of the rope is attached to a fixed point.

The bungee jumper falls vertically from the fixed point.

At time t seconds after the rope first becomes taut, the extension of the rope is

Question 16, shown as an image
(a)

A model for the motion while the rope remains taut assumes that the forces acting on the bungee jumper are

maths symbol the weight of the bungee jumper

maths symbol a tension in the rope of magnitude kx newtons

maths symbol an air resistance force of magnitude Rv newtons

where k and R are constants such that 4km maths symbol R2

(a)(i)

Show that this model gives the result

Question 16, shown as an image
Question 16, shown as an image

You do not need to find the value of A or the value of B

[6]
Reveal mark scheme Mark scheme for question 16(a)(i)
(a)(ii)
Figure for question 16
[6]
Reveal mark scheme Mark scheme for question 16(a)(ii)
(b)

A second, simpler model assumes that the air resistance is zero.

The values of k, m and g remain the same.

Find an expression for x in terms of t according to this simpler model, giving the values of all constants to two significant figures.

[4]
Reveal mark scheme Mark scheme for question 16(b)

Paper 1 June 2022, Question 1 1 mark

The displacement of a particle from its equilibrium position is x metres at time t seconds.

The motion of the particle obeys the differential equation

Question 1, shown as an image

Calculate the period of its motion in seconds.

Circle your answer.

Question 1, shown as an image
[1]
Reveal mark scheme Mark scheme for question 1

Paper 1 June 2022, Question 11 19 marks

In this question use maths symbol as 10 m smaths symbol2

A smooth plane is inclined at 30maths symbol to the horizontal. The fixed points A and B are 3.6 metres apart on the line of greatest slope of the plane, with A higher than B

A particle P of mass 0.32 kg is attached to one end of each of two light elastic strings. The other ends of these strings are attached to the points A and B respectively.

The particle P moves on a straight line that passes through A and B

Figure for question 11

The natural length of the string AP is 1.4 metres. When the extension of the string AP is eA metres, the tension in the string AP is

Question 11, shown as an image

The natural length of the string BP is 1 metre. When the extension of the string BP is eB metres, the tension in the string BP is

Question 11, shown as an image

The particle P is held at the point between A and B which is 0.2 metres from its equilibrium position and lower than its equilibrium position. The particle P is then released from rest.

At time t seconds after P is released, its displacement towards B from its equilibrium position is x metres.

(a)

Show that during the subsequent motion the object satisfies the equation

Question 11, shown as an image

Fully justify your answer.

[5]
Reveal mark scheme Mark scheme for question 11(a)
(b)

The experiment is repeated in a large tank of oil. During the motion the oil causes a resistive force of kv newtons to act on the particle,

Question 11, shown as an image

The oil causes critical damping to occur.

(b)(i)
Question 11, shown as an image
[3]
Reveal mark scheme Mark scheme for question 11(b)(i)
(b)(ii)

Find x in terms of t, giving your answer in exact form.

[6]
Reveal mark scheme Mark scheme for question 11(b)(ii)
(b)(iii)

Calculate the maximum speed of the particle.

[5]
Reveal mark scheme Mark scheme for question 11(b)(iii)

Paper 1 November 2021, Question 15 13 marks

In this question use maths symbol = 9.8 m smaths symbol2

A particle P of mass m is attached to two light elastic strings, AP and BP.

The other ends of the strings, A and B, are attached to fixed points which are 4 metres apart on a rough horizontal surface at the bottom of a container.

The coefficient of friction between P and the surface is 0.68

maths symbol When the extension of string AP is eA metres, the tension in AP is 24meA

maths symbol When the extension of string BP is eB metres, the tension in BP is 10meB

maths symbol The natural length of string AP is 1 metre

maths symbol The natural length of string BP is 1.3 metres

Figure for question 15
(a)

Show that when AP maths symbol 1.5 metres, the tension in AP is equal to the tension in BP.

[1]
Reveal mark scheme Mark scheme for question 15(a)
(b)

P is held at the point between A and B where AP maths symbol 1.9 metres, and then released from rest.

At time t seconds after P is released, AP maths symbol (1.5 maths symbol x) metres.

Figure for question 15

Show that when P is moving towards A,

Question 15, shown as an image
[3]
Reveal mark scheme Mark scheme for question 15(b)
(c)
Figure for question 15
[9]
Reveal mark scheme Mark scheme for question 15(c)

Paper 1 November 2020, Question 13 12 marks

Two light elastic strings each have one end attached to a particle B of mass 3 c kg, which rests on a smooth horizontal table.

The other ends of the strings are attached to the fixed points A and C, which are 8 metres apart.

ABC is a horizontal line.

Figure for question 13

String AB has a natural length of 4 metres and a stiffness of 5 c newtons per metre.

String BC has a natural length of 1 metre and a stiffness of c newtons per metre.

The particle is pulled a distance of fraction metre from its equilibrium position towards A, and released from rest.

(a)

Show that the particle moves with simple harmonic motion.

[8]
Reveal mark scheme Mark scheme for question 13(a)
(b)

Find the speed of the particle when it is at a point P, a distance fraction metre from the equilibrium position. Give your answer to two significant figures.

[4]
Reveal mark scheme Mark scheme for question 13(b)

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