What does Simple Harmonic Motion cover in IB Physics?

This topic introduces simple harmonic motion (SHM), a type of oscillatory motion. SHM occurs when the restoring force is proportional to the displacement from equilibrium and directed towards it. The defining equation for SHM is a = -ω^2 x, where a is acceleration, ω is angular frequency, and x is displacement.

Is Simple Harmonic Motion SL or HL?

Both. SL and HL students study Simple Harmonic Motion, and HL goes further: that a particle undergoing simple harmonic motion can be described using phase angle.

How does FourtyFive help me practise Simple Harmonic Motion?

FourtyFive has 28 Simple Harmonic Motion questions. Every answer you write is marked mark by mark, IB-style, and you see where each mark was won or lost. Every part has a hint, the AI tutor helps you through the step you are stuck on, and your Study Profile picks what to practise next.

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Topic C.1 · SL and HL

Simple Harmonic Motion: notes and practice questions

Q: How do I revise Simple Harmonic Motion for IB Physics?

Start from the core idea: this topic introduces simple harmonic motion (SHM), a type of oscillatory motion. In the exam: both papers. May 2025 HL Paper 2 TZ1 question 9 built a whole 20-mark question on a bar magnet oscillating on a spring: calculate the elastic potential energy at equilibrium (2), state and explain the direction of motion at a given time (2), describe the energy transfers over an interval (2), show the amplitude is about 0.1 m (3), calculate the maximum speed (2), determine the kinetic energy at a stated time (2), then rolled into D.4 induction for the last 7 marks. Then practise exam-style questions, easiest first, writing out every step of your working before you check it.

11 min read Jump to 20 practice questions

Summary

This topic introduces simple harmonic motion (SHM), a type of oscillatory motion.
SHM occurs when the restoring force is proportional to the displacement from equilibrium and directed towards it.
The defining equation for SHM is $a = -\omega^2 x$ , where $a$ is acceleration, $\omega$ is angular frequency, and $x$ is displacement.
Key terms include time period $T$ , frequency $f$ , angular frequency $\omega$ , amplitude, and equilibrium position.
The relationships $T = \frac{1}{f}$ and $T = \frac{2\pi}{\omega}$ are fundamental.
For a mass-spring system, $T = 2\pi \sqrt{\frac{m}{k}}$ , and for a simple pendulum, $T = 2\pi \sqrt{\frac{L}{g}}$ .
Energy in SHM (kinetic and potential) continuously transforms, with total mechanical energy conserved in the absence of damping.

How it is examined

Both papers. May 2025 HL Paper 2 TZ1 question 9 built a whole 20-mark question on a bar magnet oscillating on a spring: calculate the elastic potential energy at equilibrium (2), state and explain the direction of motion at a given time (2), describe the energy transfers over an interval (2), show the amplitude is about 0.1 m (3), calculate the maximum speed (2), determine the kinetic energy at a stated time (2), then rolled into D.4 induction for the last 7 marks. The energy parts of that question are HL-only content.

Given in the booklet

SL: $a = -\omega^2 x$ , $T = 1/f = 2\pi/\omega$ , both period formulas. HL adds the five equations above. The SL/HL boundary here is sharp and worth repeating: an SL question may ask a student to describe energy changes over a cycle, but may not ask them to calculate $E_\text{T}$ or $E_\text{p}$ from amplitude, and may not use phase angle at all.

Key ideas

conditions that lead to simple harmonic motion
the defining equation of simple harmonic motion as given by $a = -\omega^2 x$
that a particle undergoing simple harmonic motion is described using time period T, frequency f, angular frequency $\omega$ , amplitude, equilibrium position and displacement
the time period in terms of frequency of oscillation and angular frequency, as given by $T = \dfrac{1}{f} = \dfrac{2\pi}{\omega}$

At HL

that a particle undergoing simple harmonic motion can be described using phase angle
that problems can be solved using the equations for simple harmonic motion, as given by $x = x_0 \sin(\omega t + \phi)$ $v = \omega x_0 \cos(\omega t + \phi)$ $v = \pm\,\omega\sqrt{x_0^2 - x^2}$ $E_\text{T} = \tfrac{1}{2}m\omega^2 x_0^2$ $E_\text{p} = \tfrac{1}{2}m\omega^2 x^2$

Guiding questions

What makes the harmonic oscillator model applicable to a wide range of physical phenomena?
Why must the defining equation of simple harmonic motion take the form it does?
How can the energy and motion of an oscillation be analysed both graphically and algebraically?

Linking questions

How can greenhouse gases be modelled as simple harmonic oscillators?
How can circular motion be used to visualize simple harmonic motion?
How does damping affect periodic motion?
How can the understanding of simple harmonic motion apply to the wave model? (NOS)
What physical explanation leads to the enhanced greenhouse effect? (NOS)

Practice questions

28 questions · 8 easy · 19 medium · 1 hard

Showing 20 of 20

Question 1

EasyPaper 1A · calculator1 mark

An object is undergoing simple harmonic motion (SHM). Which row correctly describes the direction of the net force acting on the object and the object's displacement when its kinetic energy is at a minimum?

	Direction of net force relative to displacement	Displacement when kinetic energy is a minimum
A.	same	maximum
B.	same	zero
C.	opposite	zero
D.	opposite	maximum

Question 2

MediumPaper 1A · calculator1 mark

A block is attached to a vertical spring and is displaced from its equilibrium position. It is then released and oscillates with simple harmonic motion (SHM).

Which row of the table correctly describes the direction of the net force on the block and the position where the block has its maximum kinetic energy?

	Direction of net force relative to displacement from equilibrium	Position of maximum kinetic energy
A.	Same	At the equilibrium position
B.	Same	At maximum displacement
C.	Opposite	At the equilibrium position
D.	Opposite	At maximum displacement

Question 3

HardPaper 2 · calculator23 marks

(a)(i)

(a.i) A spring has a natural length of $25.0 \text{ cm}$ and negligible mass. Its spring constant is $3.00 \text{ N cm}^{-1}$ . The spring obeys Hooke's law when stretched and when compressed.

The spring is horizontal with its left end attached to a rigid support. A $1.50 \text{ kg}$ block is attached to the free end of the spring and can move on a friction-free horizontal surface. The block is pulled $5.0 \text{ cm}$ to the right and held in position.

Calculate the force on the block from the spring when held in position.

[2]

(a)(ii)

(a.ii) Calculate the elastic potential energy stored in the spring when held in position.

[2]

(a)(iii)

(a.iii) The block is released. Calculate the speed of the block when it returns to its original position.

[3]

(b)(i)

(b.i) The spring is turned to be vertical with the top end attached to a rigid support. The $1.50 \text{ kg}$ block is attached to the bottom of the spring causing it to extend. The block is at rest in its equilibrium position.

Calculate the extension of the spring.

[2]

(b)(ii)

(b.ii) Calculate the elastic potential energy stored in the spring.

[2]

(c)(i)

(c.i) The block is now pulled down a further $5.0 \text{ cm}$ from the equilibrium position and held in position.

State the total extension of the spring in this new position.

[1]

(c)(ii)

(c.ii) Calculate the total elastic potential energy stored in the spring.

[2]

(c)(iii)

(c.iii) The block is released and accelerates upwards, reaching the equilibrium position with a vertical speed.

Calculate the increase in gravitational potential energy gained by the block as it moves to the equilibrium position.

[2]

(c)(iv)

(c.iv) Calculate the increase in kinetic energy gained by the block as it moves to the equilibrium position.

[3]

(c)(v)

(c.v) Calculate the speed of the block when it reaches the equilibrium position.

[2]

(d)

(d) Discuss your answers to (a.iii) and (c.v).

[2]

Question 4

EasyPaper 1A · calculator1 mark

A pendulum is oscillating with simple harmonic motion.

Which graph represents the variation of the kinetic energy, $E_k$ , of the pendulum bob with its displacement, $x$ , from the equilibrium position?

Graph A showing a parabola opening upwards, with its vertex at the origin (0,0). The vertical axis is Ek and the horizontal axis is x.

Graph B showing a parabola opening downwards, with its vertex on the positive vertical axis and x-intercepts symmetric about the origin. The vertical axis is Ek and the horizontal axis is x.

Graph C showing a straight line passing through the origin with a negative slope. The vertical axis is Ek and the horizontal axis is x.

Graph D showing a horizontal line with a positive value for Ek. The vertical axis is Ek and the horizontal axis is x.

Question 5

MediumPaper 1A · calculator1 mark

A piston in a simplified engine model undergoes simple harmonic motion with period $T$ . The amplitude of the motion is $A$ .

What is the speed of the piston at the instant when its kinetic energy is equal to its elastic potential energy?

A. $\frac{\pi A}{\sqrt{2} T}$

B. $\frac{\sqrt{2} \pi A}{T}$

C. $\frac{\pi A}{T}$

D. $\frac{2\pi A}{T}$

Question 6

EasyPaper 1A · calculator1 mark

A child on a swing is gently pushed and then let go. Due to air resistance, the amplitude of the swing's motion gradually decreases. Which statement correctly describes this motion?

A. The amplitude of the swing decreases by the same amount in each successive oscillation.

B. The period of the oscillation remains approximately constant.

C. The mechanical energy of the child-swing system remains constant.

D. The frequency of the oscillation is slightly greater than the natural frequency of the swing.

Question 7

MediumPaper 2 · calculator4 marks

A cylindrical buoy of mass $m$ and cross-sectional area $A$ floats vertically in calm water of density $\rho$ . The buoy is pushed down a small distance $x$ from its equilibrium position and released.

(a) By considering the forces acting on the buoy, explain why its subsequent oscillation is approximately simple harmonic.

Question 8

EasyPaper 1A · calculator1 mark

A simple pendulum oscillates with a period $T$ . The length of the pendulum is quadrupled and the amplitude of its oscillation is doubled. Assume the small angle approximation is valid.

What is the new period of oscillation?

A. $\frac{T}{2}$

B. $T$

C. $2T$

D. $4T$

Question 9

MediumPaper 2 · calculator7 marks

(a)

A simple pendulum is used to demonstrate simple harmonic motion inside the cabin of a vertical drop tower ride.

(a) The cabin is moving upwards at a constant speed. State and explain the effect on the period of oscillation of the pendulum.

[2]

(b)

(b) The cabin is then accelerated upwards at a constant rate of $a = 4.9 \text{ m s}^{-2}$ . State and explain the effect on the period of oscillation.

[3]

(c)

(c) At the top of the ride, the cabin is released and undergoes free-fall. Describe the motion of the pendulum bob, as observed from within the cabin, immediately after release.

[2]

Question 10

EasyPaper 1A · calculator1 mark

An object is undergoing simple harmonic motion (SHM). Which graph shows the relationship between the acceleration $a$ of the object and its displacement $x$ from the equilibrium position?

A graph of acceleration a on the y-axis against displacement x on the x-axis. The graph is a straight line passing through the origin with a negative gradient.

A graph of acceleration a on the y-axis against displacement x on the x-axis. The graph is a straight line passing through the origin with a positive gradient.

A graph of acceleration a on the y-axis against displacement x on the x-axis. The graph is a parabola with its vertex at the origin, opening upwards.

A graph of acceleration a on the y-axis against displacement x on the x-axis. The graph is a sine curve passing through the origin.

Question 11

MediumPaper 2 · calculator10 marks

(a)

A small laboratory cart of mass $250\text{ g}$ is attached to a horizontal spring with a spring constant of $5.0\text{ N m}^{-1}$ . The cart is placed on a frictionless track and displaced by $0.15\text{ m}$ from its equilibrium position, then released. Its subsequent motion is simple harmonic.

(a) Calculate the maximum value of stored elastic potential energy.

[2]

(b)

(b) Calculate the maximum speed of the cart.

[2]

(c)

[2]

(d)

(d) Calculate the frequency of vibration.

[2]

(e)

(e) Calculate the displacement when the stored elastic potential energy equals the kinetic energy.

[2]

Question 12

EasyPaper 1A · calculator1 mark

A $0.40 \text{ kg}$ block is placed on a frictionless horizontal surface and attached to a spring with a spring constant of $6.0 \text{ N m}^{-1}$ . The block is displaced slightly from its equilibrium position and released.

What is the natural frequency of oscillation for this system?

A. $0.35 \text{ Hz}$

B. $0.44 \text{ Hz}$

C. $0.61 \text{ Hz}$

D. $0.77 \text{ Hz}$

Question 13

MediumPaper 1A · calculator1 mark

A mass-spring system oscillates in simple harmonic motion with a total energy $E$ . The spring is replaced with one that has double the spring constant, and the mass is halved. The amplitude of oscillation is also halved.

What is the new total energy of the system?

A. $\frac{E}{4}$

B. $\frac{E}{2}$

C. $E$

D. $2E$

Question 14

EasyPaper 1A · calculator1 mark

A mass attached to a vertical spring is oscillating with simple harmonic motion of period $T$ . The mass is released from rest at its lowest position at time $t=0$ . At what time does the mass first have maximum speed?

A. $\frac{T}{8}$

B. $\frac{T}{4}$

C. $\frac{T}{2}$

D. $T$

Question 15

MediumPaper 1A · calculator1 mark

A simple pendulum has a period of oscillation $T_E$ on Earth, where the acceleration of free fall is $g$ . The pendulum is taken to a planet where the acceleration of free fall is $0.25g$ . The new period of oscillation is $T_P$ .

What is the ratio $\frac{T_P}{T_E}$ ?

A. $0.25$

B. $0.5$

C. $2$

D. $4$

Question 16

EasyPaper 1A · calculator1 mark

A child on a swing is oscillating with simple harmonic motion of period $T$ . At time $t = 0$ the swing is at its maximum displacement from the equilibrium position.

What is the first time $t > 0$ when the swing is at the equilibrium position?

A. $\frac{T}{4}$

B. $\frac{T}{2}$

C. $\frac{3T}{4}$

D. $T$

Question 17

MediumPaper 1A · calculator1 mark

A block of mass $m$ is attached to a horizontal spring with spring constant $k$ . The block oscillates with simple harmonic motion with an angular frequency $\omega$ . The block is replaced by a new block of mass $4m$ and the spring is replaced with a new spring of constant $\frac{k}{4}$ .

What is the new angular frequency of the system?

A. $\frac{\omega}{4}$

B. $\frac{\omega}{2}$

C. $\omega$

D. $4\omega$

Question 18

MediumPaper 1A · calculator1 mark

A simple pendulum has a period $T$ when oscillating on Planet Alpha. The length of the pendulum is then doubled, and its bob mass is tripled. This modified pendulum is moved to Planet Beta, where the acceleration due to gravity is half that on Planet Alpha.

What is the period of the pendulum on Planet Beta?

A. $\frac{1}{2} T$

B. $T$

C. $2 T$

D. $4 T$

Question 19

MediumPaper 1A · calculator1 mark

A mass of $0.50 \text{ kg}$ is attached to a spring and oscillates horizontally on a frictionless surface, undergoing simple harmonic motion. The amplitude of the oscillation is $0.10 \text{ m}$ and its period is $\frac{\pi}{5} \text{ s}$ .

What is the total energy of the oscillating mass?

A. $0.125 \text{ J}$

B. $0.25 \text{ J}$

C. $0.50 \text{ J}$

D. $1.0 \text{ J}$

Question 20

MediumPaper 1A · calculator1 mark

The diagram shows a snapshot of a transverse wave propagating along a stretched string. The wave travels from right to left.

Graph of displacement y versus position x for a wave on a string. Points A, B, C, D are marked on the wave. Point A is at a crest, B is at the equilibrium position with a negative slope, C is at a trough, D is at the equilibrium position with a positive slope.

At the instant shown, which point on the string has the maximum positive acceleration?

8 more Simple Harmonic Motion questions in the app

Every answer is marked mark by mark, IB-style, and the AI tutor helps when you are stuck.

Where marks are lost

Stopping one step short of the conclusion. Two numbers and no sentence is two marks out of three.
Answering a procedure question with a platitude.
Losing precision in Paper 1B. Uniquely to this paper, quoting the right number badly loses marks.

Simple Harmonic Motion: notes and practice questions

How it is examined

Guiding questions

Linking questions

Practice questions

Question 1

Question 2

Question 3

Question 4

Question 5

Question 6

Question 7

Question 8

Question 9

Question 10

Question 11

Question 12

Question 13

Question 14

Question 15

Question 16

Question 17

Question 18

Question 19

Question 20

8 more Simple Harmonic Motion questions in the app

Where marks are lost

Practise this topic as a session

Questions,
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How it is examined

Guiding questions

Linking questions

Practice questions

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Question 2

Question 3

Question 4

Question 5

Question 6

Question 7

Question 8

Question 9

Question 10

Question 11

Question 12

Question 13

Question 14

Question 15

Question 16

Question 17

Question 18

Question 19

Question 20

8 more Simple Harmonic Motion questions in the app

Where marks are lost

Questions,answered.

Start with the IB questionbank built for you.

Questions,
answered.

Start with the IB question
bank built for you.