What does Simple Harmonic Motion cover in IB Physics?

This topic covers the quantitative description of simple harmonic motion (SHM) using phase angle and energy considerations. A particle undergoing SHM can be described using phase angle φ. Displacement is given by x = x_0 sin(ω t + φ).

Is Simple Harmonic Motion SL or HL?

Simple Harmonic Motion is HL only. SL students are not examined on it.

How does FourtyFive help me practise Simple Harmonic Motion?

FourtyFive has 6 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.

Is FourtyFive free for Simple Harmonic Motion practice?

Yes. A free account gives you 50 marked answers a month, and you do not need a card to sign up.

Can I handwrite Simple Harmonic Motion answers on an iPad?

Yes. In the FourtyFive iPad app you write your working by hand with Apple Pencil, the way you would on paper, and it is marked the same way.

Topic C.1 · HL only

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 covers the quantitative description of simple harmonic motion (SHM) using phase angle and energy considerations. 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.

5 min read Jump to 6 practice questions

Summary

This topic covers the quantitative description of simple harmonic motion (SHM) using phase angle and energy considerations.
A particle undergoing SHM can be described using phase angle $\phi$ .
Displacement is given by $x = x_0 \sin(\omega t + \phi)$ .
Velocity is given by $v = \omega x_0 \cos(\omega t + \phi)$ or $v = \pm \omega \sqrt{x_0^2 - x^2}$ .
Total energy of oscillation: $E_T = \frac{1}{2} m \omega^2 x_0^2$ .
Potential energy of oscillation: $E_P = \frac{1}{2} m \omega^2 x^2$ .
Quantitative analysis of energy changes (kinetic, potential, total) in SHM is required.
Radians must be used for phase angle calculations.

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 $T = \dfrac{1}{f} = \dfrac{2\pi}{\omega}$

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

6 questions · 2 easy · 4 medium

Showing 6 of 6

Question 1

EasyPaper 1A · calculator1 mark

A child on a swing is given a small push and released from rest. The swing undergoes oscillations that are lightly damped by air resistance. Which statement correctly describes the subsequent motion of the swing?

A. The amplitude of the oscillations decreases exponentially with time.

B. The total energy of the swing remains constant.

C. The period of oscillation decreases steadily with time.

D. The swing comes to rest at the equilibrium position in the shortest possible time without oscillating.

Question 2

MediumPaper 2 · calculator10 marks

(a)

A component in a high-precision mechanical watch oscillates with simple harmonic motion. The component has a mass $m$ . Its displacement $x$ from the equilibrium position at time $t$ is described by the equation $x = A \cos(\omega t)$ , where $A$ is the amplitude of oscillation and $\omega$ is the angular frequency.

(a) Determine an expression for the velocity, $v$ , of the component as a function of time, $t$ .

[2]

(b)

(b) Hence, show that the kinetic energy of the component is given by $E_K = \frac{1}{2} m \omega^2 A^2 \sin^2(\omega t)$ .

[2]

(c)

(c) The restoring force on the component is provided by a spring of spring constant $k$ . Determine an expression for the potential energy, $E_P$ , stored in the spring as a function of time, $t$ .

[2]

(d)

(d) Using your answers to (b) and (c), deduce that the total mechanical energy of the oscillator is constant and independent of time.

[4]

Question 3

EasyPaper 1A · calculator1 mark

A block of mass $m$ is attached to a horizontal spring and oscillates with simple harmonic motion (SHM) on a frictionless surface.

What is the phase difference between the velocity of the block and its acceleration?

A. $0$

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

C. $\pi$

D. $\frac{3\pi}{2}$

Question 4

MediumPaper 2 · calculator2 marks

A small sensor attached to a vibrating machine component undergoes simple harmonic motion (SHM) at a frequency of $250\text{ Hz}$ . The maximum speed recorded for this sensor is $1.8\text{ m s}^{-1}$ .

(a) Determine the amplitude of its oscillation.

Question 5

MediumPaper 1A · calculator1 mark

A mass-spring system is forced to oscillate by a driver with variable frequency $f$ . The system has a natural frequency $f_0$ . Curve X shows the variation of amplitude with driving frequency when the system oscillates in air (light damping). Curve Y shows the variation when the system oscillates while submerged in a viscous fluid (heavy damping).

Which diagram correctly shows curves X and Y?

Graph with two resonance curves on the same axes of Amplitude vs Frequency. Curve X is a tall, narrow peak centred at frequency f0. Curve Y is a much shorter, broader peak, with its maximum amplitude at a frequency slightly less than f0.

Graph with two resonance curves on the same axes of Amplitude vs Frequency. Curve Y is a tall, narrow peak centred at frequency f0. Curve X is a much shorter, broader peak, with its maximum amplitude at a frequency slightly less than f0.

Graph with two resonance curves on the same axes of Amplitude vs Frequency. Both curves have their peak at the same frequency, f0. Curve X is a tall, narrow peak. Curve Y is a shorter, broader peak.

Question 6

MediumPaper 1A · calculator1 mark

The graph shows the variation with time $t$ of the displacement $x$ from the equilibrium position of a mass undergoing simple harmonic motion.

A cosine curve of displacement x in cm against time t in s. The curve starts at a maximum of +4.0 cm at t = 0, crosses x = 0 at t = 1.5 s, reaches a minimum of -4.0 cm at t = 3.0 s, crosses x = 0 at t = 4.5 s, and reaches a maximum of +4.0 cm at t = 6.0 s.

What is the speed of the mass at $t = 1.0\text{ s}$ ?

A. $4.2\text{ cm s}^{-1}$

B. $3.6\text{ cm s}^{-1}$

C. $2.1\text{ cm s}^{-1}$

D. $2.0\text{ cm s}^{-1}$

Every Simple Harmonic Motion question, marked for you

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