What are simple harmonic oscillations in the context of Cambridge International A Levels Physics?

Simple harmonic oscillations refer to a type of periodic motion where the restoring force is directly proportional to the displacement and acts in the opposite direction. This is a fundamental concept in the Cambridge International A Levels Physics curriculum, often exemplified by systems like pendulums and springs.

How is the period of a simple harmonic oscillator determined?

The period of a simple harmonic oscillator, such as a mass-spring system, is determined by the formula T = 2π√(m/k), where T is the period, m is the mass, and k is the spring constant. For a pendulum, the period is given by T = 2π√(l/g), where l is the length of the pendulum and g is the acceleration due to gravity.

What is the significance of the amplitude in simple harmonic oscillations?

In simple harmonic oscillations, the amplitude is the maximum displacement from the equilibrium position. It is significant because it determines the energy of the system; however, it does not affect the period or frequency of the oscillation. Understanding amplitude is crucial for Cambridge International A Levels students studying oscillatory motion.

Can you explain the concept of phase in simple harmonic oscillations?

Phase in simple harmonic oscillations refers to the position of a point in its cycle at a given time, often expressed in radians or degrees. It helps describe the state of oscillation at any point in time and is essential for understanding wave interference and resonance, topics covered in the Cambridge International A Levels Physics syllabus.

How do damping and resonance affect simple harmonic oscillations?

Damping refers to the gradual loss of amplitude in an oscillating system due to energy dissipation, such as friction or air resistance. Resonance occurs when a system is driven at its natural frequency, leading to large amplitude oscillations. Both concepts are important in understanding real-world applications of simple harmonic motion, as taught in the Cambridge International A Levels Physics curriculum.

Why is "Forgetting negative sign in $a = -\omega^2 x$" a common mistake in Simple harmonic oscillations ?

Why it happens: Sign overlooked. How to avoid it: The MINUS makes acceleration ALWAYS POINT TOWARD equilibrium. Essential to SHM. Source: 9702 Examiner Reports 2022-2024

Simple harmonic oscillations – Study Notes & Past Paper Style Questions | Cambridge International A Level Physics 9702

Short Notes - Simple harmonic oscillations

Detailed Study Notes

Detailed notes on Oscillations for Cambridge International A Levels Physics, covering key concepts, explanations, examples, and exam-focused revision points.

Simple Harmonic Motion Study Notes — Cambridge International A Level Physics 9702 (2025-2027 syllabus)

SHM defining equation, period, amplitude. Pendulum and spring.

At a glance

$a = -\omega^2 x$ .
$x = A\cos(\omega t)$ .
$v_{\max} = \omega A$ .
Pendulum $T = 2\pi\sqrt{L/g}$ .
Spring $T = 2\pi\sqrt{m/k}$ .

What you’ll learn

Mapped to the Cambridge International A Level 9702 syllabus (2025-2027).

17.1.1 — Define SHM.
17.1.2 — Use $a = -\omega^2 x$ .
17.1.3 — Solve pendulum and spring problems.

SHM

$a \propto -x$ .

Defining equation. $a = -\omega^2 x$ .

Solution. $x(t) = A\cos(\omega t + \phi)$ (or sine). Sinusoidal.

Frequency. $f = 1/T = \omega/2\pi$ .

Velocity. $v_{\max} = \omega A$ at $x = 0$ . $v = 0$ at $x = \pm A$ .

Pendulum. $T = 2\pi\sqrt{L/g}$ (small angle).

Spring-mass. $T = 2\pi\sqrt{m/k}$ .

Cambridge tip. Always state defining equation when asked to define SHM.

Figure 1 — SHM displacement-time graph

SHM gives a sinusoidal displacement-time graph between +A and −A. The period T is the time for one complete oscillation, measured here between two successive maxima.

Figure 2 — Acceleration against displacement is a straight line of negative gradient

The defining test for SHM: a straight line through the origin on an a–x graph, with a negative gradient of magnitude ω². This is the graph examiners ask you to recognise.

Figure 3 — Displacement, velocity and acceleration against time

The dashed guides line up the three graphs. Velocity leads displacement by a quarter of a cycle; acceleration is exactly out of phase with displacement, which is another way of saying a = −ω²x.

Figure 4 — The simple pendulum and the mass-spring oscillator

The two oscillators you are expected to use. Both perform SHM, but the pendulum result only holds while the angular displacement stays small.

The oscillation at its key positions.

At the...	Displacement $x$	Velocity $v$	Acceleration $a$
Centre	$0$	Maximum, $\omega A$	$0$
Ends	$\pm A$	$0$ (momentarily at rest)	Maximum, $\omega^2 A$ , pointing back to the centre

The object is never at rest at the centre and never moving at the ends — that is the pattern every SHM question relies on.

$a = -\omega^2 x$ .
Sinusoidal motion.
Pendulum and spring formulas.

See the full worked example for simple harmonic oscillations →

Quick recap

$a = -\omega^2 x$ .
Sinusoidal solution.
Pendulum/spring periods.

Memorise this

Verbatim phrases and definitions Cambridge mark schemes credit.

SHM equation.
Pendulum and spring periods.
$v_{\max} = \omega A$ .

How it’s examined

SHM heavily tested on Paper 4. Most-tested: pendulum period (5 marks), velocity (5-7 marks).

Sources: Cambridge International A Level Physics 9702 syllabus (2025-2027); 9702 Examiner Reports 2022-2024; 9702/42 May/Jun 2024 question paper and mark scheme. Last reviewed 2026-05-11.

Take this whole topic with you

Step-by-step worked examples — Simple harmonic oscillations

Step-by-step solutions to past-paper-style questions on simple harmonic oscillations , written exactly the way a tutor would explain them at the board.

1Pendulum period (5 marks)
Extended• Adapted from 9702/42 May/Jun 2024• SHM
Question
Find the period of a 0.50 m simple pendulum. $g = 9.81$ . (5 marks)
Step-by-step solution
1. Step 1
  $T = 2\pi\sqrt{L/g}$ .
  $T = 2\pi\sqrt{0.50/9.81} \approx 1.42 \text{ s}$
Answer
$T \approx 1.42$ s.
2Maximum velocity (5 marks)
Extended• v_max
Question
An object in SHM has amplitude 0.10 m and period 2.0 s. Find max speed. (5 marks)
Step-by-step solution
1. Step 1
  $\omega = 2\pi/T = \pi$ rad/s.
2. Step 2
  $v_{\max} = \omega A$ .
  $v_{\max} = \pi \times 0.10 \approx 0.314 \text{ m/s}$
Answer
$v_{\max} \approx 0.31$ m/s.
3Identify SHM (5 marks)
Extended• identification
Question
State the defining equation of SHM and describe the motion. (5 marks)
Step-by-step solution
1. Step 1
  Defining equation: $a = -\omega^2 x$ .
2. Step 2
  Acceleration proportional to displacement, directed toward equilibrium.
3. Step 3
  Solution. $x(t) = A\cos(\omega t + \phi)$ (sinusoidal).
Answer
$a = -\omega^2 x$ . Sinusoidal motion about equilibrium.

Key Formulae — Simple harmonic oscillations

The formulae you need to memorise for simple harmonic oscillations on the Cambridge International A Level 9702 paper, with every variable defined in plain English and a note on when to use it.

SHM defining equation
$a = -\omega^2 x$
When to use
Acceleration proportional to displacement, opposite direction.
SHM position
$x(t) = A\cos(\omega t + \phi) \text{ or } x(t) = A\sin(\omega t)$
$A$
amplitude
When to use
Solution of SHM equation.
SHM velocity
$v = \pm\omega\sqrt{A^2 - x^2}, \quad v_{\max} = \omega A$
When to use
Speed at displacement $x$ . Max at equilibrium ( $x = 0$ ).
Simple pendulum period
$T = 2\pi\sqrt{L/g}$
$L$
pendulum length
When to use
Small-angle approximation.
Spring-mass period
$T = 2\pi\sqrt{m/k}$
When to use
Mass on Hookean spring.

Key Definitions and Keywords — Simple harmonic oscillations

Definitions to memorise and the exact keywords mark schemes credit for simple harmonic oscillations answers — sharpened from recent examiner reports for the 2026 Cambridge International A Level 9702 sitting.

Simple harmonic motion (SHM)
Examiner keyword
Motion where acceleration is proportional to displacement from equilibrium and directed toward equilibrium.
Amplitude
Examiner keyword
Maximum displacement from equilibrium in SHM.

Common Mistakes and Misconceptions — Simple harmonic oscillations

The traps other students keep falling into on simple harmonic oscillations questions — taken from recent Cambridge International A Level 9702 examiner reports and mark schemes — and how to avoid them.

✕Forgetting negative sign in $a = -\omega^2 x$
9702 Examiner Reports 2022-2024
Why it happens
Sign overlooked.
How to avoid it
The MINUS makes acceleration ALWAYS POINT TOWARD equilibrium. Essential to SHM.

Simple harmonic oscillations — frequently asked questions

The things students keep getting wrong in this sub-topic, answered.

Simple harmonic oscillations

Short Notes - Simple harmonic oscillations

Detailed Study Notes

At a glance

What you’ll learn

SHM

Quick recap

Memorise this

How it’s examined

Take this whole topic with you

1Pendulum period (5 marks)

2Maximum velocity (5 marks)

3Identify SHM (5 marks)

SHM defining equation

SHM position

SHM velocity

Simple pendulum period

Spring-mass period

Simple harmonic motion (SHM)

Amplitude

✕Forgetting negative sign in a=−ω2xa = -\omega^2 xa=−ω2x

Simple harmonic oscillations — frequently asked questions

Simple harmonic oscillations

Short Notes - Simple harmonic oscillations

Detailed Study Notes

At a glance

What you’ll learn

SHM

Quick recap

Memorise this

How it’s examined

Take this whole topic with you

1Pendulum period (5 marks)

2Maximum velocity (5 marks)

3Identify SHM (5 marks)

SHM defining equation

SHM position

SHM velocity

Simple pendulum period

Spring-mass period

Simple harmonic motion (SHM)

Amplitude

✕Forgetting negative sign in a=−ω2xa = -\omega^2 xa=−ω2x

Simple harmonic oscillations — frequently asked questions

✕Forgetting negative sign in $a = -\omega^2 x$

✕Forgetting negative sign in $a = -\omega^2 x$