17.1 Simple Harmonic Oscillations

2026 Syllabus Objectives

By the end of this topic, you should be able to:

  1. Understand and use the terms displacement, amplitude, period, frequency, angular frequency and phase difference in the context of oscillations, and express the period in terms of both frequency and angular frequency.
  2. Understand that simple harmonic motion (SHM) occurs when acceleration is proportional to displacement from a fixed point and in the opposite direction.
  3. Use a = −ω²x and recall and use, as a solution to this equation, x = x₀ sin ωt.
  4. Use the equations v = v₀ cos ωt and v = ±ω√(x₀² − x²).
  5. Analyse and interpret graphical representations of displacement, velocity and acceleration for SHM.

Section 1: Describing Oscillations

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What is an Oscillation?

An oscillation is a back-and-forth, repeated movement of an object around a central resting position. Think of a pendulum swinging left and right, or a mass bouncing up and down on a spring. The object keeps repeating the same movement over and over again.

The central resting position — where the object naturally sits if not disturbed — is called the equilibrium position. We label this position x = 0.

Key Terms for Oscillations

Displacement (x)

Displacement is how far the object has moved from its equilibrium position at any given moment, measured in metres (m). It is a vector quantity, meaning it has both size and direction — it can be positive (moved one way) or negative (moved the other way).

For example, if a pendulum swings 5 cm to the right of centre, its displacement is +5 cm. If it swings 5 cm to the left, its displacement is −5 cm.

Amplitude (x₀)

Amplitude is the maximum displacement — the furthest the object ever gets from the equilibrium position. It is always a positive value and is measured in metres (m).

On a displacement–time graph, the amplitude is the height of the peaks (or the depth of the troughs).

Period (T)

The period is the time taken for one complete oscillation — one full cycle of back-and-forth motion. It is measured in seconds (s).

On a displacement–time graph, the period is the horizontal distance from one peak to the very next peak (or trough to trough).

The period can be expressed in two ways:

T=1fT = \frac{1}{f}

T=2πωT = \frac{2\pi}{\omega}

Frequency (f)

Frequency is the number of complete oscillations per second. It is measured in hertz (Hz), where 1 Hz = 1 oscillation per second (s⁻¹).

The relationship between frequency and period is:

f=1Tf = \frac{1}{T}

So if a pendulum takes 2 seconds for one complete swing, its frequency is 1/2 = 0.5 Hz.

Angular Frequency (ω)

Angular frequency (symbol ω, the Greek letter "omega") is another way of describing how fast something oscillates. Rather than counting oscillations per second, it measures the rate of oscillation in radians per second (rad s⁻¹).

You can think of one complete oscillation as corresponding to a full circle, which is 2π radians. So:

ω=2πT=2πf\omega = \frac{2\pi}{T} = 2\pi f

Angular frequency is very useful because it links oscillations to the mathematics of circular motion and wave equations.

Quick summary of relationships:

QuantitySymbolUnitKey equation
PeriodTsT = 1/f = 2π/ω
FrequencyfHzf = 1/T
Angular frequencyωrad s⁻¹ω = 2πf = 2π/T

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