---
title: "Simple Harmonic Motion: the Defining Relation and Its Graphs"
description: "When the acceleration of an object is proportional to its displacement and points back towards the centre, the motion repeats with a period that does not depend on the amplitude. Displacement, velocit"
canonical: https://lightmysky.com/learn/science/simple-harmonic-motion-the-defining-relation-and-its-graphs-mt_NTDayHkdfh
source: https://lightmysky.com/learn/science/simple-harmonic-motion-the-defining-relation-and-its-graphs-mt_NTDayHkdfh.md
retrieved: 2026-09-12
---

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# Simple Harmonic Motion: the Defining Relation and Its Graphs

When the acceleration of an object is proportional to its displacement and points back towards the centre, the motion repeats with a period that does not depend on the amplitude. Displacement, velocity and acceleration then run as three sine curves each a quarter cycle ahead of the last.

Subject: Science · Area: Forces & Motion · Ages 17 to 18
Page: https://lightmysky.com/learn/science/simple-harmonic-motion-the-defining-relation-and-its-graphs-mt_NTDayHkdfh

## Ready when they can

- Tests whether a given motion is simple harmonic by checking the acceleration against the displacement
- Reads amplitude, period and phase off displacement, velocity and acceleration graphs and says how the three line up
- Predicts how the period of a mass on a spring and of a pendulum changes when mass, stiffness or length changes

## Lesson: Motion that repeats itself

A child on a playground swing and the pendulum in a grandfather clock share the same rhythm rule. A motion counts as simple harmonic when the acceleration always points back toward the middle and grows in direct proportion to how far out the object is. That single rule forces the motion into a sine or cosine wave in time. It also fixes the period, which does not depend on the amplitude.

The three graphs line up in a fixed pattern. Displacement peaks where velocity crosses zero, and velocity peaks as the object shoots through the middle, leading displacement by a quarter cycle. Acceleration is a mirror image of displacement: largest at the extremes and zero at the centre, exactly as the defining relation demands.

A mass on a spring gets its period from mass and stiffness: larger mass slows it down, stiffer spring quickens it. A simple pendulum at small angles gets its period from string length and gravity instead: lengthen the string and the swing slows, while the mass of the bob and small amplitudes change nothing.

**Example.** To test a motion, check acceleration against displacement at several points. A pendulum pulled a little further still keeps the same period, since small swings stay approximately simple harmonic. That independence from amplitude is the signature worth checking.

**Recap.** Acceleration toward the centre in proportion to displacement makes sine motion with fixed period.

## Practice

8 questions on this page, each with its working shown.

## Needs first

- [Elastic Strain Energy and Force-Extension Graphs](https://lightmysky.com/learn/science/elastic-strain-energy-and-force-extension-graphs-mt_4WnFUmR13O)
- [Small Angle Approximations](https://lightmysky.com/learn/mathematics/small-angle-approximations-mt_fIsZes5Os3)
- [Circular Motion: Angular Speed and Centripetal Force](https://lightmysky.com/learn/science/circular-motion-angular-speed-and-centripetal-force-mt_PW1G6nYStq)

## Opens up

- [Energy in Simple Harmonic Motion, Damping and Resonance](https://lightmysky.com/learn/science/energy-in-simple-harmonic-motion-damping-and-resonance-mt_3sh7E7k7j2)