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.
What a learner can do afterwards
- 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
1 · Read
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.
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.
Acceleration toward the centre in proportion to displacement makes sine motion with fixed period.
The rewritten readings were written for this dataset.
More to read
Recognising simple harmonic motion
Simple harmonic motion has acceleration proportional to displacement from equilibrium and directed toward equilibrium. Ideal springs obey this relation; small-angle pendulums approximate it. Displacement varies sinusoidally with time. At either displacement extreme, velocity is zero. Speed reaches its maximum at equilibrium. Acceleration has the opposite sign to displacement: its magnitude is greatest at the extremes and zero at equilibrium.
A spring, a circle and the motion graphs
An ideal spring exerts a restoring force proportional to a mass's displacement from equilibrium. The resulting acceleration produces sinusoidal motion. Mass and spring stiffness determine the period; amplitude does not.
Energy offers another way to analyse the motion. A projection of uniform circular motion onto an axis also oscillates sinusoidally, relating amplitude, frequency and phase. A displacement-time graph lets you determine velocity and acceleration; a single displacement value does not determine velocity's direction.
Small swings of a pendulum
A simple pendulum has a small, heavy bob suspended from a light string. At small angles, gravity's restoring action is approximately proportional to displacement, producing nearly simple harmonic motion. String length and gravitational strength determine the period. Bob mass does not, and small changes in amplitude have negligible effect. These properties make pendulums useful in clocks and experiments. A longer string gives a longer period; changing the bob's mass leaves it unchanged. Testing these predictions checks the small-angle model.
Explore more
2 · Watch
به انگلیسیTake it off screen
Where it sits
Where this leads
Jobs that lean on this skill. Follow one to see everything it is built on.
8 questions wait behind this lesson, each with its answer explained. Every answer feeds the sky: stars light as they are learned, and dim when it is time to come back.