Look around you. A pendulum swings. A guitar string vibrates. A bridge sways in the wind. An atom in a crystal lattice jiggles in place.

All of these are oscillations — motion that repeats. But one type is so fundamental, so deeply connected to the structure of physics, that it has its own name: Simple Harmonic Motion.

The restoring force

Push a block attached to a spring and release it. It moves back — past the centre — slows — turns — comes back. The spring always pulls it toward the middle.

SpringMass amplitude 80 k 100 m 1
Show spring block wall equilibrium displacement-arrow graph-x
Hide energy readouts velocity-arrow acceleration-arrow

This is SHM. The restoring force is always directed toward equilibrium, and its strength is proportional to how far the block has moved away:

$F = -kx$

The negative sign means the force opposes the displacement. $k$ is the spring constant — how stiff the spring is. This is Hooke's Law.

The motion is sinusoidal

Watch the graph. The displacement traces a perfect cosine wave:

$x(t) = A\cos(\omega t + \phi)$

Where $A$ is the amplitude — maximum displacement from centre.

Period is independent of amplitude

This is the most counterintuitive fact about SHM. Pull the block further — it moves faster by exactly the right amount to still complete one cycle in the same time.

Small amplitude:

SpringMass amplitude 40 k 100 m 1
Snapshot at extreme
Show spring block wall equilibrium displacement-arrow
Hide graph-x energy readouts

Large amplitude:

SpringMass amplitude 120 k 100 m 1
Snapshot at extreme
Show spring block wall equilibrium displacement-arrow
Hide graph-x energy readouts

Same spring, same mass. Both complete one oscillation in $T = 2\pi\sqrt{m/k} \approx 0.63$ s — regardless of how far the block was pulled.

This is the isochronous property of SHM.

Test

Static — frozen diagram, no interaction

SpringMass amplitude 0 k 100 m 2 orientation vertical
Snapshot t 0
Show spring block wall equilibrium

Animated — plays automatically, sliders only

SpringMass amplitude 60 k 100 m 2 orientation vertical
Show spring block wall equilibrium graph-x

Interactive — student drags the block live

SpringMass k 100 m 2 orientation vertical
Interactive
Show spring block wall equilibrium displacement-arrow energy KE PE E