Physics · Oscillations

Simple Harmonic Motion

Simulation loading

Open Model Lab is preparing the live lab, controls, and graph surface for this concept.

Wrap-up

What you learned

Recommended next: Open concept testCheck whether the core ideas are ready without leaving this concept.
Read next: Oscillation EnergyEnergy view

Key takeaway

Turning points mean maximum displacement, zero velocity, and the strongest restoring acceleration at the same instant.
Equilibrium crossings mean zero displacement, zero acceleration, and the largest speed.
Amplitude changes the size of the motion, while Angular frequency changes only the timing.

Common misconception

Acceleration in SHM follows displacement, not the direction the mass is currently moving.

In simple harmonic motion, acceleration is set by displacement, not by the current direction of motion.

Keep this idea moving

Open the next concept, route, or track only when you want the current model to widen into a larger branch.

Energy viewOscillations

Oscillation Energy

Watch kinetic and potential energy trade places in simple harmonic motion while the total stays fixed by amplitude and spring stiffness.

Superposition linkWaves

Wave Interference

Superpose two coherent sources, trace their path difference to phase difference, and watch bright and dark regions emerge on the same live screen.

Wave patternWaves

Standing Waves

Track fixed nodes, moving antinodes, and harmonic mode shapes on one live string while the same probe trace shows the underlying oscillation in time.

Stay with this concept

Review, test, or free-play here when you want one more same-concept pass.

More practice optionsReview on the bench · Worked examples

Practice optionReview on the benchReplay the guided setup with the key controls and graphs in sight.
Practice optionWorked examplesCompare against solved walkthroughs after you try the setup yourself.
Practice optionFree play on the benchExperiment freely with the live controls before moving to another concept.

Reference and support

Use these calmer sections when you want a fuller explanation, examples, or accessibility support after the guided flow.

Return here when you want the slower reference pass.

Open reference and support

Equation snapshotRestoring pattern · AccelerationShow 2 equations

Keep one displacement equation and one acceleration equation in sight while you read the cycle.

Restoring pattern
$x (t) = A cos (ω t + ϕ)$
Read as: position at time t equals amplitude times cosine of omega times time plus phi
The position of the oscillator as a function of time.
Acceleration
$a (t) = - ω^{2} x (t)$
Read as: acceleration at time t equals minus omega squared times x of t
The restoring acceleration is proportional to displacement and always points toward equilibrium.

Why it behaves this way

Explanation

Simple harmonic motion is motion in which the acceleration always points back toward equilibrium and grows with displacement. Move the mass to one side and the restoring acceleration points toward the center, trying to pull it back.

The mass does not stop at equilibrium because its speed is greatest there. It passes through the center, overshoots, and the same restoring rule slows it down, reverses it, and repeats the cycle.

The stage and graphs show the same instant. At a turning point, displacement is largest, velocity is zero, and the restoring acceleration has its largest magnitude. At an equilibrium crossing, displacement and acceleration are zero while speed is largest.

Use the controls one idea at a time. Amplitude sets the turning points and the energy scale, Angular frequency changes how quickly the cycle repeats, and Phase changes only the starting point in the cycle.

Key ideas

01At a turning point, displacement is maximum, velocity is zero, and the restoring acceleration has maximum magnitude.

02At an equilibrium crossing, displacement is zero, acceleration is zero, and speed is maximum.

03Amplitude sets the size of the motion, Angular frequency sets the timing, and Phase sets the starting point in the cycle.

Quick test

Loading saved test state.

Accessibility

Open the text-first descriptions when you need the simulation and graph translated into words.

The simulation shows one mass moving back and forth along a line through an equilibrium point. Optional overlays can show the equilibrium line, a short motion trail, the velocity vector, and the restoring acceleration vector.

Changing Amplitude, Angular frequency, or Phase updates the same motion, readouts, and graphs together. The stage and graphs are synchronized views of one oscillator at the same live time.

Graph summary

The graphs show displacement, velocity, acceleration, and energy as different views of the same oscillation over time.

Together they let you compare turning points, equilibrium crossings, timing changes, and the energy swap within one live cycle.

Bench tools and share links

Keep stable concept links and exact-state sharing tucked away until you actually need to relaunch or share the bench.

Try this setup

Jump to a named bench state or copy the one you are looking at now. Shared links reopen the same controls, graph, overlays, and compare context.

Saved setups

Saved setups are a Supporter study tool. Stable concept links still work for everyone.

Checking saved setup access

Open Model Lab is resolving whether this bench can save locally, sync to an account, or open Supporter-only compare tools.

Copy current setup

Exact-state sharing is part of Supporter. Stable concept and section links still stay available.

Stable links

Progress and next steps

Keep progress signals, starter-track handoffs, and review prompts available without letting them compete with the live lesson flow.

Progress

Loading progress

Loading saved concept progress for this browser or synced account before showing completion status.

Starter track

Step 1 of 3

Oscillations and Energy

Oscillation Energy is what this track opens next.

This concept is the track start.

Also appears in Waves.

See next concept