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For K-12 Students · Coupled Oscillators

Coupled Oscillators for a diving board

Built for k-12 students learning it in middle or high school. Watch the idea come alive with plain-language steps and everyday examples — perfect for projects and homework. Simulate a diving board live below — adjust the inputs and watch it respond, right in your browser.

Coupled OscillatorsLive

normal modes & beating

Controls

Presets

Outer spring k (N/m)

Coupling spring (N/m)

Mass (kg)

Two masses linked by a spring exchange energy. Any motion is a mix of two normal modes: in-phase (slow) and out-of-phase (fast). Their beat produces the sloshing you see.

▶ Run in Python

Data Inspector

Mode 1 (in-phase)0.712 Hz

Mode 2 (out-of-phase)0.955 Hz

Beat frequency0.243 Hz

Governing equation

ω_{1} = \frac{k}{m} = 4.47, ω_{2} = \frac{k + 2 k _{c}}{m} = 6.00 rad/s (k = 20, k_{c} = 8, m = 1)

Reading this result: The beat frequency is set by the coupling: at kc=8 versus k=20, energy fully sloshes from one mass to the other every 2.1 s, and stiffer coupling speeds that exchange up.

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Frequently asked questions

Is this good for k-12 students?: Yes — this version of "Coupled Oscillators for a diving board" is framed for k-12 students learning it in middle or high school. Watch the idea come alive with plain-language steps and everyday examples — perfect for projects and homework.
Do I need to install anything?: No. It runs in any modern browser, free, with no account required.