--- title: "Coupled Oscillators and Normal Modes" description: "Two masses joined by springs no longer have one frequency: the motion splits into normal modes, each a pattern of the whole system oscillating at a single frequency. Any starting motion is a combinati" canonical: https://lightmysky.com/learn/science/coupled-oscillators-and-normal-modes-mt_vnTGOIqZzP source: https://lightmysky.com/learn/science/coupled-oscillators-and-normal-modes-mt_vnTGOIqZzP.md retrieved: 2026-09-12 --- > **Agent view.** This is the Markdown twin of the page, for tools and assistants. > When to use this site, and the call that answers each job: https://lightmysky.com/agent-instructions.md > API description (OpenAPI 3.1): https://lightmysky.com/openapi.json · Authentication: https://lightmysky.com/auth.md > Pricing: https://lightmysky.com/pricing.md · Catalog: https://lightmysky.com/llms.txt · Full catalog: https://lightmysky.com/llms-full.txt > Every machine-readable file on this domain: https://lightmysky.com/.well-known/ai-catalog.json > Ask for Markdown with `Accept: text/markdown`, a `.md` address, or `?mode=agent`. # Coupled Oscillators and Normal Modes Two masses joined by springs no longer have one frequency: the motion splits into normal modes, each a pattern of the whole system oscillating at a single frequency. Any starting motion is a combination of those modes. Subject: Science · Area: Forces & Motion · Ages 19 to 20 Page: https://lightmysky.com/learn/science/coupled-oscillators-and-normal-modes-mt_vnTGOIqZzP ## Ready when they can - Writes the coupled equations for two masses and springs and finds both normal mode frequencies - Describes the in-phase and out-of-phase patterns and which frequency belongs to each - Expresses a general starting condition as a sum of normal modes ## Lesson: Two masses, two favourite dances You tuned a single swing to its own frequency. Join two masses with springs and their equations mention each other, so neither mass solves alone. Add and subtract the equations and two special combinations pop out. Each combination oscillates at its own single frequency. Those are the normal modes, and every other motion is just a sum of the two with weights set by the starting push. Frequency belongs to the whole pattern, not one mass. **Example.** Name the two patterns. In phase: both masses glide the same way together, the coupling spring never stretches, and this mode runs slower. Out of phase: they move opposite, the coupling spring stretches extra, and this mode runs faster. Each mode alone is ordinary simple harmonic motion, where acceleration points home with strength proportional to displacement. Start the system in a mixture and the two frequencies beat against each other, so energy sloshes back and forth between the masses. Pull only one mass aside and you excite both modes at once, which makes the motion breathe. Displace both masses equally in the same direction and you get pure in phase motion at the slower frequency. **Tip.** Reason mode by mode: master the single oscillator and each mode feels familiar. A mode is a pattern of the whole system, never one mass with its own private frequency. Name the pattern first, then attach its frequency. **Recap.** Coupled motion splits into two whole-system patterns, each ringing at its own frequency. ## Practice 8 questions on this page, each with its working shown. ## Needs first - [The Centre of Mass and the Motion of a System](https://lightmysky.com/learn/science/the-centre-of-mass-and-the-motion-of-a-system-mt_fHO_KrMNKo) - [Driven Oscillations, Resonance and the Quality Factor](https://lightmysky.com/learn/science/driven-oscillations-resonance-and-the-quality-factor-mt_FmpOTEZFy4) - [Combining a Sine and a Cosine into a Single Wave](https://lightmysky.com/learn/mathematics/combining-a-sine-and-a-cosine-into-a-single-wave-mt_IwQkNDRWyq) - [Eigenvalues and Eigenvectors](https://lightmysky.com/learn/mathematics/eigenvalues-and-eigenvectors-mt_TVqqaw11qa) ## Opens up - [Phonons and the Heat Capacity of Solids](https://lightmysky.com/learn/science/phonons-and-the-heat-capacity-of-solids-mt_WXZPdz_JTd)