Two Coupled Oscillators
Two pendulums + spring. Builds on SHO & Phasors.
Two identical pendulums hang from a horizontal rod, connected by a weak spring at their midpoints. You pull the left pendulum to one side and release it from rest, while the right pendulum starts at rest.
What happens over the next minute?
The energy flows completely from left to right, then completely back! The left pendulum gradually slows to a stop while the right one builds up to full amplitude. Then the process reverses. This back-and-forth energy transfer repeats indefinitely (in the absence of damping). It’s like two normal modes — symmetric and antisymmetric — beating against each other at a frequency equal to their difference.
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Pull one pendulum and release. Watch the amplitude slowly transfer to the other pendulum. The time for a complete transfer depends on the coupling spring stiffness.
ω₁ = √(g/L), ω₂ = √(g/L + 2k/m)
The antisymmetric mode is faster because the spring stretches more, adding to the restoring force. The beat (energy transfer) frequency is ω₂ − ω₁ ≈ k/(mω₀) for weak coupling.
Mount two identical tuning forks on a shared base. Strike one and wait. The other will start vibrating as energy transfers through the base. When the first fork goes silent, the second is at full amplitude.
In molecules like CO₂, two C=O bonds act as coupled oscillators. The normal modes (symmetric stretch, antisymmetric stretch) determine which infrared frequencies the molecule absorbs — the basis of IR spectroscopy.
Two LC circuits linked by mutual inductance behave identically to coupled pendulums. Energy sloshes between circuits at the beat frequency. This is the principle behind coupled-resonator filters in electronics.
A quantum particle in a double-well potential oscillates between wells in exact analogy to coupled pendulums. The tunneling frequency is the analog of the classical beat frequency.
“Coupled oscillators reveal a profound truth: when systems talk to each other, energy doesn’t just spread out — it can flow coherently back and forth, like two friends passing a ball in perfect rhythm.”
Coupled Oscillator Lab
Build a system of two coupled pendulums. Adjust the coupling spring stiffness and initial conditions. Watch energy transfer in real time. Excite individual normal modes or create arbitrary superpositions.
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Start by pulling just the left pendulum. Watch the energy gradually transfer to the right. Increase the spring constant — the energy transfer speeds up. Now try exciting the symmetric mode: pull BOTH pendulums the same way. They swing together forever with no energy transfer. Try the antisymmetric mode: pull them in opposite directions. Again, no transfer! These are the pure normal modes. Any other initial condition is a superposition of these two modes.
Normal modes are the natural ‘languages’ of coupled systems. Any motion, no matter how complex, can be decomposed into normal modes. In each mode, all parts oscillate at the same frequency with fixed phase relationships. This idea generalizes to systems with any number of oscillators — and ultimately to waves.