NORMAL MODES
Two carts, two normal modes — ω_s=√(k/m) in step, ω_a=√((k+2k_c)/m) opposed. Start one and their motion envelopes trade prominence and return.
Use the simulation above to change the variables and play through the guided stages. The explanation below describes the default starting values; the simulation updates its explanation as you experiment.
Setup
Two identical carts, three springs in a row, a wall at each end. On its own each cart is a simple oscillator — but the middle spring lets them feel one another, and that changes everything about how the pair behaves.
Two modes
There are exactly two motions that stay perfectly sinusoidal. If the carts move together the middle spring never stretches, and they oscillate at ω_s = √(k/m). If they move oppositely it stretches double, and they run faster at ω_a = √((k+2k_c)/m).
Superpose
Every other start is a blend of the two normal modes. A one-cart release gives equal mode-coordinate amplitudes, but the faster antisymmetric mode stores more energy whenever the coupling is nonzero; the calculator prints that split explicitly.
Slosh
The bars track each cart’s displacement-amplitude envelope, not a local energy assignment. At the first complete envelope swap one cart’s displacement envelope vanishes while its instantaneous kinetic energy need not; the animation ends there continuously and labels the distinction.
Audit
Audited: the two modes are pure SHM to twelve digits, total energy is conserved as it crosses over, and the beat period is exactly 2π/(ω_a−ω_s). Independently confirmed by integrating both coupled equations directly — the measured mode frequencies land on √(k/m) and √((k+2k_c)/m).