Problem — Two coupled swings

Problem

Two identical swings, side by side, are connected by a weak horizontal spring. Initially only the first one oscillates. What is observed over time? Discuss the phenomenon of beats between coupled oscillators.

Solution

The coupled system has two normal modes: the in-phase one (frequency $\omega_+$) and the anti-phase one ($\omega_-$). Since the spring is weak, the two frequencies are close ($\omega_- \gtrsim \omega_+$).

The initial state (only the first swing oscillating) is the superposition of the two modes with equal amplitude. The two modes, having slightly different frequencies, slowly drift out of phase over time: the energy MIGRATES from one swing to the other.

The first swing slows down until it stops while the second reaches maximum amplitude, then the process reverses. It is a beat between oscillators, with exchange period:

$T_b = \frac{2\pi}{\omega_- - \omega_+}$

A weaker spring $\Rightarrow$ $\omega_-$ and $\omega_+$ closer together $\Rightarrow$ slower exchange.

$\ev{\text{energy migrates back and forth between the two swings with } T_b = 2\pi/(\omega_- - \omega_+)}$

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Topics: Oscillazioni e moto armonico Concepts: Modi normali · Battimenti · Pendolo semplice Objects: Pendolo semplice