Coupled pendulums and normal modes
Couple two pendulums with a spring and their energy "moves house" back and forth.
Two identical simple pendulums are joined by a weak spring. When they swing in phase, the spring is never stretched, so the frequency equals that of a single pendulum, ω₁; when they swing in opposite directions, the spring is repeatedly stretched and compressed, and the frequency rises to ω₂. These two collective motions, which keep their form unchanged, are called normal modes. When only one pendulum is pulled aside, the motion is a superposition of the two modes with slightly different frequencies, so beats appear: the energy passes completely from pendulum A to pendulum B and then back again.
Angles of the two pendulums
The envelopes trade off: when A comes to rest, B swings the most.
Think about it
The softer the spring, is the time needed for energy to transfer from pendulum A to pendulum B longer or shorter?
Longer. The smaller k is, the closer ω₂ gets to ω₁, so the beat period 2π/(ω₂ − ω₁) grows longer and the energy transfers more slowly; at k = 0 the two pendulums do not affect each other and the energy never transfers.
Model assumptionsSmall angle (8°) · 1 kg bobs · spring attached between the two bobs; both pendulums hang vertically when it is at its natural length · damping ignored
Data log
Set the parameters and click “Record data”. Record several rows with different parameters to plot their relationship and fit a straight line; pause and record at different times to study how a reading changes with time.