Two Joined Pendulums and the Two Rhythms Hiding in Them
Last timeThe One Rhythm Nearly Everything Falls Into
Join two pendulums with a spring and the motion looks irregular, with the swing migrating from one to the other. Two combinations of them turn out to be perfectly regular.
Hang two identical pendulums side by side and join them with a spring. Pull one
back, leave the other hanging, and let go.
What happens next is not what either pendulum does alone. The one you started swings
a few times and then begins to die away, while the one you left alone begins to
swing. After a while the first is nearly still and the second has the whole of the
motion. Then it reverses, and the swing comes back.
Nothing is being lost. The swing is moving between them through the spring, and it
will keep moving as long as the pair keeps going.
total energy of the pair0.053% of it gone by the end, at every setting of every slider
- stepped by
- v1 → v1 - 0.1*(x1 + c*(x1 - x2)) v2 → v2 - 0.1*(x2 + c*(x2 - x1)) x1 → x1 + 0.1*v1 x2 → x2 + 0.1*v2
- each rule reads
- whatever the rules above it have already produced
- started at
- v1 = 0.0, v2 = 0.0, x1 = 1.0, x2 = 0.0
- held steady
- total energy of the pair = 0.5*(v1*v1 + v2*v2) + 0.5*(x1*x1 + x2*x2) + 0.5*c*(x1-x2)*(x1-x2)
- drift
- 0.053% over the run, against 0.20% allowed; swings 7.3% and returns
Two combinations that refuse to be complicated
Now do something that looks like bookkeeping and turns out to be the whole answer.
Instead of watching the two pendulums, watch two numbers made out of them: their sum
and their difference.
- how far each pendulum is from hanging straight down
- the sum. This is large when both pendulums are on the same side and zero when they are opposed
- the difference. This is large when they are opposed and zero when they are in step
- the first pendulum rebuilt out of the two combinations, which is what makes the change of description reversible
Here is the same calculation as the figure above, nothing recomputed, with the two
combinations drawn instead of the two pendulums. There is one more thing to set
this time: the second pendulum can be pulled aside before release as well as the
first, and the sliders left alone give exactly the run above.
The lesson stops here
3 more paragraphs to go
You have read the opening. The rest of the argument, the problems that check whether it landed, and the lines worth keeping at the end all come with a plan.
The first lesson of every course in the library reads the whole way through, free, so you can see exactly what the rest of them are.
See the planThe contentsThis is the reading half
Starting the course gives you your own copy of it. Every idea on every page has problems standing under it, marked with a reason rather than a tick, and any sentence you do not believe can be opened and argued with. None of that can happen on a page nobody owns.
The contents