Pushing at the Rate a Thing Already Has
Last timeA Shape That Moves While Nothing Travels With It
Push an oscillator at its own rate and small pushes accumulate into a large swing. The response is sharp or blunt depending on how fast the system loses what it is given.
A child on a swing is pushed gently and ends up going high. Nothing about any one
push explains the result, because the pushes are small and the swing is large. What
explains it is their timing.
- the restoring pull proportional to displacement, which is what makes this an oscillator and was the whole of the first lesson
- the leak: a resistance proportional to speed, which removes energy from the motion. Without it nothing in this lesson has a finite answer
- the push, applied at a rate the pusher chooses rather than one the system has. The whole question is what happens as that chosen rate is varied
- the rate of pushing. This is the number along the bottom of the figure below, and the system's own rate is a separate number entirely
The response, and what the leak does to it
Push at a rate well below the system's own and it simply follows the push, with a
small swing. Push far above and it cannot keep up, and the swing is smaller still.
Push at its own rate and the pushes arrive in step with the motion that is already
there, each one adding to it, and the swing becomes large.
That is worth stating as a trade rather than as two facts. A system that holds on to
energy builds up an enormous response, but only when the rate is almost exactly
right, because the long memory lets a small mismatch accumulate into opposition. A
system that loses energy quickly cannot tell a slightly wrong rate from the right
one, because it has forgotten the earlier pushes by the time the mismatch would have
mattered.
The lesson stops here
5 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