The Quantity That Turning Keeps
Last timeWhat a Collision Keeps
A third conserved combination notices circulation rather than travel. A force aimed at a point cannot change it, which is why a skater spins faster with her arms in.
Momentum is blind to going round. Set a planet on a circular orbit and its
momentum points first one way and then the opposite way, averaging to nothing
over an orbit, while something about the motion is obviously steady. That
something is a third combination.
- where the body is, measured from the point you have chosen to take the angular momentum about
- how it is moving, in the same two directions
- the part of the motion that is sideways to the line from the chosen point, times how far out the body is. Largest for a body going straight past at a distance, zero for one heading straight in
- the total, which is one number in a plane. Change the point you measure from and you get a different number, so the point is part of the quantity
The condition for this to be conserved is strikingly weak. The force has to
point at the chosen centre. That is all. It may fall off as the square of the
distance, or rise with it like a spring, or be constant like a string being
pulled through a hole in a table. None of that matters, because a force aimed
along the line from the centre has no sideways part at the body's position, and
a force with no sideways part cannot twist.
An orbit that is not a circle
Below is a body moving under a pull toward the origin that falls off as the
square of the distance, started with a sideways push too small to make a circle.
It swings in close and far out again. The shape is drawn as a path, not as a
history.
The lesson stops here
4 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