Ropes, Wheels and Forces That Do No Work
Last timeWhy a Spinning Book Tumbles
A rope ties two bodies into one number, and the pull in the rope does no work, so it never appears in the energy total and never has to be found.
Hang a weight on each end of a rope and lay the rope over a wheel. Let one weight
be heavier. Everything about what happens next can be had without ever finding out
how hard the rope is pulling, and the reason is worth more than the answer.
Start by counting. Two weights, each able to move up and down, looks like two
numbers. But the rope has a fixed length, and it is over a wheel, so if one weight
descends by a hand's width the other rises by exactly a hand's width. That is not
an approximation or a special case. It is true at every instant of the motion, and
it is one equation relating the two positions.
- how far each weight hangs below the wheel, measured along the rope
- the length of the rope, which does not change, and that is the entire content of the constraint
- what the constraint says about motion rather than position: one weight's speed is the other's with the sign turned round
One equation on two unknowns leaves one unknown. This is the first thing a
constraint does, and it is purely a matter of bookkeeping: each relation that holds
for all time removes one of the numbers you would otherwise have to track. A bead
threaded on a wire has one number instead of three. A rigid body has six instead of
three for every atom in it. Mechanics is mostly the study of systems where the
constraints have done most of the work before the physics starts.
The force nobody needs
Now the part that is not bookkeeping. Write the energy of the machine: the two
weights are moving, so there is a moving part, and they are at heights, so there is
a height part. The rope also pulls on both of them. Does the rope's pull belong in
the total?
Follow the energy it moves. The rope pulls upward on the descending weight while
that weight travels downward, which takes energy out of it. The rope pulls upward
on the rising weight while that weight travels upward, which puts energy into it.
The two pulls are the same size, because it is one rope and the wheel is ideal. The
two distances are the same size, because that is what the constraint said. So the
two amounts are the same size with opposite signs.
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