Re: Atwood's machine problem

[John Denker <jsd@AV8N.COM>]

Ken Caviness wrote:

[snip analysis of dead-end approach]

> You are right, we don't assume that r is straight down.
> Its downward component is L, its horizontal component is R (the radius of the
> pulley).

That's the key.

> No external torque means angular momentum is conserved.

Yup.

> Wait, what if we don't use a pulley, just a frictionless hook?  If R=0, the
> angular momentum is already zero without putting any constraint on v.

The R=0 limit is pathological.  The angular momentum argument fails
when R=0.

> Oh, but notice that if this problem really needs conservation of angular
> momentum to do it correctly, it shouldn't be appear in the book in the section
> dealing with linear momentum.

First of all, "needs" is too strong a word.  I visualize science
as a huge lattice of facts (the nodes) interconnected by logic
(the struts).  The structure is highly over-constrained, so that
any particular node can be reached in many different ways.

It is an excellent exercise to see how many different solutions
you can find to a given problem.

Secondly, I disfavor of the notion that the questions at the
end of a given chapter should be restricted to issues covered in
that chapter.  Students should be taught from Day One that they
should (unless otherwise stated) attack every problem with *all*
the resources at their disposal.  (There are exceptions, but they
should be rare, and should be explicitly tagged as exceptions.)


> Perhaps the suggested analogy to two iceskaters
> pulling on a rope is the easiest way to handle the problem after all.

Yes, I prefer that approach, but I was careful to call it my preferred
approach, not the only valid approach.

=====

If the following tangential remark doesn't appeal to you, just
ignore it:

The monkey/bananas problem is not very practical, but there *is*
a rather similar set-up that is 100% practical.

In analog electronics, there is such a thing as a _current mirror_.
The circuit has two branches, which are hardware-wise highly
symmetrical.  But in operation they are anti-symmetrical, in the
sense that if the current in one branch increases, the current in
the other branch must decrease.  Current mirrors are a very common
building-block, useful for building differential amplifiers.  Think
for instance of the differential inputs on an op-amp.

My favorite references for such things are
 -- Horowitz & Hill, and
 -- Tietze & Schenk.

I know most people buy op-amps and don't worry about how to build
them ... but if you wind up in a physics research lab, you're always
pushing the limits, and sooner or later you'll wind up in a
temperature range or a frequency range or some such where you
can't buy what you need, and it's nice to know you can build it if
you have to.

Also, *somebody* has to design the next generation of products, so
there will always be some market for designers.