Re: conservation in rotating frames

[John Denker <jsd@MONMOUTH.COM>]

At 12:36 PM 10/21/99 -0400, Bob Sciamanda wrote:
> If I try to play a game of billiards on a carousel, I will encounter
> fictitious centrifugal and coriolis forces.  "Fictitious" because there is
> no agent for these forces, momentum is not conserved, etc.  Eg., if I
> carefully set a cue ball down on the table (even an "air table") it
> mysteriously acquires "un-caused" horizontal momentum.

I'm going to make a rather subtle point here.  If you don't follow the
argument, don't worry -- it's not super-important.

The point is that more things are conserved in rotating frames than you
might at first guess.

1) Let's start with a rotating frame comoving with the center of mass of
the whole system (billiard table, balls, and everything).

1a) There will be a centrifugal field, but we can treat that almost the
same way we treat the gravitational field.  We can re-orient the billiard
table so that its surface is an equipotential, to a good-enough
approximation.  We define this new orientation to be "horizontal".  That
means we can set a ball on the table and it will not undergo any un-caused
movement in the redefined horizontal direction.

1b) Now set two billiard balls on the table, with a firecracker sandwiched
between them.  Nothing is moving in the rotating frame, so there are no
Coriolis forces.  When the firecracker goes off, the two balls zoom off in
opposite directions.  Each experiences a Coriolis force.  The interesting
thing is that the two Coriolis forces are equal and opposite.  There is a
law of "conservation of coriolicity".

2) In a frame not comoving with the center of mass, i.e. a frame where the
system as a whole has nonzero momentum, then there is an overall "global"
Corolis term, but that can be recognized and subtracted off analytically,
and everything that remains obeys a conservation of coriolicity law.

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At 11:39 PM 10/20/99 -0400, Bob Sciamanda wrote, regarding nonconservation:
> This is why centrifugal and other inertial forces are called
> "fictitious" in traditional Newtonian mechanics.

Since I disagree about the role of nonconservation, I ought to give an
alternative explanation for why the "centrifugal force" is sometimes called
a fictitious force.

My answer is that all too often, the thing called "centrifugal force" is
not a force at all.  It's an acceleration.  If you want to stay out of
trouble, you might want to steer clear of the term "centrifugal force".
The usual remedy is to call it the centrifugal field.  This supports the
analogy between the centrifugal field and the gravitational field.
(Gravitation is not a force-field either -- it's an acceleration -- force
per unit mass).

______________________________________________________________
copyright (C) 1999 John S. Denker             jsd@monmouth.com