Re: [Phys-l] Coriolis effect puzzlement

[John Denker <jsd@av8n.com>]

Consider a subsystem that is /closed/ with respect to momentum.
That is to say, there is no momentum flow across the boundary
of the subsystem.  Equivalently, that means that for every force
within the subsystem, there is an equal and opposite force
*within* the subsystem.

More-or-less equivalently, we can restrict attention to a
subsystem where every torque takes the form of a /couple/ i.e.
two equal-and-opposite forces acting on the ends of a lever
arm.

For such a subsystem, there is a fundamental theorem that says
the torque around any pivot-point X1 is the same as the torque
around any other pivot-point X2.  These points can be anywhere
in the universe, not necessarily within the boundary of the
subsystem.

  More generally, the theorem says the torque about X1 is
  equal to the torque about X2, plus a term involving the
  amount of unbalanced force and the lever-arm between X1
  and X2 ... but that is more than we need to know for
  present purposes.

> > > A skater stands way off center on a rotating ice rink, partaking only of the 
> > > platform motion, and holding weights in her outstretched hands.
> > > If she brings in her arms, will she begin to spin about her platform 
> > > position?

On 12/02/2011 12:50 PM, Paul Nord wrote:

> That one's got Mythbusters written all over it!

Gaaack.  I hope not.

===============

The aforementioned theorem gives us another way of answering
the question.  Rather than considering the angular momentum 
(L) itself, consider the torque (dL/dt).

Consider the subsystem consisting of the skater (including
the weights).  This subsystem is closed with respect to
momentum.  No rocket thrust crossing the boundary.  No
applied magnetic fields or anything like that.

Therefore the dL/dt of this system is independent of what
we choose as the axis for defining L.

=====================

Here's another way of looking at it:

Consider the skater's view of the world, including her
view of the spectators, and her view of the fixed stars,
and her view of the gyroscope that hangs from her
necklace.  In the initial situation, where she partakes
of the platform motion, it is totally obvious to her 
that she is spinning.

If the platform is bowl-shaped in just the right way
("hydrostatic equilibrium") she won't be aware of the
centrifugal force, in which case she won't know or care
where the pivot-point is ... but she will *always* know
that she is spinning.  She is spinning relative to any
inertial frame.

===

Similarly, the moon /orbits/ the earth once per month.
It also /spins/ on its axis once per month.  If you
are standing on the moon, it is very easy to figure
out that the moon is spinning.

The skater in question is like the moon in the sense
that her orbital rate is the same as her spin rate.
Relative to the rotating platform, she is not spinning,
but I was asked to analyze the system in an inertial
frame ... and in *any* inertial frame she is spinning.
She can't tell by looking at the platform that she is
spinning, but she can tell if looks at almost anything 
else, including a gyroscope or any inertial reference
frame.

====================

This is a famous result in fluid dynamics.  If you have
a bunch of fluid that is undergoing rigid-body rotation,
every parcel of that fluid is spinning.  There is a
constant _density of circulation_ throughout that bunch
of fluid.  There is a constant density of vortex lines
(lines per unit area).

Mathematically, this is closely analogous to having a
region of constant magnetic field, in which case there
is a constant density of field lines (lines per unit
area).

As Feynman was fond of saying, the same equations have
the same solutions.

A good reference for all of this is Feynman volume II
chapter 40, "The Flow of Dry Water".

============

Suppose you have two bowling balls in outer space, connected
by a string, orbiting around the system CM like a barbell, 
once per second.  I claim that each ball is also spinning 
around its own center, once per second.  I further claim
that if the string snaps, each ball moves away along a
straight line, no longer orbiting the system CM ... but
still spinning!

===============================================

You can begin to see why I thought this result was obvious.
I have at least five, maybe ten ways of knowing it, based
on fluid dynamics, based on the moon analogy, based on
the change-of-pivot theorem, et cetera.

I feel bad about situations like this.  I hurt my own
feelings.  I just hate it when knowing the material
interferes with being able to explain the material.  I
really should have been more careful in this case.  All
the warning signs where there.  For starters, if somebody
asks a question and you cannot imagine where the question
is coming from, it's a warning that simply answering the
question is not going to be good enough.  There's some
piece of background knowledge missing, maybe five or ten
pieces of background knowledge;  otherwise the question
would never have come up.  The correct strategy is to 
peel the onion layer by layer until you figure out what 
the underlying issue(s) might be.

This is not easy in person one-to-one, and it is even
harder in a many-to-one classroom setting, and harder
still via email, when the interaction is more clumsy and
more delayed.