Re: Centrifugal Force

[David Bowman <dbowman@tiger.gtc.georgetown.ky.us>]

A. R. continues:
> > ...
> > I don't know if it is so much a commitment to inventing mystical forces, as
> > it is the absence of a rigid doctrinaire commitment to one particular
> > absolute true definition of a true force.
>
> Ah, and all this time I thought a single definition would be useful.  But
> no one is saying definitions have truth value, I hope.  I have repeatedly
> said definitions are free, but it seems to me we should settle on one if
> we are to communicate among ourselves and to our students.  It seems to me
> that force is too important and fundamental a physical concept to be left
> in the murky state being proposed. 

Apparently I wasn't clear in communicating what I meant above.  I did not
mean to seem to advocate free usage of multiple definitions of force (esp.
in front of introductory students).  I'm in favor of using a single
precise definition of force.  Rather, I meant to not be committed to a
particular definition (namely, yours, to be blunt) of a force with a
particularly high qualification standard which takes no (or little) notice
of the functional roles that various quantities play in the mathematical
descriptions of phenomena.

> > It is rather a holding to a looser
> > concept of a force based on utility and functional role in the mathematics
> > of a given description of the situation.
>
> I see.  This is very illuminating, and clears up exactly where the
> difficulty has been in all this lengthy discussion.  We must allow a
> looser concept of force.  That should clear up all the other discussions
> going on, too.  Allow a looser concept of current, charge, mass, weight,
> Etc.  It is a great problem solver.

Apparently my choice of the word "looser" was ill-advised since you have
taken my meaning in a way that I did not intend.  I *want* a precise
definition of force.  I meant I want it to be *easier* for a quantity to
qualify as being a force under my definition than under yours.  I also
meant that the definition chosen ought to more strongly consider the
values of utility and mathematical function than your definition does.

Since you brought up those other perennially divisive physics concepts,
let me state my philosophy of defining physics concepts, especially, since
I recently argued on the side of a higher (i.e. less loose) definition
regarding the definition of 'current' in the displacement current
discussion.  To me the main guiding principal values to be consulted in
defining the various concepts of physics are: 1) logical consistency,
2) utility (e.g. ease of initial student understanding, simple
extensibility to more sophisiticated definitions of more advanced
generalizations of the concept at a more advanced level with a minimum of
relearning and reorientation, having a useful conceptual niche in the
theoretical structure of the theory, lack of needless but possessing
useful qualifications or restrictions, etc.), and 3) functional role in
the mathematical formulation of the theory.  I do not necessarily take
'ease of qualification' as one of these main guiding values.  The
stringincy or height of the bar of qualification in a particular case will
depend on the other values listed above.

In the case of displacement current, I argued for a higher bar in the
definition of 'current' because the easy low bar definition, IMO, violated
some of these values (e.g. #3).  The displacement current in E&M does not
really mathematically act like how normally accepted currents function
(despite a deceptive appearance as a source of a magnetic field).  What
makes a quantity function as a current is a carrying quality whereby some
quantitatively accountable stuff (however abstract or concrete) is being
spatially transported.  A useful mathematical role for a current is as a
particular transport term in a generalized continuity equation for the
stuff.  (Acting as a source for some other field is not necessarily
mathematically characteristic of a generic current, just consider a
probability current, an entropy current, a fluid current, etc.)

In the case of the definition of 'force' my low-bar definition, IMO,
obeys the above guiding values better than A. R.'s high-bar definition.
In any event, this debate, as most of the other perennially favorite
debate topics, seems to revolve around semantic esthetics rather than
around physics.

> A pedagogical question: Do you not introduce Newton's laws I & II to your
> physics majors  with the qualification "relative to an inertial frame"
> included?

Yes, especially, at first.  Later when discussing accelerated frames I
tend to point out that one can formulate generalized versions of these
laws whose domain of applicability is extended to such frames via the
introduction of those frame-induced forces that you dislike so much.

> ...                                      If you do include such a
> qualification (and I obviously do), then not calling frame-induced
> accelerations forces results in no violation of the laws.

Good point.  I should have been more careful with my use of the term
'violate'.  Rather than saying N1 and N2 are violated in non-inertial
frames using your definition of force, I should have said that they cease
to apply since the hypothesis of an inertial frame is not satisfied.
The real experimental problem of distinguishing an inertial from a
noninertial frame is present in both of our ways of discussing the
situation.  The main difficulty is with our ablity to be sure to recognize
and account for all possible physical interactions that can possibly exist
in a given situation.  For instance, suppose that there is a "5th force"
which is a long range physical interaction that couples to an object's
net baryon number.  Such a force (being essentially directly proportional
to the mass of an object) will tend to be hard to distinguish from a
gravitational interaction or a frame-induced 'fictitious' force, and
identifying just which frame is an inertial one could be problematic for
either of us in such an instance.

BTW, you keep calling the frame-induced noninertial terms that appear in
the equations of motion for objects described in accelerated frames as 
"accelerations".  They may under certain circumstances may act as
an acceleration (the circumstances being that all other terms in the
equation of motion solved for the object's acceleration happen to cancel
out), but they are not accelerations in general.  Consider the previous
case of the whirling mass on the string as described in the frame in which
the mass is at rest.  In this frame the centrifugal term (force,
acceleration, whatever you want to call it) cancels against the string
tension so that the system mass experiences *no* acceleration.  Yet the
centrifugal term is present.  How can this legitimately be called an
acceleration?  Maybe you mean that it is a bare tendency to produce an
acceleration in the absence of other terms.  But if so then such a
description sounds awfully like a force to me.

> I thought progress was being made, but now I'm not so sure.  When a "quack
> test" for forces was proposed, I got my hopes up, because that usually is
> very easy to apply.  But when I saw that the proposed test allowed
> Coriolis acceleration to masquerade as force, even though it has been
> agreed that a) it does no work, b) it has no third law counterpart, and c)
> it does not cause anything to deviate from inertial motion, my hopes were
> dashed.  To teach such would be like teaching students to call a cat a
> duck, even though the feline does not waddle, has no feathers and can't
> quack.  

Nice try.  But your definition of force would presumably disqualify the
magnetic force between moving charges on criteria a) and b) as a force
as well.  Criterion c) is a red herring anyway since my definition of a
force does not require that forces exist in inertial frames since the
general transformation properties of forces are very complicated and any
force can be made to vanish under appropriate transformations.*  Rather,
does it tend to produce an acceleration in an object (which is otherwise
left to itself) in whichever frame is being used at the time?  Does it
appear as an appropriate derivative of the Lagrangian/Hamiltonian for the
system in whatever frame is being used?  Does it act to change the
momentum (again defined via a differentiation of the Lagrangian/
Hamiltonian) in whatever frame coordinate system is being used at the
time?  According to these questions the Coriolis force has quacks, has
feathers, and waddles.  Your criteria a)-c) are more related to details as
the color of the neck band, web structure of the feet, tail feather
markings, etc.

*(Consider that the Hamiltonian vanishes identically under a canonical
transformation generated by the characteristic function of Hamilton-Jacobi
theory.  In this case all the new coordinates and momenta for the system
are constants of motion and all forces vanish no matter how complicated
the physical interactions are among the constituent particles.)

> The argument does in the end seem to be over definitions -

Agreed.

>                                                           - apparently
> whether they should be loose or not  -

Not quite. Rather it seems to be over how they should incorporate
conflicting visions of esthetics and utility (and maybe ontology).

>                                       -  and I conclude there is still a
> lot of room for advancement in the fundamentals of our profession.

This may be true, but I don't conclude that from our discussion.

David Bowman
dbowman@gtc.georgetown.ky.us