Re: [Phys-l] Dynamic equilibrium in mechanics

[John Denker <jsd@av8n.com>]

On 09/25/2006 10:32 AM, Savinainen Antti wrote:
> I have occasionally seen the term "dynamic equilibrium" describing situations in which the object
> is moving at constant velocity and the net force on the object remains zero. 

That's all good.

> Furthermore, when
> object is at rest the equilibrium is called "static". 

Yes.

> I'm not sure, however, that there is much
> physical insight in this distinction since both cases include zero acceleration. 

That's in interesting question.  The answer is slightly tricky.

I hope we all agree that equilibrium usually means "forces in
balance".

As others have pointed out, a spinning wheel is a fine example of
nontrivial dynamic equilibrium.

It is nontrivial because it either fits or doesn't fit the usual
definition of equilibrium, depending on what reference frame you
choose.

Consider a parcel of material near the rim of the wheel.
 a) In the lab frame, this parcel is subjected to a net unbalanced
  force -- a force of constraint -- causing it to move in a non-straight
  line.
 b) In the frame corotating with the wheel, this parcel is subjected
  to a centrifugal force which just balances the aforementioned
  force of constraint.  As a result, the parcel is at rest and remains
  at rest in the rotating frame.

> Any opinions whether the distinction ... is useful in teaching
> mechanics? 

Yes and no.  It's a moving target.  Certainly dynamic equilibrium
is not a suitable starting point, not suitable for HS students in
September.

On the other hand, it would be nice if students learned enough to
be able to handle this by the end of the school year.

The issue is complicated by the fact that all-too-many texbooks
are very unhelpful on the topic of rotating reference frames.
I get pretty tired of reading claims that there is no such thing
as centrifugal force.  It's OK to say rotating frames are beyond
the scope of the course, but it's not OK to say they are beyond
the scope of physics.

Denying the existence of rotating frames just brings physic class
into ill repute.  Students are not stupid.  They have enough
experience with cars and with playground merry-go-rounds that
they *know* centrifugal forces exist.

> ... inertial frame of reference ...

Beware that "inertial frame" has two distinct meanings.
 -- In high school settings, the lab frame is often classified as
  an "inertial" frame.
 -- From a modern-physics point of view, certainly including general
  relativity but also in many cases special relativity, the term
  "inertial" is much more restrictive, referring only to /unaccelerated/
  frames.

My policy is to sidestep the terminological wrangle by avoiding the
term "inertial" as much as possible.  Instead I say "unaccelerated"
or equivalently "freely falling" when that's what I mean, and I say
"Newtonian" to refer to a frame that is either unaccelerated or
subject to an overall uniform acceleration (such as the usual lab
frame).

> From a modern point of view, the lab frame is an accelerated frame,
which gives rise to what we call gravitation.  A rotating frame is
another type of accelerated reference frame, only very slightly
more complicated than gravitational acceleration.  This gives rise
to centrifugal and Coriolis effects.

  http://www.av8n.com/how/htm/motion.html
and in particular:
  http://www.av8n.com/how/htm/motion.html#sec-centrifugal-intro

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

Folks who are not interested in tangents and messy details should
stop reading now.

There is an even messier notion of dynamic equilbrium.  Consider
a /damped/ harmonic oscillator, perhaps a mass on a spring with a
dashpot, or perhaps a sound mode in an organ pipe, taking viscosity
into account.

Harmonic motion is not terribly different from the rotational
motion considered a moment ago.  There is sinusoidal motion in
one dimension, rather than in two dimensions.

Consider a driving force F1 that suffices to drive the oscilator
at a given amplitude, A1.  This can be called dynamic equilibrium.
The concept here is more one of energy-budget in balance, rather
than forces in balance.  (It is not helpful to think about this
in terms of "average" forces in balance, averaged over a cycle.
Any periodic force and any periodic motion will average to
zero, even if the force isn't matched to the amplitude.)

There is a way to think about oscillators in terms of rotating
frames, but that is getting too far afield for now.  I will
however note that NMR and ESR theory makes heavy use of this
trick.