Re: [Phys-l] Sharing a problem for students

[Ludwik Kowalski <kowalskil@mail.montclair.edu>]

Dec 21, 2007, at 11:37 AM, Bob Sciamanda wrote:

> If you do a search (Google, Wikipedia, etc) you will find "dynamic
> equilibrium" usefully defined
> only for chemical and biological interactions.  In Mechanics a
> somewhat related set of  useful concepts is stable vs unstable
> equilibrium.
>
> Note that in mechanics we say that an object at rest under the action
> of a set of forces which sum to zero is in a state of static
> equilibrium.  This same object could be said to be in a state of
> dynamic equilibrium when viewed from a different frame, in which the
> object is in motion.  This distinction does not add much and AFAICT is
> not used.

Consider a cone with its flat base in contact with a horizontal table 
top. Try to tilt it slightly and notice that a restoring torque is 
produced. That is why we say that the equilibrium is stable. The 
equilibrium is said to be unstable when the cone is turned upside down. 
In this configuration, even a tiny angular tilt of the axis will 
produce a destructive torque. And the equilibrium is said to be neutral 
when the cone is resting on its side. That how most of us were 
introduced to the concept of stable and unstable equilibrium.

A weight at rest, supported by a vertical spring, is also said to be in 
stable equilibrium. Try to displace it slightly, either up or down, and 
a restoring force appears. The equilibrium, we say, is stable in the 
position of minimum potential energy. The same is true for a pendulum, 
or for a more complicated system. I am looking at Figure 8-22 in 
Giancoli’s textbook; it shows a potential energy curve with two minima 
(two possible stable equilibrium positions). Referring to one minimum, 
x0, the author writes: “If the object at rest at x=x0 were moved 
slightly to the left or right, a nonzero force would act on it in the 
direction to move it back toward x0. An object that returns toward its 
equilibrium point when displaced slightly is said to be at a point of 
stable equilibrium. Any minimum in the potential energy curve 
represents a point of stable equilibrium. . . . Points like x4, where 
the potential energy curve has a maximum, are points of unstable 
equilibrium.” An object is said to be in neutral equilibrium, when 
small virtual displacements do not result in a change of its potential 
energy.

Yes, I know that this is common knowledge. That is why it is a good 
starting point. Consider a solar system in which a single planet (mass 
m) revolves the sun (mass M>>m) along the circular orbit of radius R. 
The potential energy U(r)=-G*M*m/r increases (becomes less negative) 
when r becomes larger (U becomes zero at infinity). Thus the r=R does 
not represent a minimum. And yet, we know, that the orbit is stationary 
(durable, stable, unchanging). How to reconcile this undeniable fact 
with a statement that stability calls for a minimum of potential 
energy? A tiny change in r would not create a restoring force toward 
r=R. It would simply changes the orbit, from circular to elliptic. The 
concept of “dynamical equilibrium” was probably invented to describe 
stability of orbits.  I have a heretical idea on how to reconcile the 
apparent paradox. But first I want to know what others think. I do not 
to want to be embarrassed again.

By the way, the U=-G*M*M/r is also applicable to a double star system 
(two identical stars of mass M on a circular orbit of radius R). This 
system seems to be simpler than a solar system; the center of mass is 
always at rest, not only when M>>m. In rest with respect to what? With 
respect to an absolute frame of reference invented by Newton.
_______________________________________________________
Ludwik Kowalski, a retired physicist
5 Horizon Road, apt. 2702, Fort Lee, NJ, 07024, USA
Also an amateur journalist at http://csam.montclair.edu/~kowalski/cf/