Re: ENERGY WITH Q

[John Mallinckrodt <ajmallinckro@CSUPOMONA.EDU>]

On Sat, 1 Dec 2001, John S. Denker wrote:

> OK, I see how the math works out in this case.
>
> Now I have some different questions:  First, it appears
> that the result here (namely the m/F v^2 term) depends
> on finding a way to express a second integral in terms
> of the square of a first integral.
>
> One cannot in general find such an expression.  In
> this case it was possible because we were given a
> precisely constant force.  Very precisely constant.

I am glad that we seem to be making progress and I sincerely want
to solidify those gains, but I honestly don't understand the
above.  The only thing I can think of is that you are somehow
suggesting that the derived relationship between "center of mass
work" and the change in bulk translational kinetic energy results
from some kind of square of the impulse-momentum theorem, but I
could well be misinterpreting your remarks and I would appreciate
clarification if I am mistaken.

For now let me just say that, while the specific problem I posed
is indeed, as you say, rather simple with its constant external
force (not to mention its endpoint in which the CM of the gas has
shifted backwards L/4 and is at that specific time moving at the
same speed as the container), the equations I used to analyze the
problem are *completely* general.  The *only* caveat is that we
restrict ourselves to the realm of classical mechanics.

The proofs are carefully and explicitly detailed in the paper I
have so often (and so shamelessly) plugged here:  "All about
Work," Am. J. Phys., 60, 356-365, (1992).  In that paper we
1) decompose the system of interest into equivalent zero internal
degree of freedom constituents (i.e., "particles"), 2) identify
seven useful work-like sums of integrals, and 3) rigorously prove
the fully general connections between those seven work-like
quantities and changes in various types of system energy.  I would
again refer you to that paper for those proofs and would sincerely
welcome having anyone find any fault whatsoever with those proofs;
not one has ever been brought to my attention.

> Now suppose that in contrast, we were considering a
> frictional force.  Imagine the apparatus in question
> sliding along a table.  We might know the average
> macroscopic force, but we would not in any practical
> situation know the instantaneous microscopic force
> accurately enough to have any hope of applying the
> method suggested here.  Note that the spring and
> dashpot are under my control;  I could choose them
> to resonate with some high-frequency component of the
> frictional scritching and scratching.

And all of this is rigorously accounted for by the methods I have
employed.  I acknowledge the fact that we might in many cases not
know the microscopic forces and might, therefore, not be able to
make *practical* use of some of the work-energy relationships, but
that certainly does not detract from their validity. This kind of
thing happens all the time in physics.  Equations that are
perfectly general often--perhaps even *usually*--can not be
applied to any practical advantage in specific problems.  For
instance, try deriving the numerical value for the bulk-modulus of
steel starting with the "Standard Model."

> I stand by my assertion that the integral contains
> a signifcant oscillatory component.  It seems utterly
> obvious to me based on a glance at the diagram above,
> that if you start shoving on the apparatus at t=0
> the contents will shosh for a while.

Without question.  Moreover, it is interesting to see what the
various work-energy connections have to say about such a sloshing
situation.  See, for instance, example 2 in the above referenced
paper which analyzes the transient period in the free expansion of
a gas.

> I've learned something, but I am still skeptical
> that this approach is applicable to real-world
> situations.

I do appreciate this comment and I hope that I may be able to
reduce your skepticism.

John Mallinckrodt      mailto:ajm@csupomona.edu
Cal Poly Pomona          http://www.csupomona.edu/~ajm