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Re: CONSERVATION OF ENERGY



Ludwik writes,

You are correct, Bob, for a single particle MODEL. But the reality of the
situation is more complicated. Two internal non-conservative forces are
involved in a process by which internal kinetic energy K (macroscopic)
is converted into thermal energy (microscopic). And if my understanding
of Bruce Sherwood is correct (his AJP article was mentioned here) it is
not at all obvious what fraction of the really-observed x should be used
to calculate the work correctly.

Ludwik,

As Bob Sciamanda, Al Bachman, and others (including myself) have
often pointed out here, the equivalence between what is often
called "pseudowork" (i.e., net external force x displacement of
the center of mass) and the change in "bulk" (or "center of mass")
kinetic energy is a purely mechanical result that is readily
derived directly from Newton's second law for *any* system; this
result is *not* restricted to point particles.

Pseudowork *must* be distinguished--as I have repeatedly argued
here and in my AJP article with Harvey Leff--from other useful
definitions of work, in particular including what I have called
here "external work" (i.e., the sum of the works done by all
external forces times the related motions of their *actual* points
of application.)

The central point of Sherwood and Bernard's article is the fact
that kinetic frictional forces always act over a distance that is
*different* from the observed displacement of either object's
center of mass. As a result the external work (which, in the
absence of heat, can be shown to be equal to the change in the
*total* energy of the system) is different from the pseudowork.

I *really* believe that the original "sliding cube on stationary
plate" problem--and, by direct analogy--the new meteor problem
have been *fully* analyzed and explained here several times.
Nevertheless, I will take one last, differently worded, stab at
it.

In the case of the cube, the negative pseudowork (force of
friction x sliding distance) is identically equal (as it must
*always* be) to the change in the cube's bulk kinetic energy.
However, since the points of contact repeatedly break free and
reestablish contact, *their* displacements are *less* than the
sliding distance. As a result, the external work is *less*
negative than the pseudowork and, therefore, the *total* energy of
the cube does not decrease by as large an amount as does the bulk
kinetic energy. That is to say, *some* of the initial bulk
kinetic energy remains with the cube (in the form of internal or,
if you really must, thermal energy.)

In the case of the always stationary plate, the pseudowork is zero
and is identically equal (as it must *always* be) to the change in
the plate's bulk kinetic energy. However, since the frictional
force displaces elements of the surface of the plate in the
direction of the sliding cube, *positive* external work is done
thereby increasing the *total* energy of the plate. (This energy
shows up as internal or, if you really must, thermal energy.)

In the case of the composite system, negative pseudowork is done
(by the single external force that holds the plate stationary) and
is identically equal (as it must *always* be) to the change in the
system's bulk kinetic energy. (Work this one out and see that it
is so.) However, no external work is done and, therefore, the
*total* energy of the system does not change.

John
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A. John Mallinckrodt http://www.intranet.csupomona.edu/~ajm
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