Re: CONSERVATION OF ENERGY

[John Mallinckrodt <ajmallinckro@CSUPomona.Edu>]

[From my end this appears to have been lost in the aether so I have 
taken the liberty of making a few revisions and reposting.  Apologies 
if it shows up elsewhere as a duplicate. ]

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 translational" (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 due to external 
forces with each one calculated as force times the related motion 
of the *actual* point of application.)

The central point of Sherwood and Bernard's article is the fact 
that kinetic frictional forces always act over distances that are 
*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 pseudowork (force of friction x 
sliding distance) is negative and 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 effective 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 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, the pseudowork (done by the 
single external force that holds the plate stationary) is negative  
and 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.

Finally, for the composite system, it also turns out to be useful 
to consider what I have called the "*relative* external work" 
(i.e., the sum of the works due to external forces with each one 
calculated as force times the related motion of the actual point 
of application *relative* to the center of mass.)  This work is 
always (in the absence of heat) equal to the change in what I like 
to call the "internal energy" (i.e., the total system energy minus 
its bulk translational kinetic energy.)  In this case the relative 
external work on the system is equal and opposite to the 
(negative) pseudowork.  Thus, the *decrease* in the system's bulk 
kinetic energy is exactly accounted for by an *increase* in its 
internal energy.

John
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