Re: ENERGY WITH Q

[John Mallinckrodt <ajmallinckro@CSUPOMONA.EDU>]

On Wed, 28 Nov 2001, Carl E. Mungan wrote:

> Several people on this list have suggested alternate forms of the
> first law of thermodynamics. I would like to call on these folks to
> make up and post an introductory-level homework problem that could
> *best* be done using their alternate form. If possible, include
> numerical values and make us calculate something. The only constraint
> is that if your form of the first law has "X" in it (eg. thermal
> energy) that the problem not specifically ask us to calculate X. It
> may use X to get at something else, but shouldn't artificially force
> us to use X just because of how the problem is worded.

Thanks, Carl.  I've been working on just such a problem and
appreciate the invitation.  (I confess that this is not an really
an "introductory-level" problem and I would not give it to
introductory students.  I think--nay, I hope--that the fundamental
points of disagreement in this thread are too subtle to be
discussed in an introductory course.  Nevertheless, I think we as
"practicing physicists" should have significantly deeper grasp of
the subject and I, like Carl, am interested in seeing how others
deal with these issues.)

Here is my problem:

Consider a long narrow cylindrical container of mass M filled with
a monatomic ideal gas with a total mass equal to that of the
cylinder.  The cylindrical container has an inner length L.  The
gas particles are initially distributed uniformly throughout the
volume of the cylinder.  Both the container and the particles are
initially at rest in a region devoid of any "gravitational field."
You begin pushing with a constant force F on one end of the
cylinder along a direction parallel to its axis of symmetry. All
ensuing collisions between the particles and the cylinder are
elastic.  Some time later it is found that the speed of the
container is v.  To recap, the "givens" are M, L, F, and v.

a) Can you determine the distance that the container has moved?
If so, what is it?

b)  Can you determine the work done on the system consisting of
the container and its contents?  If so, what is it?

c) Can you determine the internal energy of the gas?  If so, what
is it?

d) Can you determine the thermal energy of the gas?  If so, what
is it?

If your answer to any of these is still "cannot determine" here is
some more information:  At time t, the center of mass velocity of
the gas is the same as that of the cylinder.

e) Now can you determine the distance that the container has
moved?  If so, what is it?

f)  Now can you determine the work done on the system consisting
of the container and its contents?  If so, what is it?

g) Now can you determine the internal energy of the gas?  If so,
what is it?

h) Now can you determine the thermal energy of the gas?  If so,
what is it?

If your answer to any of these is still "cannot determine" here is
some more information:  At time t, the center of mass of the gas
lies at a distance L/4 from the end on which you are pushing.

i) Now can you determine the distance that the container has
moved?  If so, what is it?

j)  Now can you determine the work done on the system consisting
of the container and its contents?  If so, what is it?

k) Now can you determine the internal energy of the gas?  If so,
what is it?

l) Now can you determine the thermal energy of the gas?  If so,
what is it?

m) If your answer to any of these is still "cannot determine"
please suggest a piece of information that would remove any
remaining ambiguity.