I didn't get any takers on my problem. (Not surprising, I suppose;
my students also cringe at my "m part" problems.) At any rate,
for those few who might be interested here are my answers
(interspersed with the problem statement):
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?
No.
b) Can you determine the work done on the system consisting of
the container and its contents? If so, what is it?
Not by any definition of work that makes sense to me.
c) Can you determine the internal energy of the gas? If so, what
is it?
No.
d) Can you determine the thermal energy of the gas? If so, what
is it?
Not by any definition of thermal energy that makes sense to me.
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?
No.
f) Now can you determine the work done on the system consisting
of the container and its contents? If so, what is it?
I can determine the quantity that some people call
"pseudowork," that others call "center of mass work," and that
still others (who, I believe, are usually inconsistent in their
definition of "work") call simply "work." In any case I can
determine only the integral of the net force on the system dotted
with the displacement of the *center of mass* of the system.
I'll call it "center of mass work," W_cm. It is simple to show
(by integrating Newton's second law as is done in most
introductory textbooks) that W_cm is equal to the change in the
system's bulk kinetic energy. Thus, W_cm = Mv^2.
g) Now can you determine the internal energy of the gas? If so,
what is it?
No.
h) Now can you determine the thermal energy of the gas? If so,
what is it?
Not by any definition of thermal energy that makes sense to me.
If your answer to any of these was "cannot determine" here is some
additional 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?
Yes. The container has moved a distance Mv^2/F + L/8.
(IMO, the proof of this is not at all trivial without a clear
understanding of a variety of work-energy relationships.)
j) Now can you determine the work done on the system consisting
of the container and its contents? If so, what is it?
I'll say! Now we can determine not only W_cm but also the
values of two other useful "work-like" quantities. W_ext (the sum
of the integrals of all external forces on the system dotted with
the displacements of their points of application in the lab frame)
and w_ext (the sum of the integrals of all external forces dotted
with the displacements of their points of application in the
center of mass frame--sometimes, as in this case, a noninertial
frame.) It can be shown that W_ext is equal to the change in the
total energy of the system and that w_ext is equal to the change
in the internal energy of the system.
We find w_ext = FL/8 and W_ext = Mv^2 + FL/8.
k) Now can you determine the internal energy of the gas? If so,
what is it?
Yes. The internal energy of the gas is FL/8.
l) Now can you determine the thermal energy of the gas? If so,
what is it?
Not by any definition of thermal energy that makes sense to me.
m) If your answer to any of these is still "cannot determine"
please suggest a piece of information that would remove any
remaining ambiguity.
Tell me that "thermal equilibrium has been reached", if so,
then I would be willing to say (in this case) that the thermal
energy of the gas is equal to the internal energy of the gas,
i.e., FL/8.