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Re: ENERGY WITH Q



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.

I guess since I started this, I should start answering (assuming no
one else already has).

a) Can you determine the distance that the container has moved?

No. Ditto through question h.

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?

Let this distance be x. It is the distance the container's cm has
moved. Gas's center of mass has moved x - L/4. Thus system's cm has
moved x - L/8.
net pseudowork on system = F*(x-L/8)
delta(K) = M*v^2 where K = K_container + K_gas

thus x = M*v^2/F + L/8

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

net external force = F
displacement of point of application of force = x

thus W = M*v^2 + F*L/8

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

Q = 0
first law => W = delta(K) + delta(E_int)
E_int_initial = unknown => please specify amount of gas and initial
temperature, or initial pressure and cross-sectional area of cylinder

but delta(E_int) = F*L/8

However, you cooked up this problem because it's supposed to reveal a
problem with this form of the first law, so I imagine you're going to
tell me something's wrong with this "solution."

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

I'm still not sure of the definition, but perhaps E_int?

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

Please specify (N,T0) or (P0,A) so I can calculate E_int_initial.
--
Carl E. Mungan, Asst. Prof. of Physics 410-293-6680 (O) -3729 (F)
U.S. Naval Academy, Stop 9C, Annapolis, MD 21402-5026
mungan@usna.edu http://physics.usna.edu/physics/faculty/mungan/