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Maybe it would help to cite a specific example (that involves actual
numerical calculations not just concepts) that can be done using your
favorite method x but not using conventional textbook methods. Do you
have such an example?
What about the Rumford experiment, for example? If
you want it to be numerical (as well as conceptual),
let a cannon with heat capacity C rotate through an
angle theta while subject to a torque tau; your
mission, should you decide to accept it, is to
calculate delta T.
I'm a bit mystified. Perhaps part of this is because I haven't read
the Rumford citation you have on your webpage (but it's on my "to
read" list).
But this is not conceptually different than kinetic friction of a
block sliding to rest on a table, right?
(a) Suppose I put both the block and table (cannon and boring tool)
in the system in Step 2 below. Further, I assume C = m1*c1 + m2*c2
(where m = mass and c = specific heat capacity of each object in the
system).
(b) In the first step, I calculate work on the block as -friction *
linear displacement. I do not claim this to be thermodynamic work;
call it center-of-mass work if you feel a distinguishing adjective is
necessary. But for simplicity and consistency with mechanics books,
label it W. In the second and third steps, I claim that a standard
chain of reasoning (see below) shows W = C * delta T. I would not
invoke Q for the system of block + table. (If the system is the block
alone, things are not so simple and we need to consider the timescale
for thermal equilibration and possible subsequent
conduction/radiation to the air, to cool parts of the borer or table,
etc.)
(c) I would certainly claim W also equals delta K for the block.
To me, all of this is standard stuff, basically what any textbook
would do. If you insist on the cannon boring problem, I can put
together a similar line of reasoning for it. But I believe the basic
ideas are here. If you have a different solution to this problem, I'm
all ears.
** Details of the chain of reasoning **
Step 1. LOOK IN MECHANICS CHAPTER
Apply work-kinetic-energy theorem to block AS A WHOLE => -umgx = -m*v0^2/2.
Step 2. LOOK IN CONSERVATION OF ENERGY CHAPTER
Apply delta(E) = 0 assuming an isolated block+table system => 0 =
-m*v0^2/2 + delta E_int. Note that I have *not* invoked any
thermodynamics so far.
Step 3. LOOK IN THERMODYNAMICS CHAPTER
Apply definition of heat capacity C = delta E_int / delta T.