I hope I'm not opening too much of an old can of worms here. I just
want a way of deriving the various work-energy relations which is
simultaneously sound and reasonably simple, if that is possible (ie.
I don't necessarily want to get every detail right on the first go,
but no big errors either: an overview one could actually present in
class).
So here's my stab at it:
Start from the W-K theorem for a particle: W_net = delta K proven in
any textbook.
Since this is based solely on N2 plus definitions, I think we would
all agree it's perfectly general for a particle. (Nitpicks: N2
requires an inertial frame - assume the lab is. I haven't defined
what a particle is - take it to be electrons, protons, and neutrons
if you like and we'll ignore relativistic particles including
photons, okay? **)
Sum over all particles making up all objects in the system: sum
W_net_i = sum delta K_i.
Split the kinetic energy term into the collective (translational and
rotational) kinetic energy of macroscopic objects (define this to be
simply delta K) and the remaining microscopic kinetic energy of the
constituents relative to their collective parts (delta K_micro).
Split the work term into work done by external forces and that done
by internal forces.
Split the external work into work done by "collective" forces (define
this total to be simply W) and that done by "random" forces (defined
to be Q). My thought here is that heat is just work done by random
molecular impacts, in contrast to the organized force on a piston,
say. But I could use help clarifying this. **
Split the internal work into work done by conservative forces and
that done by nonconservative forces (call the latter quantity
W_int,NC).
Split the conservative internal work into the interactions between
the macroscopic objects (define this to be simply -delta U) and the
remaining microscopic interactions (defined as -delta U_micro).
Collect terms and rearrange to get:
W + Q = (delta K + delta U) + (delta K_micro + delta U_micro) - W_int,NC
Define K + U to be the mechanical energy E_mech (ie. the bulk energy
of the macroscopic blocks, springs, earth, etc.).
Define K_micro + U_micro to be the internal energy E_int into which
we neatly sweep all the complicated microscopic motions, chemical
bonds, nuclear interactions, etc.
Argue that W_int,NC = 0 by considering examples such as kinetic
friction between a block and a table, inelastic collision of a lump
of putty with a table, etc.**
End up with W + Q = delta E_mech + delta E_int, the general result.
Special cases:
(1a) Q and delta E_int both zero => W = delta E_mech ("work-energy
theorem for a system" applicable when there's no heat or change in
microscopic energy)
(1b) special subcase of (1a) W = 0 => delta E_mech = 0 ("conservation
of mechanical energy" applicable to an isolated system whose
microscopic energy is constant)
(2) delta E_mech = 0 => W + Q = delta E_int ("first law of
thermodynamics" useful when we are not interested in macroscopic K or
U of the system)
(3) W and Q both zero => delta E_tot = 0 ("conservation of energy"
applicable to any isolated system)
Okay, fire away. Is this overview basically okay or completely
flawed? Consider in particular the three points marked with a double
asterisk.
--
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/