> I reject the loaded names "pseudowork" and "real work."
Bingo. A light just came on for me.
Seeds planted by John M. and Bob S. have sprouted.
When it comes to KE (Kinetic Energy) I thought I
knew "the" definition but alas I was just looking
at part of the elephant. Sorry.
Let's see if I can now give a more-complete
description of the whole elephant.
There is a range of different concepts all of
which are sometimes called "the" KE.
a) At one extreme there is what we might call the
microscopic or ultramicroscopic viewpoint, where we
take into account the motion of every microscopic
particle in the system:
KE = sum .5 (p_i)^2 / m_i
where the sum runs over all particles; p stands
for momentum and m stands for mass.
b) At the opposite extreme there is what we might
call the macroscopic or holoscopic viewpoint, where
we look only at the common-mode motion of the whole
object, using only the total momentum P := sum p_i
and the total mass M := sum m_i, so that
KE = sum .5 P^2 / M
c) There are innumerably many viewpoints between
these extremes. We might call these mesoscopic
viewpoints. Among these is the thermodynamic
viewpoint, as exemplified by a spinning flywheel:
normally when we calculate "the" KE of the flywheel
we don't include the KE of the ultramicroscopic
thermal agitation, but we do include the rotational
KE, which is nonzero even though the center of mass
isn't moving.
================
A more-general approach would be to specify a
length-scale "lambda" specifying the resolution,
i.e. how closely we are going to look at things.
Then we partition the object into cells of size
lambda, and define
KE[lambda] := sum .5 (p_k)^2 / m_k
where the sum runs over all cells.
a) When lambda is infinitesimal, we recover the
ultramicroscopic KE.
b) When lambda is larger than the size of the
object, we get a single cell and we recover the
holoscopic KE.
c) Intermediate values of lambda are useful,
too.
d) You can have a hierarchy of lengthscales.
The cells of size lambda1 can be subdivided
into subcells of size lambda2 and so on.
Remarkably, the value of KE[lambda] is not
very sensitive to the choice of lambda, over a
wide range, as we now discuss.
Consider a flywheel in the form of a solid cube
of size L = 1 meter on a side. Choose lambda =
1 cm; that is, partition the object in to a
million cubelets each 1 cm on a side.
We can make a scaling argument. The moment of
inertia scales like r^2 m. Since the cube and
cubelets all have similar shape (similar in the
strict sense of Euclidean geometry), we don't
need to worry about dimensionless factors in
front of the scaling formula.
The moment of inertia of each cubelet scales like
lambda^5 ... three factors of lambda for the
mass and two factors for the r^2 in r^2 m.
The number of cubelets scales like 1/lambda^3,
so when we sum over cubelets we find that the
total KE[eps] tied up _inside_ the cubelets
scales like lambda^2. That is,
KE[eps] - KE[lambda] ~ (lambda/L)^2 KE[eps]
In our numerical example, lambda = 1 cm, so
KE[1cm] ~ 99.99% KE[eps]
where eps is some length-scale small compared
to lambda but large enough to wash out any
ultramicroscopic motions (e.g. thermal agitation).
================
For each lengthscale lambda, we can also define
a work[lambda] and establish a
work[lambda]/KE[lambda] theorem. Specifically,
for each cell, work[lambda] is the total force
applied to the cell, dotted with the distance
moved by the center of mass of the cell. The
total work[lambda] is just the sum over cells
in the obvious way. The theorem states
delta KE[lambda] = work[lambda]
Pseudowork is work[lambda] for large lambda.
==========
Another bit of terminology that may be helpful:
For any object (or cell or subcell):
-- Define "common mode" motion to refer to the
motion of the center of mass. The common-mode
momentum is P = sum p_i. The common-mode KE is
KEcm = .5 P^2 / M.
-- Define "differential mode" motion to refer
to the motion relative to an observer comoving
with the center of mass. In the lab frame, the
center of mass has velocity
V = P/M.
The differential-mode momentum of the ith
particle is
p'_i = p_i - V m_i.
The differential-mode KE of the particle is
.5 (p'_i)^2 / m_i and the differential-mode KE
of the object is
KEdm := sum .5 (p'_i)^2 / m_i
This is useful because
KE = KEcm + KEdm
which in turn helps you understand the
work[lambda]/KE[lambda] theorem.
====================
All of the above is restricted to nonrelativistic
mechanics only. I tried generalizing it in the
obvious way ... without success.
=====
None of the above is sufficient to salvage the
ghastly "thermodynamic" formula
delta E = W + Q
That formula is a Bad Idea and no amount of
tinkering with the definition of W will fix it.