Most of the following has already been said in one
form or another, but just to pull it all together:
There are multiple different energy-like principles
that can be invoked.
1a) By far the most fundamental is plain old conservation
of plain old energy. That is satisfied in this case as
follows: The total energy of the system is unchanged.
The rotational KE of the system increases at the expense
of the biochemical potential energy. No energy flows
across the boundary of the system.
1b) This is also satisfied on a subsystem-by-subsystem
basis as follows: Considering each handheld weight
as a subsystem, the KE of the subsystem increases
and the total energy of the subsystem increases.
This is consistent with energy flowing across the
boundary of the subsystem in the form of F • dx
work.
2) There is also large family of Work/KE theorems.
These have to be stated very carefully.
2a) At the most macroscopic level, no work is done
on the overall system, and the KE associated with
the *center of mass* of the overall system is
unchanged. (The rotation does not affect the CM.)
2b) At the subsystem level, the KE of each handheld
weight increases. Again this is consistent with
F • dx work done on the subsystem.
Pedagogical remark: I use the fundamental notion
of conservation of energy all the time. However,
in the real world, I use the work/KE theorems rarely
if at all. If I didn't see them in introductory-
level textbooks I don't think I'd see them at all.
On the rare occasions when I "could" use some sort
of Work/KE theorem, it is virtually always easier
to use plain old conservation of energy directly.
And it's not just me. Not too long ago a friend
who is a professor at a Big Name university called
me up. She was teaching the introductory course
and ran into the so-called "Work/Energy" theorem
for the first time in her life. She called me up
to ask what the #&!! the book was talking about.
My point is, I'm not at all sure that it's worth
teaching Work/KE theorems in the introductory class.
YMMV, but to me it seems messy and not worth the
trouble. For starters, the very definition of KE
versus PE depends on the choice of coordinates,
as we recently discussed in conjunction with the
"air spring" i.e. PE due to gas pressure.
If anybody has an argument (or even an example)
for why the Work/KE theorems are worth the trouble,
I'd be interested to hear it.