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I'm confident you agree that for an object with or without internal
degrees of freedom and having total mass M, the net force (which is
of course external) on the object equals M times the acceleration of
its center of mass (CM). Now dot both sides with a differential
displacement of the CM and integrate. That gives the W-K thm.
Applied to your cart example. If F is the *only* external horizontal
force on the cart, I hereby publicly guarantee it will gain KE,
regardless of what clock mechanism is inside it.
If you meant that there are other external horizontal forces, please spell
out a free-body diagram for your clock-cart a bit more clearly.
I assume this is simply a miscommunication and that you are too busy
to look over my long document more carefully.
I strongly object to texts which consider examples (in the mechanics
chapter) of say clay hitting the floor and then say (in passing, no
less!) that the floor does no work because it doesn't move. Is there
an introductory student alive who is supposed to make sense of this
before having been introduced to thermo?