= Internal energy and system boundaries
For an ordinary stationary laboratory sample, a useful convention is
$$
E_{\rm total}=U+K_{\rm bulk}+\Phi_{\rm ext},\qquad
\Delta U+\Delta K_{\rm bulk}+\Delta\Phi_{\rm ext}
=Q_{\rm in}+W_{\rm on}.
$$
The bulk <kinetic energy> $K_{\rm bulk}$ includes organized whole-body motion; for pure translation it is $Mv_{\rm cm}^2/2$, with total <mass> $M$ and <center of mass> speed $v_{\rm cm}$. The external <potential energy> $\Phi_{\rm ext}$ includes elevation in an imposed <gravitational field>, approximately $Mgz$ in a uniform field. Accelerating a sealed bottle or lifting it without otherwise changing its state changes these mechanical terms, rather than its <internal energy>. Molecular <kinetic energy> is measured relative to the chosen bulk motion.<University Physics Volume 2 by Samuel J. Ling and others/3.2 Work, Heat, and Internal Energy>[§3.2, “Internal Energy”: molecular speeds relative to the center of mass; moving-train and roof examples]{ref}
State the convention rather than applying this separation blindly. If an experiment separately tracks a rotating apparatus or an organized flow, its mechanical <kinetic energy> must not also be counted in $U$. In <Joule's paddle-wheel experiment>, waiting for the apparatus and fluid to come to rest removes such an endpoint ambiguity. Including the vessel and paddle changes whose <internal energy> is measured; including the falling weights additionally puts their changing <gravitational potential energy> inside the enlarged system's total <energy> balance.
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