Ron_Hardin comments on The Second Law of Thermodynamics, and Engines of Cognition - Less Wrong
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But the more important point: Suppose you've got an iron flywheel that's spinning very rapidly. That's definitely kinetic energy, so the average kinetic energy per molecule is high. Is it heat? That particular kinetic energy, of a spinning flywheel, doesn't look to you like heat, because you know how to extract most of it as useful work, and leave behind something colder (that is, with less mean kinetic energy per degree of freedom).
Systems in thermal contact (by radiation of nothing else) come to the same temperature. That makes it pretty objective if one of the systems is a thermometer, whether it's heat or not.