What is the difference between free energy (over unity) and perpetual motion?
Please provide some examples, both real world and theoretical.
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"Free energy" is a term used either by science or pseudoscience, and it means totally different things.
In pseudoscience, "free energy" is energy for free that can be created out of nothing and it is the same thing as a (source of) perpetual motion, and it's an impossible concept. Conspiracy theories exist that civilizations are suppressing evidence that energy may be created for free.
In actual science, free energy is the energy with an extra term, meaning the total energy that may be converted to work. Free energy is $$A=U-TS$$ which is the Helmholtz version or $$G=E+pV-TS$$ which is the Gibbs version. None of those energies is for "free" in the layman's sense - they're just different ways of counting energy and its transfer (including work) in which energy is ultimately conserved.