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I also had hard time to picture what Feynman wanted to explain. It feels like my confusion arose from lack of proper definitions of system state, perpetual motion, and reversible machine. The examples he used are also not quite clear of the mechanical apparatus used, not sure if by design he wanted to abstract away that from the reader, but if so, I would ...

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The third plate will be attracted into the gap between the two outer plates so you get work, $E_\text{in}$, as the third plate moves into the gap. Then you have to do work, $E_\text{out}$, to pull the third plate out of the gap again. If the two outer plates stay fixed then the two amounts of work are equal: $$E_\text{out} = E_\text{in}$$ and energy is ...

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Conservation of energy works pretty well. Assume that instead of a squirting fountain and a paddle wheel, you had a cylinder and a piston to allow the compressed water to do work. The work done would be $\int PdV$. At the end of the expansion the water in the lower container is at atmospheric pressure. Now we can either repressurize by opening the red ...

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You will not be able to produce any useful energy with this system. The previous answers have picked up on a number of the issues... Water is nearly incompressible, so the gain in pressure that you are counting on is too small to do any significant work. Don't forget that the work required to fill a 1km tall pipe with water is part of the system. The ...

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In order to send it back to the container you have to do work against gravity. The energy obtained by rotating wheels would be from the kinetic energy of the fluid which came into existence because of the work we did. The energy from wheel would equal the work we did to pour the fluid back in. So energy gain would be 0. Moreover the efficiency of the wheel ...

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There are two possibilities: water (or whatever fluid you're using) is compressible or it's incompressible. If it is incompressible, when the red valve is closed the tank pressure drops to zero, and opening the yellow valve produces no effect on the generator. If it is compressible, its pressure remains constant when the red valve is closed, and it ...

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Some energy would come from the part where "the water would be collected and sent back into the container", which you somewhat glossed over.

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