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In Feynman's Lectures on Physics, in the lecture on the conservation of energy, he makes the following argument:

Consider weight-lifting machines—machines which have the property that they lift one weight by lowering another. Let us also make a hypothesis: that there is no such thing as perpetual motion with these weight-lifting machines. (In fact, that there is no perpetual motion at all is a general statement of the law of conservation of energy.

We must be careful to define perpetual motion. First, let us do it for weight-lifting machines. If, when we have lifted and lowered a lot of weights and restored the machine to the original condition, we find that the net result is to have lifted a weight, then we have a perpetual motion machine because we can use that lifted weight to run something else. That is, provided the machine which lifted the weight is brought back to its exact original condition, and furthermore that it is completely self-contained — that is, it has not received the energy to lift that weight from some external source.

We imagine that there are two classes of machines : those that are not reversible, which includes all real machines, and those that are reversible, which of course are actually not attainable no matter how careful we may be in our design of bearings, levers, etc.

We suppose, however, that there is such a thing — a reversible machine — which lowers one unit of weight (a pound or any other unit) by one unit of distance, and at the same time lifts a three-unit weight. Call this reversible machine, Machine A.

Suppose this particular reversible machine lifts the three-unit weight a distance X. Then suppose we have another machine, Machine B, which is not necessarily reversible, which also lowers a unit weight a unit distance, but which lifts three units a distance Y. We can now prove that Y is not higher than X; that is, it is impossible to build a machine that will lift a weight any higher than it will be lifted by a reversible machine.

Let us see why. Let us suppose that Y were higher than X. We take a one-unit weight and lower it one unit height with Machine B, and that lifts the three-unit weight up a distance V. Then we could lower the weight from Y to X, obtaining free power, and use the reversible Machine A, running backwards, to lower the three-unit weight a distance X and lift the one-unit weight by one unit height. This will put the one-unit weight back where it was before, and leave both machines ready to be used again! We would therefore have perpetual motion if Y were higher than X, which we assumed was impossible.

With those assumptions, we thus deduce that Y is not higher than X, so that of all machines that can be designed, the reversible machine is the best.

My understanding of what Feynman means by a reversible machine is one which will continuously lower and raise the 3 unit weight over and over by lowering and raising the 1 unit weight. (Basically an ideal seesaw-like machine which oscillates the weights upward and downward, over and over again, never losing any energy via friction etc.)

I do not follow how this property of the machine is used in his argument. That is, assume our ideal reversible machine A were replaced simply with another non-ideal machine A', which lowers a 3 unit weight a distance X by raising the one unit weight. Couldn't we simply replace machine A with A', and follow the exact same construction? A' returns the 1 unit weight to its original position, at the same time lowering the 3 unit weight distance X, creating the net result of raising the 3 unit weight?

Clearly there is some flaw in my reasoning, since if that were the case, we would have to conclude that all such weight lifting machines, reversible or not must lift the 3 unit weight the same distance. Feynman's argument must make use of some property of the reversible motion machine A, however, I just can't quite seem to figure out what property that is.

There have been quite a few questions on this topic asked on this site including: Feynman Lectures: Why a non-reversible weight lifting machine cannot lift higher that a reversible one? What Feynman meant in description of reversible machine and levers

After reading these, I still don't understand what it is about machine A's role in the perpetual motion machine that requires that it be reversible (after all, machine A only ever runs backwards).

I think my problem comes from not understanding exactly what he means by reversible and irreversible machines.

Apologies if the answer is extremely obvious, but after searching for quite a while, I've still been unable to figure it out.

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We are trying to prove by contradiction that ‘B (any machine) can be more efficient than A (reversible)’. To do that we must assume as a premise that this is false and then derive contradictory consequences from it, thus proving its negation (i.e., our original claim) true.

If you pick A to be any machine, you do not have a guarantee that it can be run backwards, so the whole reasoning that follows is incorrect (remember the reversibility of A i used in the argument). Therefore the derivation of the contradictory consequences is flawed and you have proved nothing. This is why it is essential to assume A to be reversible.

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  • $\begingroup$ I understand the logic of "reductio ad absurdum", but why does machine A have to run backwards? Neither Feynman's explanation nor your answer clarifies this. $\endgroup$ – Paul Razvan Berg Nov 22 '20 at 22:28
  • $\begingroup$ Because we are assuming that it is reversible. A reversible machine must work backwards, or it would not be reversible. $\endgroup$ – Guillermo BCN Nov 23 '20 at 12:10
  • $\begingroup$ I'm trying to understand this visually but Feynman introduces the term "backwards" seemingly out of nowhere. Is it that we implicitly assume that Machine A ran forwards at the same time that machine B lifted those weights to distance Y? $\endgroup$ – Paul Razvan Berg Nov 23 '20 at 14:24

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