# Question regarding reversible machines - Feynman Lectures Vol. I Chapter 4.2

In chapter 4.2 of the first Feynman Lecture, Feynman talks about deriving the formula for gravitational potential energy through facts and reasoning. He talked about weight-lifting machines, and explained that an irreversible machine that lowers a certain weight some distance 'd' to lift another weight will always be less efficient than a reversible machine that can lower that same certain weight the same distance 'd', because that would be a violation of the conservation of energy since it would mean that there is perpetual motion. Here is what he said about it:

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 Y . 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 4-4 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.

I understood the reasoning about why an irreversible machine will always be less efficient than a reversible machine if they both lower the same weight by the same distance. Then Feynman said this:

We can also see that all reversible machines must lift to exactly the same height. Suppose that B were really reversible also. The argument that Y is not higher than X is, of course, just as good as it was before, but we can also make our argument the other way around, using the machines in the opposite order, and prove that X is not higher than Y. This, then, is a very remarkable observation because it permits us to analyze the height to which different machines are going to lift something without looking at the interior mechanism. We know at once that if somebody makes an enormously elaborate series of levers that lift three units a certain distance by lowering one unit by one unit distance, and we compare it with a simple lever which does the same thing and is fundamentally reversible, his machine will lift it no higher, but perhaps less high. If his machine is reversible, we also know exactly how high it will lift. To summarize: every reversible machine, no matter how it operates, which drops one pound one foot and lifts a three-pound weight always lifts it the same distance, X. This is clearly a universal law of great utility. The next question is, of course, what is X?

What I am struggling to understand is what he meant when he said "but perhaps less high" (which is written in the bold lines). It's reasonable two reversible machines lower one unit weight by one unit distance to lift a three unit weight, that three unit weight would be lifted by the same distance with both machines always regardless of the difference in the complexity of the machines. This would be because the energy from lowering that unit weight through a unit distance is the same for both machines. But then, how could the three unit weight be lifted through a smaller height by using one machine instead of the other machine, if both get the same energy to lift the three unit weight?

Maybe the complex machine with the series of levers may be less efficient because not all the energy from lowering the one unit weight is used to lift the three unit weight exclusively (maybe the levers themselves impose that inefficiency as they themselves would have mass), and that would explain why if the distance lifted is not the same, for one machine it must be lower than for the other machine, but it cannot be higher than the machine that lifts the three unit weight through the highest distance possible.

If my question is not clear, please let me know. I would really appreciate if someone can explain to me this.

• "perhaps less high" was meant for dissipative forces which breaks energy conservation, such as friction, air drag, converting part of energy into heat upon collision, etc. Jan 1 at 20:46