## Buoyancy Motor No. 4 comments by Donald Simanek.

A couple of critics object to the inclusion of this device in our collection because it forces us to push materials to the limit. We are accustomed to giving inventors viscous-free liquids, frictionless materials, and in this case we'd also have to do something about that pesky surface tension. With real liquids a buoyant object would have to have considerable velocity to break through the surface and go completely underwater long enough to make it around the bend in the tube. Then it would have some difficulty popping up above the surface. Picky! Picky!
It seems to me that this is one of the sneakiest and subtlest PM proposals I've seen in a while. Here are some thoughts for your consideration. Warning; we will discuss the trivial objections first, before revealing the fundamental reason why devices of this sort always fail no matter how ingenious the details.

The work done by and against gravitational force adds to zero around the loop, so we needn't be concerned about that. The system is "primed" by (1) raising the water level on the right, which requires work done by an external agent. It also requires work by an external agent to force the ball to the bottom of the liquid. Thus, the system begins with stored energy. It also begins with the system components (the liquid) located far from its equilibrium position.

Some extra work is required to push the ball floating at the surface down until it is entirely below the surface. The ball will recover this amount of energy as it breaks through the surface again on the way up. But then it will lose that amount as it crashes through the lower surface on the left. Still, after the first half cycle, these losses and gains add to zero.

So, allowing the idealization of zero viscosity, etc., the ball should execute more than one cycle around the loop. Does this continue forever? Can we extract a bit of energy from it during each cycle, forever?

I, with Dave Carvel's permission, have simplified the gates. Instead of using "flap" or hinged gates, I use these neat iris diaphragms that do not "push around" much water. Yet still, the gates present a problem more subtle and fundamental than mere inertia of water.

Niket Padwardhan was the first to email me [12 Sept 2002] this one absurdly simple reason why a device with such gates, even perfect ones, will not work.

The system will eventually run out of energy by lowering the water level. This will happen even with perfectly rigid components [and perfect gate seals]. Consider what happens when the ball goes through a gate. The total volume in the two spaces connected by the gate must remain constant, so if the ball goes up, a volume of water equal to the volume of the ball must come down. In each iteration through the loop one ball's worth of liquid effectively moves from the top section to the bottom section. The upper level stays the same—until the ball pops up from the top surface, at which point the level drops by one ball volume worth.

The bald assertion in our description of the device—that the gates maintain the water level—was simply a lie. If the lower gate were designed so ingeniously that it did not allow liquid transfer as the ball moved through, then the ball would simply not move through this gate.

It seems that nature always has a "gotcha" to thwart the cleverness of the perpetual motion machine inventor.

I was reluctant to include this device for reasons stated at the top of this page. So I was surprised to discover that the idea wasn't new, and is part of the long and sordid history of failed devices. So much ingenuity has been wasted on perpetual motion machines over the last few centuries that it's rare that anything really new comes along. This figure illustrates that this idea is at least as old as 1825. It appeared in the June 1825 issue of The Mechanic's Magazine and is discussed in Phin's book. But (why am I not surprised?) Phin didn't give, or didn't know, the basic, insurmountable flaw explained above. The person who submitted the machine to the magazine writes:

I beg leave to offer the prefixed device. The point at which, like all the rest, it fails, I confess I did not (as I do now) plainly perceive at once, although it is certainly very obvious. The original was this—to enable a body which would float in a heavy medium to sink in a lighter one, to pass successively through the one to the other, the continuation of which would be the end in view. To say that valves cannot be made to act as proposed will not be to show the rationale (if I may so say) upon which the idea is fallacious.

The author recognizes that engineering difficulties related to the valves is irrelevant, and distracts attention from the more fundamental flaw in the idea.

The glass tubes contain water on the left, air on the right. A water reservoir is below, but isn't strictly needed. Balls less dense than water are used. The apparatus has valves at points 2, 3, 4, and 5. It is just like the version we have discussed above, except for the neat addition of the curved portion of the tube (4, 5, 6) on the right. This is adjusted for timing so that the balls take the same time to fall on the right as it takes them to rise on the left. The valves have springs and are opened by the balls themselves, then closed by the springs. The balls must be timed so that the two valves on the left aren't open at the same time. They therefore "sustain the whole column of water."

Also note the "pile-up" of three balls between E and F. These provide the weight sufficient to push the bottom ball of this stack into the water so it can enter the left tube and then rise to the top. Clever!

Perpetual motion machine inventers often acknowledge flaws found in their devices. Typically their response is to try to "fix" the flaw by redesign. Suppose we fix the flaws noted above by eliminating one of the gates, and substituting a "black box" B at the bottom that not only sustains the water column on the right, but cleverly allows the ball to move from air to water without letting any water out. This gate mechanism prevents the water on the right from ever being depleted. This eliminates the flaw explained above.

Removing that flaw reveals another flaw, even more fundamental, which prevents any buoyant device of this type from working. The energy gain of the ball falling through air on the left is mbgH where mb is the mass of the ball. The energy required to push the ball from air into the bottom of the water column is equal to the potential energy of the water volume of height h and cross-sectional area A at the top of the water column, i.e., mwgH where mw is the mass of that water. Forcing the ball into the liquid raises the liquid level a height h, equivalent to a volume Ah equal to the volume of the ball. Archimedes' principle has thwarted our hopes again.

An equivalent way to analyze this is to note that pushing the ball into the water requires work to overcome the pressure difference between air and water. The pressure at the bottom of the water is rwgH. The work required is VD P = r wVgH. The energy the ball has after falling distance H from rest is mgH = r bVgh. This isn't enough to push the ball into the water.

But wouldn't the ball rising in water gain energy, an additional energy that might be enough to keep this system cycling? We are generously assuming perfect components, so we can disregard friction and viscous drag. Under these ideal conditions the ball in water experiences a force upward of size B - W, and accelerates upward, gaining kinetic energy. How much energy gain? It gains (B - W)H, where W is the weight of the ball and B is the buoyant force on it. This is just the amount of energy we put in when we pushed the ball to the bottom of the liquid (priming the device, so to speak)! It's also the additional amount of energy needed to actually push the ball to the bottom of the liquid by any means. So we conclude that if we prime the device, putting in energy by pushing the ball to the bottom of the liquid, then it can cycle continuously, going on forever, but producing no excess useful work. We have called this a device of type (1).

This is a nice example of a "primed" device of type (1), which turns continually. Whenever we eliminate dissipative processes like friction, we get, at best, a system that cycles continually, but produces no useful work.

We also saw in Dave Carvell's clever puzzle device how gates in liquid systems (even highly perfected gates) can allow liquid transfer that eventually depletes the priming of the machine, bringing it to a stop.

We have indulged ourselves in this discussion and tried the reader's patience. First we swept aside irrelevant considerations such as friction and viscosity, then looked for more fundamental flaws. The first ones we found exposed others. Analyzing a perpetual motion machine proposal in this way is like peeling an onion—it makes you cry (when you realize that the fatal flaw is insurmountable). To "correct" such a flaw would require violation of very basic laws of physics.

So the bottom line is this: The work required to push a ball into the bottom of the tube by any method requires that work be done against the pressure difference between inside and outside. This is (at least) as large as the work the ball does in rising to the top of the tube. Once this is realized, all other Baroque embellishments in the device, no matter how clever, become completely irrelevant. For another example of fruitless design details, read the following example from the patent literature.

## This just in. Yet another version arises. Discredited ideas never die.

Just when we thought this one was wrapped up, our German colleague Hans-Peter Gramatke discovered this "World Patent" WO 9631696 Buoyancy motor dated 1996. The "perpetual" in perpetual motion simply means that dumb ideas get recycled perpetually.

This inventor seemed to realize that the gate-valves are a problem, and spent considerable effort in their design (24, 25, 26). He also must have realized that some water would transfer across the gate (26) as it opened and closed so he provided a valve (41), which somehow takes the excess water and pushes it up a tube (42) to reservoir (43). It's unclear what provides the work to do that. He also has some kind of linkage (37) between gate (26) and the ball release at (32). It must prevent the balls from piling up somwhere, though it's not clear whether (32) triggers (26) or vice versa. This linkage (37) seems improperly drawn, for it seems to me that the dotted and solid lines should be swapped. But perhaps I just don't understand the subtle and mystical principles employed by the inventor.

With all that attention to these irrelevant details, the inventor has completely overlooked the real reason (discussed at length above) why this thing won't work. We often tell students to sweat the details. But this inventor was so focused on engineering details that he forgot to attend to the basic principles of physics.

The flaw is a simple one, shared by countless patented systems that suppose that buoyancy generates energy for free. The potential energy gained as a lighter-than-water object rises to the top of a water column is no greater than the energy required to insert that object at the bottom of the column. However the ball is forced into the bottom of the water tube, it must pass from air pressure outside to higher liquid pressure inside, and that requires work to accomplish. I continue to be amazed that so many patents have been granted worldwide for devices whose inventors overlooked this simple fact.

"Aha!" someone is sure to say. "What if we use air in the tube and balls lighter than air. Then the bottom input valve will be no problem and will require no work." This modification will transfer the problem to that of "How do you get those lighter than air balls back to the bottom without doing work on them?" And that's (at least) the same amount of work as they gain as they float to the top. Nature has gotcha again. The very fact that such devices continue to be submitted to patent offices (and sometimes the patents are granted) shows how clueless some people are about how nature works.

All of the fussy mechanical details the inventor included in this picture do not in any way improve the performance of this mechanism. They only distract us from noticing its fundamental flaw. The work required to push a ball into the bottom of the tube by any method requires that work be done against the pressure difference between inside and outside. This is (at least) as large as the work the ball does in rising to the top of the tube.

Rube Goldberg would have loved this device. You have to admit it's a magnificent-looking contraption.