It would be better all round to roll them down a mountain (albeit at the greater cost of getting them up there in the first place - but that’s the problem with any of these schemes)
Scenario: a vessel weights 1001 kilos and has volume of one cubic meter at sea level. You generate and store energy diving. At 1000 meters you start electroysis in the ballast water. At some stage you have displaced 2 kilos of water with hydrogen. At any state hydrogen should weight less than water. Some of water must leave the tank through that open hole. Won’t the system become buoyant - ever?
Yes. But the work of cracking the water to create that buoyancy is always going to be greater than the combined work you can extract from burning the hydrogen and from the upward motion of the now-buoyant object.
You can be quite confident that the energy stored on the way down will be insufficient to accomplish enough electrolysis.
ETA: What Mangetout said.
katunari, in this house we obey the laws of thermodynamics!
We seem to be hung up on two competing definitions of “perpetual motion”. So (some) folks are talking past each other.
The strict physics definition of Perpetual Motion includes the entire universe as its system. And with that definition, A) Perpetual Motion is impossible, period, amen, don’t even start with me/us. And B) the OP’s setup isn’t even a good attempt at it.
A more practical definition, and (IMO) probably the one the OP meant, is a system which extracts energy from some much larger, *practically *inexhaustible source. So a satellite gravitationally slingshotting around Jupiter, a solar panel soaking up sunlight, or a submarine moving due to boyancy are all examples of (very) weak
[gigantic bold flashing purple scare quotes]Perpetual Motion[/gigantic bold flashing purple scare quotes].
I think the OP misused the term; almost anyone else using the words “perpetual motion” means the first, strong, definition and only that strong definition. But that doesn’t mean we can’t usefully discuss situations like the OP’s idea which are examples of the weaker sort.
The very short answer to the OP (IMO), is that at the theoretical level of two half-buzzed college kids messing at a whiteboard you would indeed get something for nothing, i.e. weak PM under my definition above. And because strong PM is impossible, the something is actually coming from the earth’s total thermal & gravitational energy supply, therefore making it not true strong PM. Your something isn’t really coming from nothing; it just seems that way to a not-very-observant observer.
Once you had some real professionals look at the whiteboard, they’d find that any benefit apparently emerging from a part of your system would be totally swamped by heat losses, friction, random ocean motion, and other inefficiencies.
The amount of electricity you generate on the way down will be a tiny fraction of what you need to come back up. You lose a lot of the energy using the water flow to turn a propeller. You lose more converting that to electricity. More when charging the battery you’ll need to crack the water at the bottom. More when you take it out of the battery. A lot more during electrolysis. A lot more creating the pressure necessary to displace the water.
Perpetual motion isn’t just hard - it’s impossible, period. Tapping solar energy or the Earth’s rotational energy or geothermal energy is hard but possible. It’s also expensive and doesn’t scale well. And electrolysis will not be involved in the most efficient way to do it.
When I was young I had the idea of coupling an electric motor to an electric generator and plugging the motor into the generator’s outlet. Once you started it moving it will turn forever, and you could plug your house into the other outlet on the generator! My dream died when I understood why it couldn’t be done.
I don’t think so - the OP’s scenario isn’t attempting to extract energy from some massive natural source - it consists of pushing things up a hill, then hoping to get more energy out of them on their way back down - that’s perpetual motion in the impossible sense.
I have to say I can’t really understand what the OP is asking; there are too many parts left to the imagination and his further clarifications don’t clarify much. You may well be right.
I was commenting mostly on the two very different and disconnected conversations going on about what I labeled weak & strong PM. Those are different problems with different debunkings.
am i missing something here (other than the obvious violations of thermo) ??
you have a sub. the ballasts are… in a vacuum? otherwise “gravity” won’t be able to sink it in the first place, right? so you sink to the bottom. you open up the ballasts. water goes in. you zap the water. your tanks are now filled with superdense hydrogen and oxygen. where is the buoyancy coming from?
Here’s the proposal, as I understand it:
[ul]
[li]You flood the tanks, your sub starts to sink.[/li][li]As you sink, you can generate some electricity using turbines driven by the water you’re falling through.[/li][li]When you hit the bottom, you use that electricity to crack water, producing hydrogen and oxygen, which expels the water in the ballast tanks.[/li][li]You float back to the surface - again, you’re moving through the water, so you can spin your turbines and generate some electricity.[/li][li]At the surface, you vent the hydrogen and oxygen into something that burns it, generating some more electricity.[/li][/ul]
Why it won’t work:
[ul]
[li]The work you can extract from burning the hydrogen cannot be more than the work you expended in cracking the water.[/li][li]The work of cracking the hydrogen is made harder by the pressure at extreme depths - it’s made harder by an amount exceeding the work you can extract from your turbines on the way up and down.[/li][li]Products of electrolysis of seawater at extreme pressures may not be particularly buoyant.[/li][/ul]
if you just electrolyze water into gas inside the ship, the density of the ship will not change, so no going up. To go up you need to expel the gas out of the ship (or the water without splitting it) to reduce the mass of the ship while keeping the volume constant (hence reducing density and increasing buoyancy). And you will find out that you have spent just as much energy doing this expulsion of gas as you have gained by going up. That’s in the ideal case - in reality you will lose a lot more. Many people have built buoyancy related rube goldberg style contraptions - none of them have ever worked 
In any event, it has long been recognized that in the realm of mechanically based systems perpetual motion is not just unfeasible but also very easy to disprove conceptually. That’s why the frauds and the “alternative scientists” switched to building engines that violate the more abstruse laws like 2nd Law of Thermo and similar, where to the average investor both the sales pitch and the expert refutations from physicists sound as equivalent gibberish.
The OP’s proposal doesn’t seem to require extrenal input, as some have suggested – it seems to be a closed system, and by the laws of thermo, it oughtn’t to work. I think Mangetout’s explanation is correct, but he doesn’t sufficiently stress one of the weak points in the OP – just because you generate gases by electrolysis doesn’t mean that you’ll produce them in enough quantity or pressure to “blow out” your ballast tanks. More likely than not the pressure of the sea water will push back against whatever minimal pressure you generate, and will perhaps even inhibit your electrolysis – look up Le Chatelier’s principle
This is a very good book on the topic, and even covers those cases of “apparent” perpetual motioon, where you’re actually drawing on the forces of the earth, such as tidal and atmospheric pressure variations:
There aren’t that many parts in this machine. A room for gases and brine, dynamo and a battery. I quess that the hole in the bottom is the most important. It goes down, but must it come up? I cannot calculate the difference between gravity and buyoancy - is it 10 grams per kilo or less?
er thanks mangetout but i was just being snarky in asking a question i already knew the answer to. code grey put it more explicitly. the point i was getting at was, the crux of the argument - the buoyancy - doesn’t even exist.
i didn’t even catch the turbine argument. REALLY? USING THE WATER FALLING BY THE SUB TO POWER… TURBINES? I’m pretty sure hamsters on wheels would turn out more Joules than a turbine powered by a sinking submarine. harnessing the awesome power of me urinating onto a waterwheel would churn out more Joules. hooking up masturbaters to one of these: http://www.amazon.com/Dynamo-Illuminator-Wind-up-LED-Flashlight/dp/B0009H50OA would be a bigger energy producer…
actually that last one could be quite substantial, but you get my point. i’ve partaken in some half-baked bullsessions in my day but this idea… it’s almost as bad as that trans-saharan desalination project proposed a week ago.
EDIT:
and 02. so you have a tank full of gas, but it’s super pressurized. you have a hole at the bottom (poor location but i’m on a time crunch so i’ll explain later). the gas won’t leave the hole because it’s at equal (or less) pressure as the water outside. if anything, water will further rush into the tank. just because it’s gas doesn’t automatically mean it floats. a soda can is mostly C02 and water but it sinks like a stone. you open it up at the bottom of a lake, the c02 fizzes out, but it’s still at the bottom. yeah?
You could put that sub in a 7 mile deep box filled with water, so it is a closed system. It’s not the sub but sub+ocean, which is closed, so that’s not a issue.
The problem I see is the pressure at depth. The proposal seems to indicate just make the falling/rising time as long as possible - increasing the generator charging time, but the energy it’s recovering is due to a increasing pressure difference. This pressure will require more 2H2+O2, to displace the water at depth, the deeper you go the more gas you have to produce to fill the tanks to lift the sub.
Pardon me for bringing down the level of the discussion here, but this is how I see it. Blah, blah, water past a propeller, blah, blah. Essentially, you have a windmill. Free energy for the guy who doesn’t have to make the wind blow but no perpetual motion machine. I think this is what’s tripping the OP. Not all “free” energy means perpetual motion machines.
This might be adaptable to underwater sea vents with a net positive, though that’s for power generation not for perpetual motion.
If you have a 1 litre container full of air, it will generate an upthrust (when fully submerged) basically equal to the weight of 1 litre of water (minus the weight of the air and container).
ETA: the important point is that this happens because when you push the container into the water, you are displacing (i.e. lifting) the litre of water out of the way.