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Let's say you have a fixed pipe, linked at one end to the surface and the other submerged connected to a detachable chamber as shown below:

The pipe is fixed in its position by a barge of some sort but the chamber is buoyant unless it is weighted down by magnets (Chamber: grey, Magnets green)

The chamber is also collapsable but sealed with a rubber bellow of some type to prevent leakage when the chamber condenses. Two magnets are placed in the otherwise buoyant chamber to weigh it down to a set position in which it connects to the submerged end of the pipe.

When in position the chamber is allowed to compress, sea pressure forcing the magnets which are arranged with their north poles facing one another together.The magnets naturally resist being compressed but the depth pressure is overpowering condensing them akin to a spring.

Once the magnets are condensed they are dragged up through the pipe by a winch. Upon reaching the surface the magnets are allowed to separate, pushing a rack along a pinion connected to a gearbox and generator setup. The chamber disconnects from the pipe, floating to the surface as it is no longer weighed down by the magnets. Once the chamber reaches the surface the magnets are placed inside again and the cycle repeats.

This is essentially a perpetual motion machine and thanks to the laws of conservation of energy this can't produce electricity but I don't really understand why not in this particular arrangement and would be grateful for a clarification.

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  • $\begingroup$ Energy has to put into the system when the winch drags the magnets back up to the surface. $\endgroup$
    – M. Enns
    Nov 13, 2017 at 20:01
  • $\begingroup$ Yes, I just can't see how that would cost that much energy, I mean can't you have extremely strong magnets that are pretty light? $\endgroup$ Nov 13, 2017 at 20:31
  • $\begingroup$ If the magnets are very light, how can they weigh down the chamber and make it sink? $\endgroup$ Nov 13, 2017 at 20:34

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Suppose energy $E_1$ is recovered from releasing the compression of the magnets, and energy $E_2$ is expended in raising the chamber to the surface. If $E_1 \ge E_2$ then the perpetual motion machine works.


For simplicity replace the magnets with an ideal spring (ie there are no losses due to hysteresis).

Initially the chamber is at the surface with the spring in its relaxed state. The weight of the chamber is greater than the buoyant force on it so it sinks. As it does so the spring is compressed. The volume of the chamber decreases, so the buoyancy force decreases.

When the chamber reaches depth $h$ the water pressure is $\rho gh$ so the force compressing the spring is $F=\rho ghA$ where $A$ is the cross section area of the chamber. The energy stored in the spring is $E_1=\frac12 Fx=\frac12 \rho ghxA$ where $x$ is the amount by which the spring is compressed at this depth.

The spring is then locked in position, and the chamber is raised back to the surface. The work done to raise the chamber against gravity can be recovered from the work gravity does on the chamber as it is lowered. The net work done in raising the chamber is solely due to the reduced buoyancy force caused by the decrease in volume.

The difference in the buoyancy force is $\Delta B=\rho g \times \text{difference in volume of the chamber}=\rho g xA$ . The overall work done to lower then raise the chamber is therefore $E_2=h\Delta B=2E_1$.

Twice as much work is done raising the chamber against the reduced buoyancy as is recovered from the spring. So the process is guaranteed to lose energy regardless of how efficient it is.


Where did the missing energy go?

The situation is similar to a weight $W$ being attached to a spring hanging vertically. When the weight has been lowered gently to its equilibrium position the spring has extended by $x$. The gravitational PE lost by the weight is $Wx$. The elastic PE gained by the spring is $\frac12 Wx$. We've lost half our GPE!

The explanation is that if the weight had been released suddenly and allowed to fall, the weight would reach the equilibrium position with KE of $\frac12 Wx$ while the spring gains elastic PE of $\frac12 Wx$, giving a total energy of $Wx$, which is exactly the gravitational PE lost at this point. (See Where is the energy lost in a spring?)

In this case the missing energy was taken away by the device (or human hand) which lowered the weight gently, removing the KE it would have gained if it had fallen freely. In the case of the sinking magnetic chamber the KE of the piston was (we must presume) removed ("dissipated") by viscous forces in the water, or by friction between the piston and cylinder.

To see that this is correct, imagine the spring is locked at its relaxed length at the surface before it descends to the required depth. No energy has been lost (dissipated) yet. Now the locking mechanism is released suddenly. What happens? Just like the weight released on the spring, the magnetic spring reacts to the sudden force on it and starts oscillating. At its equilibrium compression the chamber has elastic PE of $E_1$ and also KE of $E_1$, giving it a total energy of $2E_1=E_2$. If this KE could be recovered before the spring is clamped at its equilibrium position, then the overall energy $E_2$ expended in raising the chamber back to the surface would equal the energy $2E_1$ extracted. There would be no overall loss or gain of energy.

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Simple: it takes energy to lift the device out of the water, because the device is less buoyant on the way up and it therefore weighs more.

The energy accounting of the lifting needs to be done carefully, because while it takes energy to lift the device out of the water, you can also get some of that energy back when you put it back in. Because of that, the easiest thing is to take two devices at a time, one going into the water and one coming out, joined together by a pulley system that pulls the one on the left as it goes out using the weight of the one on the right as it goes in.

So, what is the problem? well, as perceived by the pulley system, the one on the left is heavier, so it takes work to move the system. Why is it heavier? because your protocol requires it to compress, and that means that its volume has decreased, so it displaces less water and, by Archimedes' principle, the buoyant force on the compressed device is smaller, and it cancels out a smaller fraction of the weight.

And, it should go without saying, if you somehow manage to operate this system without losses, then whatever energy you get from separating those magnets will be exactly enough to power the winch that pulls the compressed systems out of the water.

In general, both hydrostatic pressure and electromagnetism are conservative systems. We know this because we understand the basic laws that govern their behaviour (you're using them explicitly) and we can prove rock-solid theorems that forces with those characteristics, and all the systems built on top of them, will conserve energy. You can try and build a system so complex that you'll manage to confuse yourself into overlooking a spot where energy needs to come in, but there is an easy way out: if the underlying dynamics are conservative, the whole system will be conservative.

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    $\begingroup$ All of this is already explained in sammy's answer, but I reckoned a simpler and punchier explanation was in order. $\endgroup$ Nov 14, 2017 at 0:26

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