Seeing a ship canal lift in TV spawned the following thought:
"Oh my, this sure is a heavy ship and it can be lifted by simply adding water. What a marvellous machinery. After all, I could lift that ship with my teacup and a bit of water in it"
Which in turn spawned the following thought experiment:
Given some basin of know area, I can pour my teacup filled with water into it. The water level will be raised by a certain amount, depending on the volume and area. Seems legit so far.
The water level will always rise by that amount, no matter what is floating in the basin. I have my rubber duck floating in my basin and it is raised by the teacup amount after pouring. Now let's add a big lead duck (or a ship if you want, but ducks are that much cooler, aren't they?) that floats in my basin. I still raise all that floating stuff by the teacup amount, even if I added the ship.
And this is where things started to feel a bit dubious.
How is it possible that I "invest" a (constant) teacup worth of potential energy, yet "gain" the potential energy of rising an arbitrary big mass that is floating in my basin by the constant amount caused by the water from the tea cup?
That right there doesn't sound dubious any more, that sounds plain wrong to me. Because wouldn't that allow the teacup ship lifter gain more potential energy out of the process than he needs to re-rise the water into his teacup?
Where did my thought experiment took a turn for the worse?
I try to describe my problem in more detail: A big ship needs more space, so the basin gets bigger, right?
I'm afraid that's not the case and pretty much the core of my problem: The geometric properties seem to be independent enough to allow a "gain" in potential energy.
Here's how: Let's say the basin area is constant. This means the water level rises a constant amount (depending on how much water is in the teacup) Now how does the constant basin support "arbitrary large masses"? By varying its depth. In order to make a bigger mass float, it has to displace more volume (of water). As the area is fixed, the ship/boat/duck can extend its volume in the third dimension (down) to displace more water, thus supporting more mass to float, yet being raised to the same height as the basin area stays constant.