A Battleship Would Not Float In A Bathtub

Update on 2020-07-31: I was wrong. My mistake was in thinking that Archimedes' Principle required there to be a direct mechanical balance between the weight of the floating object and the weight of the water that is being held above what was the level of the water before the object was inserted. I had thought that when an object is inserted into the water, it will sink to the point where it has displaced upward an amount of the existing water equal to its weight. And that happens to be true any time there is more water than the object can displace.

But fluid mechanics makes no distinction between the existing water and the battleship. The battleship can be modeled as a weightless transparent arbitrarily-thin battleship-shaped water bowl that is filled with water to the battleship's waterline. Seen this way, it's obviously not just the water outside the ship-shaped bowl that is contributing to the final water volume in the tub. Each virtual column of water within the footprint of the ship has to be in balance with every other virtual column of water in the tub. For the ship to settle below its waterline (i.e. for the ship to sink to the bottom and maintain a cavity above it), it would require that a taller column of water outside the ship's footprint is balanced by a shorter column of water inside the ship's footprint. That's impossible. So the ship will stop sinking when the water in the bathtub -- no matter how small its original volume -- rises to the level of the ship's waterline.

Archimedes' principle states that "the upward buoyant force that is exerted on a body immersed in a fluid is equal to the weight of the fluid that the body displaces."  It does not mean "the fluid that the body has displaced".  It means "the fluid that the body is taking the place of" i.e. "the fluid that would fill the cavity created by the body". It doesn't matter if there isn't enough of the original fluid at hand, because the body itself acts like a very large particle of the fluid.

Credit for enlightening me goes to the explanations at https://physics.stackexchange.com/questions/304245/can-a-ship-float-in-a-big-bathtub, especially the diagram at http://www.wiskit.com/marilyn/battleship.jpeg.

Original post from 2012-09-08:

The world thinks (e.g. here, here, here, here, here) that a battleship could float in form-fitting tub of water after it displaced all but a tiny fraction of the water in the tub.  These writers cite Archimedes' Principle that the upward buoyant force exerted on a body immersed in a fluid is equal to the weight of the fluid the body displaces. They ignore Archimedes' assumption that the body of water has enough water and enough unused volume to combine together to balance the weight of the immersed body.

A floating ship is hydraulically balanced against the mass of the top layer of water that the ship has displaced upward in the body of water on which the ship floats.  A ship can only float if the body of water can contain that top layer of water and that water has a mass equal to that of the ship. If that water escapes or is otherwise not present, the equilibrium fails and the ship sinks.

Another way to think about it is to ask whether the battleship in the empty bathtub could be floated simply by pouring in the water to fill the bathtub around it. Battleship floaters claim this would work with an arbitrarily small amount of water.  But there is no free lunch -- you can't do the enormous work of lifting a massive ship merely by balancing it against a small mass of water.

Some floaters point to canal locks (e.g. the Miraflores in Panama) that can float a ship with just a foot of clearance on the sides (and allegedly the bottom).  However, they ignore the clearance at the front and back. After a ship enters a canal lock, you can bet that there is a new top layer of water (relative to the prior water level) whose mass is equal to that of the ship.

Floaters tell naive skeptics that when an object is floated in a full tub, the system doesn't remember that some water overflowed when the object was added, and that it floats just as well when it is taken out and then added back to the tub -- which now will not overflow at all.  However, what the floaters don't notice is that the no-overflow system has just enough room at the top to hold the mass of water that balances the object.

Some floaters point out that large machines like telescopes are often "afloat" on a thin film of oil. However, these lubricants are kept pressurized in a sealed system, and the pool of oil is not open to the atmosphere. It's a safe bet that such a telescope cannot be levitated (i.e. lifted) by a thin film of oil unless there is some balancing mass of oil (or some other way of pressurizing the oil).

It seems the physics folks who promote the bathtub battleship meme need to talk to some hydraulic engineers.