How much water do you need to keep a cruise ship afloat? Less than you'd think. Archimedes' Paradox lets you float a huge object in just a gallon or two of water. (It also shows why you'd need a massive dam to hold back just a few inches of water.)

Archimedes, who took history's most famous bath (narrowly beating out Jean-Paul Marat), noted that an object would float if it displaced its own weight in water. If the object weighs a pound, and is so light and large that it displaces a pound of water when it's only half-submerged, it will bob on top of the water. A dense object, that will only displace half a pound of water when fully submerged, will sink to the bottom.

The hulls of ships are designed to displace a lot of water, and keep the ship riding high on the ocean, but they still need to be out in the ocean, in deep water. Right? Nope. You should, theoretically, be able to float a cruise ship in a few gallons of water — you just have to get the right container. To prove this, we turn to Blaise Pascal.

The Hydrostatic Paradox


Pascal was a child prodigy who went on to become a mathematician, scientist, and philosopher. He noticed what he considered to be paradoxical behavior in liquids and built a water receptacle to test this apparent paradox. It was a long, clear pipe that connected the bottoms of multiple containers. One container was a small cylinder, and could only hold a little water. One container was a large cone, like a vase, and could hold a lot more water. Others were square containers, or jug-like containers. They all held vastly different amounts of water, and they all were connected by the one pipe. Pascal filled the overall structure with water, so that each container had the same water level. And the water, in all of the containers, stayed still.

Remember, these containers were connected, and remember that gravity was pulling down on that water. If a gallon of water in the jug-like container was straining towards the floor, and directly connected to the small cylinder of water, which only held a few spoonfuls of water, shouldn't the pressure of the water in the gallon jug cause the tiny cylinder to overflow? And yet, no matter how much water one structure contained, and how little the other one did, they stayed in equilibrium as long as the water levels were the same in each container. It was like watching a rabbit and an elephant get on alternate sides of a teeter-totter and having it stay balanced. This has come to be called the hydrostatic paradox.

The Resolution


Of course, it's not an actual paradox. It's just something that confounds our expectations. We, and Pascal, can reason it out like this. Take a look at two particles at the same level, in two different containers. It seems like the particles in the smaller container one should be squirting out like intestines of a frightened sea cucumber, but they're not. They're both staying still. If there were any pressure on them, from any direction, they'd be flowing. Since they're not, we have to conclude that these particles of water are under the same amount of pressure from every direction, horizontal and vertical.

We might not be able to calculate the horizontal pressure, but we can calculate the vertical — it's the pressure of the water (and atmosphere) directly above that water particle. Since this varies by height, we conclude that it's the pressure of the water above it (and nothing else) that determines the pressure on any given water molecule.

Archimedes' Paradox


This helps us float our cruise ship. Imagine the ship floats in an ocean; suddenly, the ocean around it turns to ice, except for a thin layer of water just skimming over the hull. (The water doesn't have to turn to ice. It could be turned to concrete, or glass, or any other solid substance.) Although the ship is now floating in just a bucket or two of water, it doesn't crash to the ice. The water that exists is inside a very oddly-shaped "container," but it's still a very deep container. That means that the water at the bottom has the pressure of all the water above it - and that determines the upward pressure on the boat. There doesn't have to be a lot of water to float the ship, just a lot of depth. As long as the ship would have floated on the ocean, it will float in these few gallons of water. This is often known as Archimedes' Paradox.

The paradox works both ways. If you were to build a water container only half an inch wide, but a mile deep, you would have to support it with all the pressure brought to bear to keep back a lake a mile deep. The horizontal pressure should be the same.

Top Image: Jos Van Zetten.


[Via Archimedes Principle and the Hydrostatic Paradox, Web Archive Debate, Kenyon, Scubageek]