# Can a Battleship Float on a Gallon of Water?

Ask Marilyn ® by Marilyn vos Savant is a column in Parade Magazine, published by PARADE, 711 Third Avenue, New York, NY 10017, USA. According to Parade, Marilyn vos Savant is listed in the "Guinness Book of World Records Hall of Fame" for "Highest IQ."

In her Parade Magazine column of June 6, 2004, Marilyn claims that in theory, with a container of the right size and shape, you could float a battleship on a gallon of water.

## Sorry Marilyn

Ken Cliffer <kcliffer@k12.com> was the first to contact me regarding Marilyn's error. In his letter to Marilyn, he stated that "A floating object displaces an amount of water with a mass equal to the mass of the object."

Jonathan E. Jobe <jejobe@townsend.com> wrote two days later with some interesting numbers: "An Iowa Class battleship displaces 45,000 tons. So you would need at least 45,000 tons of water to float a battleship. The container would have to be able to hold at least 90,000 tons of water. You couldn't even wet more than a few square yards of a battleship's hull with a gallon of water."

James Whitmore <jwhitmor@muscanet.com> was the third to write. He adds: "One gallon of water (no matter how you arrange it) can only support 8 lbs. The ship would sink to the bottom of the tub and the water would be displaced."

## An argument in Support of Marilyn

The laws of physics say that any object lighter than water will float in the water, no matter how little water there is.

Please take a minute to look at this drawing, and I am sure you will agree with me.

## Ken Cliffer Responds

After looking over your "proof," I decided that my analysis of the battleship floating problem was too superficial. Indeed, I decided you are right, but I am more right. That is, if you had a battleship with a completely smooth and perfectly water-repellant hull, so that it was so flat that it varied no more than the equivalent of a couple of hundred molecules of water anywhere, and a container with equally smooth walls, to the same molecular precision, exactly the same shape as the hull of the battleship, but a fraction of a micrometer larger, and the ship were balanced perfectly so that it could not shift more than the width of a bacterium, and the container and ship didn't vibrate at all, and you had an atmosphere in which the water would not evaporate, perhaps you could "float" a battleship in a gallon of water. However, that degree of precision, balance, and stability is practically inconceivable, so I am forced to conclude that one could never actually, nor even feasibly theoretically, float a battleship in a gallon of water. Theoretically, almost (but not really) maybe. Practically, definitely not.

My conclusion is that Marilyn was trivially theoretically almost right, but only in a sense in which theory ignores reality entirely. She should have qualified her answer by saying that it would be theoretically possible if it were done with an inconceivable degree of precision. Furthermore, she implied that anything that floats on any amount of water could be floated on any small amount of water. You yourself claim: "The laws of physics says, that any object lighter than water will float in the water, no matter how little water there is." This is not correct. You'll see from extending my argument that at some point of a diminishing amount of water, the water would be spread so thin that there would not be enough molecules of water to support an item that would need to be submerged a given amount to float. I would contend that a gallon of water has reached that point for a battleship for any reasonable considerations. If you reduced the amount of water to any less than 1/1000 of a gallon, there would clearly be an insufficient amount to float the battleship under any circumstances, theoretically, inconceivably precisely, or otherwise (the water would be less than one molecule thick on average).

Here is the argument:

For the sake of illustration, let's use some statistics for the battleship Massachusetts. According to the following website, it took 3,000 gallons of paint to paint the 130,000 square feet of its underwater hull (at least to the date of the report): Battleship Massachusetts at Battleship Cove. Already, if it took 3,000 gallons of paint to paint the hull, you can imagine that it is highly unlikely that one gallon of water could be around the hull to a thick enough layer to consider the ship floating. However, let's see what it would take to succeed.

Here's the strategy: We'll calculate how much water would cover a square meter in area at one molecule thick - one layer, then calculate how many layers the surface area of the battleship's hull would have if evenly covered with a gallon of water, then calculate how thick that layer would be. We'll compare that thickness to the size of a bacterium, and ask whether it's credible that the ship would be actually floating.

First, to work in the metric system, let's calculate the surface area of the hull in square meters. There are 0.0929 square meters per square foot. So there are 130,000 x 0.0929 = 12077 square meters of hull to consider.

Now, let's see how much water it would take to cover a square meter with a single layer of water molecules. We'll do it using the number of molecules. One liter of water has a mass of 1000 grams, which means it has (1000 grams / 18 grams/mole) x 6.02 x 10^23 molecules / mole = 3.34 x 10^25 molecules of water. A liter can be fit into a cube 10 cm on a side. Therefore, the cube root of the number of molecules in a liter gives you the number of molecules that fit along 10 cm., on average. This is 3.22 x 10^8 molecules. Since there are 100 cm in a meter, the number of molecules that would fit along one meter would be 10 times the number along 10 cm, and the number to cover one square meter in a layer one molecule thick would be (3.22 x 10^8 x10)^2 = (3.22 x 10^9)^2 = 10.4 x 10^18 = 1.04 x 10^19 molecules per square meter.

Now let's see how many molecules of water are in a gallon. One gallon is 3.79 liters, so, using the number of molecules in a liter from above, a gallon has 3.79 x 3.34 x 10^25 = 1.27 x 10^26 molecules of water.

At 1.04 x 10^19 molecules per square meter (calculated above), the gallon of water would cover 1.27 x 10^26 molecules / 1.04 x 10^19 molecules per square meter = 1.22 x 10^7 square meters at one molecule thick.

To figure out how many molecules thick this amount of water would be over the submerged hull of the ship, we must divide the total area it could cover at one molecule thick by the submerged area of the hull: 1.22 x 10^7 square meters at one molecule thick / 12077 square meters = 1.22 x 10^7 /1.2077 x 10^4 = 1.01 x 10^3 molecules thick, or about 1000 molecules thick.

To figure out how thick this is, we can divide by the number of molecules per meter we figured out before: 1.01 x 10^3 molecules / 3.22 x 10^9 molecules / meter = 10.1 x 10^2 / 3.22 x 10^9 meters = 3.14 x 10^-7 meters = 3.14 x 10^-4 mm = 0.314 micrometers

Therefore, the gallon of water would coat the hull of a battleship to about 1/3 of a micrometer thickness. A typical E. coli bacterium is about 1 micrometer long. The coating of the hull by the gallon of water would be less than the length of a bacterium. In order to float the ship in a gallon of water, one would need to have the container fit so well that that amount of water would be evenly distributed around the hull, 1000 molecules thick. Any imperfection of the length of a bacterium or so anywhere, in the hull or in the container, would allow the water to flow around the imperfection while the ship rested on the imperfection, not floating, but supported at least partly on the container. Not only that, but the ship would have to be perfectly balanced so that its weight would be perfectly centered above the container, and could not shift and let a few thousand molecules of water slip from one side to the other, letting the ship rest on the other side of the container. Credible? Theoretically, perhaps almost, but not quite. Practically, I think not a chance.

Having floated this argument, I rest my ship ? er, case.

## Another Explanation

Quentin Cobb <cobbq@spots.ab.ca> explains the solution as follows:

Marilyn is NOT wrong! Theoretically of course - not practically.

Ken Cliffer, Jonathan E. Jobe and James Whitmore are all confusing displacement with the amount of water remaining - a ship floats just as well in a canal lock as it does in the open ocean. The displacement is the volume of water that would otherwise be in the hole the ship makes - it's the water that's not there that causes the ship to float!

Now imagine a container shaped like the hull of the battleship but a tiny fraction larger. If you fill the container with water and place the battleship in the container it will displace almost all the water in the container which will pour over the side.

What's left can float the battleship - even if it's considerably less than the displacement of the vessel.

Of course it's in no way practical! Surface irregularities on the surface of a real ship would require much more than a single gallon and ultimately a you cannot get a film of water thinner than one molecule - but 4 litres of water (1 gallon) of water can cover an area of 20 million square metres one molecule thick.

## Floating and Bouyancy

Mark Whitis argues that the battleship does not float as a result of bouyancy.

She is wrong. At least on this planet. Perhaps on a planet of a thousand gravities she would be right. Even if you had the financial and engineering means to create her ideal conditions, the battleship would not be suspended by bouyancy. Not because the theory is wrong but because it is the wrong theory to apply to those conditions. The point she was apparently trying to make was valid but by going to extremes (1 gallon) and tunnel vision, she snatched defeat from the jaws of victory.

In my explanation, float (without quotation marks) refers to an object being held up by bouyancy but "float" refers to superficially similar conditions where bouyancy does not apply or is not the dominant source of levitation. For example, you can "float" one magnet over another, if the poles are in the appropriate orieintation and the "floating" magnet is constrained from sideways motion but it is not actually floating.

Here is an experiment: Take two disposable plastic beer pint cups - these are thin and designed to nest together without too much wasted space. Poor 4 ounces of water into one cup. Pour 16 ounces into the second. Gently lower the 16 ounce cup into the 4 ounce cup.

What happens? It floats. Twirl the cup as one test that it is floating. A second test is to push down on the cup (you lose some water in this test).

On the other hand, if you try to float the cup with 16 ounces in a pie pan with 32 ounces of water, it will not float because it bottoms out first.

If you take your cup with 16 ounces inside the cup with 4 ounces and put it in the freezer long enough, the inner cup will still be held up but it won't be floating anymore. The water will be a solid and not a liquid.

Now if you carry this to extremes of floating a polished battleship in an exact mold of itself, there are some forces besides buoyancy that come into play. At some point, the compressive strength and intermolecular fources will outweigh the buoyancy effect.

Wet the bottom of a flat plate of steel and place it on a flat surface. Does it "float"? yes. Does it float? no. Various effects keep the water trapped there including friction with the surfaces above and below, cohesion between water molecules, etc. As a thought experiment to show how complicated things get when carried to extremes, If you press down on the plate hard enough with a press you can force some of the water out but eventually (if the plate remained perfectly flat and didn't curve due to uneven application of force) you might end up with a layer 1 molecule thick. At this point, downward force would not result in sideways force squeezing the water out towards the edges. Essentially, the water would no longer be a fluid. Of course, with a normal steel plate, the water would just be absorbed in the surface defects. But in real world situations, you end in situations where a thin layer of fluid effectively ceases to be a fluid.

Every hydroplaned a car? The amount of water displaced by the tire was not greater than the weight of the car. The inertia, viscosity, "surface tension", etc. of the water keep it from getting away fast enough. Your car "floats" but it does not float.

Well lubricated Bushings (like ball bearings without the balls) or hydraulic bearings "float" a shaft on a thin film of oil. Many of the worlds largest telescopes "float" on a thin film of oil. A compressor injects oil into the main bearings at all times. Again, this is not buoyancy.

Buoyancy is an oversimplificatin that applies in normal situations where the water can move freely but when the water is reduced to a thin film it doesn't really apply.

If you take a cylinder/piston combinatin and pump it full of water you can support a huge amount of weight, limited by the strength of the seals and the tensile strength of the cylinder. But in this case, it is not floating. The water is not free to move up the sides of the piston ("ship").

In a thin film of liquid, the interaction between the liquid and the nearby solid objects can overwealm the forces between the liquid molecules and the rules which we normally apply to the behavior of liquids become less relevant. Most of the rules of physics we commonly use are approximations that only work within a limited range of parameters.

In some situations the gravitational forces on the water and the "floating" object could be supplemented or overwelmed by other forces such as the air pressure pusing down (or in the case of vacuum, not pushing down) on the water.

Capillary action is another area where microscopic forces can overtake macroscopic forces.

So, even if you polished your battleship and placed it in a polished mold of the battleship with a gallon of water, it would not be floating even if it "floats". The vast majority of the force "floating" the battleship would not be buoyancy. I am not sure how thin the layer of water needs to be to be before less than 50% of the lift is provided by buoyancy. So, it is really not correct to say that even under ideal non-real conditions that you could float (in the strict buoyancy meaning that apparently was used in the original article) a battleship on a gallon of water.

What you can say is that you can float a battleship in a volume of water that weighs less than the battleship itself. But that is not true for sufficiently small volumes of water and certainly not true for 1 gallon of water. She apparently (I didn't read the original article) came close to making a valid point but invalidated it by taking it to extremes where the original forces under discussion were no longer the dominant forces.

An expert could do the calculations (or a simulation) of all the various forces acting on a battleship and the water and calculate for what volume of water 50% of the lift would come from buoyancy. If the water was 1 inch thick, the ship would probably be floating. At one thousandth of an inch it probably would not be. Even one inch is very small compared to the size of the hull of a battleship so a water molecule attempting to scoot out from under the ship would encounter a lot of friction (or shear forces) by the time it reached the edge of the hull. But 1 inch is prettly large compared to the scale of microscopic forces.

It is even possible that the ship would sink, depending on its shape and weight and considering all the forces involved portions of the ship might be able to force the thin layer of liquid out of the way.

An interesting use of buoyancy is in large telescopes. When the telescope points away from zenith (straight up), gravitational forces pull the mirror "sideways" (downward) from the axis of the telescope. An intertube full of mercury is used to float the mirror in this situation back to the center.

http://www.wiskit.com/marilyn/battleship.html last updated September 5, 2004 by herbw@wiskit.com