Floating a Battleship in a Bathtub

Gold Member
I had always thought that things float when they displace an amount of water that weighs more than them. Bur obviously this is wrong because I recently read about "Floating a Battleship in a Bathtub" were the author of a physics book said that as long as there's enough water to fit between the boat and the tub, even a millimeter thick, the boat will float.
So what determines if it can float or not? He mentioned pressure on the boat.
Also, what is hydrostatic pressure and how does it have to do with this?
Thanks.

Mentor
You have to understand what is meant by amount of water "displaced". It means the volume taken up by the object compared to the surface of the water, wherever that happens to be. So, assuming the bathtub is big enough to float the battleship, the water level will rise to some point on the tub. The volume of water that you'd have to replace if you removed the ship (to keep the water level at the same raised point in the tub)--that's the amount of water "displaced". That (large) amount of water will weigh as much as the ship.

Gold Member
Why does that amount of water keep the ship up?

Mentor
It can be shown that the net upward force due to the water pressure on the object--the buoyant force--will equal the weight of the displaced fluid. That's called "Archimedes' Principle". If the buoyant force equals the weight of the object, the object floats. (If the buoyant force is less than the object's weight--it sinks like a stone.)

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Mentor
Here's another way to picture the amount of water displaced that might make more sense. Imagine your giant bathtub filled to the brim with water. You carefully lower your battleship into it. The amount of water that overflows the tub--that's the amount of water displaced by the ship. Note that the amount of water that remains in the tub can be quite small--but the amount displaced must weigh as much as the ship does if the ship floats.

pallidin
I am not convinced of this so-called effect of floating a battleship with, say, only 1 gallon of water, even after "filled bathtub displacement"
Here's why:

A bathtub IS NOT a closed system. As the battleship is lowerd into the water, water moves upwards along the sides of the tub. In order for the battleship to not touch the bottom of the tub, there must be an equal force opposing it.

Where is that force? I don't see it. IF something were to STOP the water from rising, than yes. But there is nothing but atmospheric pressure and the capillary effect to impede the rising water; surely much less than the force of a ship lowering.

So, the bottom of the heavy battleship will touch the bottom of the tub.

But, what about "special" circumstances? I've heard that a "V" shaped hull in water-filled "V" shaped container will produce pressure forces which allow this to happen with a minimal amount of water. Not sure if it's true.

Homework Helper
pallidin said:
Where is that force?
It's the pressure differential within the water versus the lower pressure of the air at the surface. Water pressure applies a force on the sides and bottom of the ship, but not on the top, so you have a net upwards force due to the pressure differential. The pressure in the water increases with depth as well, which also increases the pressure diffferential, and the net upwards force.

Gravity is supplying the original force that produces the pressure in the water and the downforce of the ship.

I've heard that a "V" shaped hull in water-filled "V" shaped container will produce pressure forces which allow this to happen with a minimal amount of water.
Pressure of water is related to the depth of the water, not the volume.

Mentor
pallidin said:
A bathtub IS NOT a closed system. As the battleship is lowerd into the water, water moves upwards along the sides of the tub. In order for the battleship to not touch the bottom of the tub, there must be an equal force opposing it.
The rising water, and thus increased water pressure difference along the surface of the ship, provides that buoyant force. The tub need only be deep enough so that the ship can float and not touch bottom.

Realize that this is just a thought experiment to dramatize Archimedes' principle. Lots of luck arranging things so that you can float a real battleship in a gallon of water. But the principle is quite real.

Looks like Jeff Reid beat me too it!​

Gold Member
pallidin said:
Where is that force? I don't see it. IF something were to STOP the water from rising, than yes. But there is nothing but atmospheric pressure and the capillary effect to impede the rising water; surely much less than the force of a ship lowering.

So, the bottom of the heavy battleship will touch the bottom of the tub.

Just to clear up a possible point of miscommunication/misunderstanding - we are all agreed that:

for a hypothetical battleship that typically has (say) 10 feet of draft in open water, you WILL need a bathtub that is 10 feet deep

right?

As Doc put it:
Doc Al said:
The tub need only be deep enough so that the ship can float and not touch bottom.

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Homework Helper
Doc Al said:
Lots of luck arranging things so that you can float a real battleship in a gallon of water. But the principle is quite real.
Taking a USA aircraft carrier through the Panama Canal with about 1 foot of clearance on the sides and not much on the bottom (by design), comes about as close as you'll get to a battleship in a bathtub.

Mentor
An excellent example!

pallidin
I have seen this debated all over the web. Yet, both DocAl and Jeff Reid seem to provide both the theoretical dynamics and real-world example of this effect. Nice job.

I shall now meditate upon this issue while floating in my bathtub filled with only 16oz of beer.
Ok, I probably WILL sink, 'cause I will drink the beer while floating. ICK!

mangurian
anyone out there have the math on this?

Homework Helper
anyone out there have the math on this?
What maths?

mangurian
a formal physics solution. relevant forces in terms of densities, gravity, etc.

Mentor
buoyant force = weight of fluid displaced

Not sure what exactly you are looking for beyond that...

mangurian

The link actually has some physics...Shouldn't The battleship in the tub solution calculate the approx pressure gradient due to gravity and show how the differential pressure on the top versus the bottom of the ship will float it. Relevant parameters include ship mass, density of fluid, fluid volume, bathtub volume, and idealized ship dimensions (like a box: x by y by z).

What am I missing here?

Mentor
Could make an interesting exercise, but is not necessary since Archimedes.

Besides, you will get the same result.

mangurian
ah well...I am not interested in answers. My interest lies in how to get them.

Just saying F=MA or in this case B= W displaced is very boring (to me) and provides zero insight. :zzz:It is just a workbook exercise. As in: "I have a formula. That's all I need."

It leaves out the beauty of physics.

Kepler told us how to compute lots of stuff about planetary motion...useful but BORING.
Newton provided the insight and the "why" behind the answers (ie. the beauty).

Mentor
This issue is so simple (a certain volume of water weighs what it weighs) that I don't see how anything is gained by your approach. But if you want, I'm sure you could calculate the weight of a block of water by integrating pressure gradient with depth, if you want. All that tells you though is the same thing that stacking weights on a scale tells you: the force on the scale is the sum of the weights of the weights stacked on it. By stacking 1lb weights one at a time, you can read the scale and verify that 1=1, 1+1=2, 2+1=3, etc. That's an integration force vs height of the stack. It really seems to straightforward of a concept to bother with to me.

But here's how to do it: multiply the cross sectional area (a single number if uniform, a function if not) by the integral of the weight density vs depth.

mangurian
I will now let this topic go in that you seem not to get my point.

Gold Member
It seems that one of the difficulties with this thought expereient is how a single gallon of water can hold up a battleship. This seems counterintuitive.

But of course, it's not merely the single gallon of water, is it? That gallon of water is braced against many feet of concrete, which is braced against an unlimited amount of earth. And that can definitely hold up a battelship.

Not convinced this is a relevant factor? Consider what would happen if you decided to eliminate this bracing. Eliminate the dirt, then reduce the concrete to one foot thickness. It would, of course, explode into wet rubble with a beached battleship wallowing in the middle. The nature of the walls of the tub definitely factor into the solution.

Consider a second scenario in which we have a battleship at moorage, supported by the ocean. None of us have any problem at all with imagining this. Now, start filling the ocean bottom around the battleship with concrete. Continue filling the ocean with concrete, molding it very carefully to within one millimeter of the hull until the entire hull is enveloped in concrete at a distance of 1mm. Anybody see a point where the battleship would suddenly lose bouyancy?

dovvod
Not convincing to me.

Suppose you have 100 + 10 tons of water in a tub. Put in a 100 ton battleship. It displaces 100 tons of water which pours out of the tub. Let's say the boat floats in the remaining 10 tons of water.
Is it floating because it displaced 100 tons of water?
Obviously not.
If you start out with 10 tons of water and put in the battleship then the final scenario is the same as above. I presume if it floated above it will float here. But here there is no displacement of 100 tons of water.
Some of the other explanations (air pressure, hydrostatic pressure, etc) might make sense and be correct by the laws of physics. But they are not intuitive.
The need for 'displacement of water' seems to be an attempt to make the explanation intuitive. But it fails for me.

Gold Member

Suppose you have 100 + 10 tons of water in a tub. Put in a 100 ton battleship. It displaces 100 tons of water which pours out of the tub. Let's say the boat floats in the remaining 10 tons of water.
Is it floating because it displaced 100 tons of water?
Obviously not.
If you start out with 10 tons of water and put in the battleship then the final scenario is the same as above. I presume if it floated above it will float here. But here there is no displacement of 100 tons of water.
Some of the other explanations (air pressure, hydrostatic pressure, etc) might make sense and be correct by the laws of physics. But they are not intuitive.
The need for 'displacement of water' seems to be an attempt to make the explanation intuitive. But it fails for me.

The 10 tons of water is supported by the ground around the bathtub that's supporting the battleship. Basically, the battleship has displaced 100 tons of dirt.

But you are right in that you could build a bathtub on stilts with a strong enough support scaffolding that displaces no water or dirt, and still support it in a mere 10 tons of water.

Mentor

Suppose you have 100 + 10 tons of water in a tub. Put in a 100 ton battleship. It displaces 100 tons of water which pours out of the tub. Let's say the boat floats in the remaining 10 tons of water.
Is it floating because it displaced 100 tons of water?
Obviously not.
Why not? You even used the word "displaced" in your explanation!
If you start out with 10 tons of water and put in the battleship then the final scenario is the same as above. I presume if it floated above it will float here. But here there is no displacement of 100 tons of water.
When you put the battleship into a pre-formed tub with 10 tons of water, the water rises to fill the gap around the ship. No, it hasn't moved 100 tons of water out of the way, but there is a ship-shaped space where 100 tons of water would fit if the ship wasn't there.

Molydood
we are talking about one particularly stupid looking bathtub, I know that's beside the point but I think the original statement should be 'float a ship in a bathtub of water', not just 'float a ship in a bathtub'

I know some people seem to find this concept bizarre to the point of unbelievable, but although I did too when I first read it, the way I got around this was with the following thought process. Envisage a ship floating in completely undisturbed water (ie in complete and constant equilibrium) then have divers slowly build a watertight frame around the ship underneath the water just 1 mm away from the surface of the ship... as the last brick goes in, nothing happens... but you could at that point drain out all the ocean water and still be floating the ship, except in minimal water

just my 2c.

btw, agree about the simple math comments - as an example, the 'bathtub volume' as mentioned by Mangurian would be merely an indicator of your manufacturing tolerances of how accurately you could match the (hyperthetical) bathtub dimensions to the ship dimensions, so not even anything to do with mathematics (that particular variable would be an indicator of manufacturing capability, to pick just one example)

Molydood
oops, just noticed there is a page 2 - and what is quite bizarre is that Dave has made an almost idenitcal comment to mine - dun dun duuuuuuh...

so apologies for the duplication. but quite interesting never the less!