Tuesday, February 02, 2010

Buoyancy or Why do ships float

This topic was initiated by DaBoss with an apparently simple question - If I weigh X now and lose 10 pounds if I strap a 10 pound barbell to myself and go swimming will it be the same as not losing the weight?

It seems that the principles of buoyancy were something that was either not taught or quickly glossed over and forgotten.

In an effort to explain I took a quick gander at Wikipedia and then turned away, the explanation is there, but well wrapped up in either mathematics or dense text so here's a simplified expanded version.

Water has weight, to keep matters simple a 1cm cube of water weighs 1g so a 10cm cube weights 1kg (10x10x10 1cm cubes). If I take a cube of metal that weighs 1.5kg and submerge it in water the amount of water it displaces (i.e. pushes away) would form a 10cm cube. As stated that amount of displaced water weighs 1kg, the metal cube weighs 1.5kg so it sinks.

If I try a similar 10cm cube weighing 0.5kg it displaces exactly the same amount of water weighing 1kg so it floats.

Finally a cube weighing 1kg displaces exactly the same weight in water so when released stays exactly where I left it.

So the 0.5kg cubes floats, but by how much? The answer is until the two weights balance and essentially cancel each other out. So if I place that cube halfway into the water the amount it displaces is half a 10cm cube and that weighs 0.5kg the same as my metal cube. So my 0.5kg cube will float along the half-way line

But all that assumes that the metal in the cubes is distributed evenly, what if the bottom half of my 0.5kg cube is empty and the top half full? Say the bottom half weighs 0.1kg and the top 0.4kg, if I drop in the 0.1kg side what happens?

Well now it gets slightly more complicated. In this simplified case there are two forces acting on the cube, the pressure of the water upwards and gravity (weight) acting downwards. If we place our lopsided cube into the water such that the centre of gravity both halves lie above each other than this appears as a single 0.5kg force downwards which in turn is countered by our 0.5kg of displaced water acting upwards so it'll sit there as before.

What happens if I start to rock it slightly? Instead of having one face of the cube pointing straight down let's say I tilt it so we have one edge pointing down. We're still displacing our 0.5kg of water, but now I have only half the 0.1kg and half of the 0.4kg halves in the water. So half of 0.1 plus half of 0.4 is 0.25kg. That's lighter than the water so our cube should rise. But we've changed the relative positions of the centres of gravity so they no longer lie directly above each other. Now we've the equivalent of a 0.4kg force and a 0.1kg force acting downwards. The displacement of the 0.1kg half is creating an upward force of 0.25kg as is the 0.4kg half. So the lighter half wants to rise and the heavier half wants to fall. So our cube topples and the heavier side ends up face down in the water and we appear to be back where we started with it floating along the halfway line.

But that's for solid cubes, why is it if I take the same amount of metal in our 1.5kg cube and flatten it out into a curve does it float? Well lets try it and see what happens. I'll take my 1.5kg cube and turn into into a hollow hemisphere with a radius of 10cm. Now when I start to drop my hemisphere into the water the amount of water displaced is equal to that of the metal shell and the amount of air our hemisphere contains up to the water level. Air is lighter than water and in this case I'll call it a negligible weight, that means I can ignore it and just use the metal's weight.

If it were a perfect sphere it would have a volume of four-thirds pi r cubed so as it's a hemisphere it's half that. That's roughly 2094 cubic centimetres and it still weighs 1.5kg. If I push it into the water to the point where it's almost submerged it's displacing a little over 2kg of water; it weighs 1.5kg so it floats.

Now if  I pushed it down further so water starts to fill our dish or started putting things into it then the weight increases; if it goes above 2kg then the dish will sink.

So this is why certain things sink, but will float if you simply change their shape.

So to answer DaBoss no it won't be the same strapping a 10 pound barbell to yourself because before you lost the weight it was distributed along most of your body (high volume) whereas a barbell would concentrate that weight into a small volume. However if the weight was somehow kept out of the water it would be closer to how it was before, but still not quite the same because of the imbalance.


Orphi said...

To summarise several pages of text: No, it wouldn't be the same. You'd have the same mass, but a different volume, and hence your bouyancy would not be the same.

Regardless, you should still lose 10 pounds, fatty.

(Oh, now he's going to ask about using a megaphone, isn't he?)

FlipC said...

I'm just surprised at how many people don't know this and how poorly Wikipedia tries to explain it (yes I could alter it, but pfft).

Reinforced Balcony a must for every critic; I knocked off Gloria's theatre last night. Lost only one projection level too so I was impressed with myself; hell I lost 7 just trying to get the last figment in Basic Braining the one floating around the log run - yeesh.