As you might have noticed, some objects float and some others don’t. Here below I have a rendition of a boat and a cube of ice floating.
Today, I will go on a bit about flotation. As a matter of fact, some of you might remember a puzzle with an egg I wrote down a while ago. Of course, most of you have probably heard of Archimedes Principle as describing flotation, so I will explain some aspects of how that principle comes about.
So the first thing you have is a fluid, and secondly you place an object on top of the fluid and you can ask if it will float or not. But this can depend on the shape of the object that is floating. If someone asks wether iron floats on water or not, the answer is that iron sinks to the bottom. Except that we have iron boats (nowadays they are mostly made out of steel). Indeed, Metal boats were a XIX century wonder and many thought they could not possibly float. Nowadays we are so used to it that we don’t stop to see the wonder of it.
In the end, the important concept is pressure. In particular, the liquid exert pressure on every object inserted in it. The pressure is defined as a force per unit area. It is exerted uniformly in all directions. This is a property that liquids have. So if you have an area that encloses some volume, you can expect that there is some pressure exerting force on the corresponding area.
However, the force is perpendicular to the area in question, so if you have some complicated object, force will be exerted in all kinds of directions, because the direction of the surface of the object is changing. So let us stick to a rectangular -shaped like a box- object as it is more simple. We can also make it as small as we like.
On each area of the box there will be some force pushing it. If we look at the vertical direction, there is a force from the bottom pushing it up and there is also a force from the top pushing it down. In the vertical direction these are equal to
Where P is the local pressure at various points, A is the area of the base and h is the height. There will be similar expressions for the horizontal directions. So if we divide the formula by the volume, we find that there is a force density per unit volume equal to the gradient of the pressure. This is
In particular, consider a liquid in equilibrium in the presence of gravity. That means that the liquid is not moving and it is not experiencing acceleration. We have two forces acting on the liquid: gravity and the pressure gradients. There should be a balance of forces between these two. We this find that
where is the mass density of the fluid.
Thus, we can solve for the pressure at various depths. We find that
So if we submerge a box, the top of the cube feels less pressure than the bottom. The total force is
where we see that the right hand side is the weight of a box of the liquid (the displaced liquid as in Archimedes Principle).
This explains why a metal boat can float. So long as the displaced volume of water weighs more than the boat, the boat will float. This requires one to have a shape that guarantees that the boat is in the end lighter than the water. We do that by adding a lot of empty space filled with air.
Incidentally, since ice floats, the volume of liquid water that it displaces weighs more than the ice itself. This means that ice is less dense than water. So when water cools off below freezing: it expands. This expansion can cause excess pressure on water pipes. After all, the ice will be trying it’s best to expand to full size and may cause the water pipes to break.
There is one more thing to consider. Most objects get a bit smaller when compressed, but water is pretty incompressible. Thus, if you try to keep an object at a fixed depth and it goes a bit further down you will see that it will most probably start sinking, whereas if it goes a bit up it will most probably start to rise. This actually happens to fish and many have developed a swim bladder to help with this. The fish ‘empty’ the swim bladder if they go too high and fill it up if they are sinking too much. Thus, any equilibrium in depth they have is dynamical. Sharks on the other hand rely on hydrodynamic lift instead to accomplish this equilibrium.