I want now to go into more detail discussing fluids, both...
... static fluids
Here the question is if the body will float or sink into the fluid and to what extent. Books say that everything depends on the relative density of the body. That looks like pointing at numbers: it seems like the winner should be the one that encapsulates higher numbers within a given volume, but that is not enough, we still need to know how many of those guys will get involved.
a) Let us start with the case of a body (like a log of balsa wood) that is less dense than water.
This object will float, yes, but the truth is it starts sinking... The reason is that the wood is solid and so there are rigid connections between its molecules. Thus when the first row of wood molecules clashes with the first more massive layer of water, wood soldiers will call the rest of their solid army through their connections and persuade them to take part in the interaction, to the extent needed. Instead in the water band the intermolecular cohesive forces barely serve to keep the particles side by side, but there is no solidarity among them: if one is pushed aside, it will slide over its colleagues, which will not participate in the interaction, until they are pushed themselves. This explains that the balsa wood commences by penetrating the water: it does because it outnumbers its "involved" opponents.
That does not mean, however, that the fight is over. The reason is that, although water molecules do not get "involved" in the fight until they are displaced, from that moment onward they do keep all of them involved. Hence the wood log will only be winning the battle until it has displaced and thus involved in the interaction water molecules amounting to its own mass, i.e. (in a sloppy way) until it is equal-numbered by involved water molecules. This will happen when it is only partially submerged, precisely because it is less dense than water, so the latter manages to pack up the equivalent of the body's mass in only the submerged volume.
b) If the test body were of the same density as water, it would only be equal-numbered when it became totally submerged, with its top side at the level of the surface of the water, because precisely at that moment the involved (displaced) water mass matches its own mass.
c) And if the body were denser than water, it would sink to the bottom because its mass will always outnumber the involved water mass opponents, which are as usual those displaced by its volume and hence of less mass than its own mass.
That is Archimedes principle, explained in terms of numbers or, if you want, in terms of "the winner is at each moment whoever involves more mass", taking into account that by definition the solid body involves 100% but the fluid involves the displaced mass.
As natural as this rule looks (involved water molecules are those having been displaced), there is a story behind it.
The reason you read in books is that (i) pressure depends on depth, (ii) pressure acts in all directions, (iii) pressure on each side of the body neutralizes, (iv) we are left with pressure from below acting upward minus pressure from above acting downward and (v) force is pressure per area, in this case, area of the sinking body, so the net force is (density of water * g * height of body * area of body) = m of displaced water/V displaced of * g * h*A = m/V * g * V = m * g, i.e. you get the weight of the displaced water mass.
Isn't this as clear as cold and arid?
In trying a more colorful alternative, I would start with noting the experimental fact that "water floats on water". In general, a fluid floats on itself. This means that it is at equilibrium: the weight of a column of water (gravity force exerted on it by the Earth) is matched and counteracted by support force exerted by what is below. This accounts for the fact there is a force upward equal to the column of water until the bottom surface of the body.
But why does upper water float and not penetrate the slightest into water? It does because it is of the same density, but with a difference with respect to what we said about solids: wood of the same density as water partially sinks because, being a solid, it does win the battle against the first layers of water by calling into service all its molecules; instead, water cannot do that, so it does not penetrate into neighboring water, but what it does do is pushing such adjacent molecules and thus transferring pressure downward, until it meets the ground. And the ground is another thing. The ground is a solid that exerts a constraint force: it is very massive and rigid and thus able to call into service as many soldiers as needed to avoid penetration and match whatever rests on it. That is why the support force sustaining the patch of water at each level of depth is equal to its weight: because it is held by the ground, with the water in between acting as a neutral messenger between above and below.
.... and dynamic fluids
Here the question is... I would say that in practical terms it boils down to whether a body may suddenly be pushed because of a pressure difference that causes the fluid to rush from some area to another.
Again this will happen because one side outnumbers the other, due to some physical configuration, of which there are two basic types.
One type is those situations where there is some sort of wall that produces this difference: bigger numbers on one side than the other. For example:
- Narrowing: when a pipe narrows, the molecules at the wider section outnumber those at the narrow one, pressure is behind greater than in front, so the more advanced molecules are accelerated and hence acquire more velocity. Thus the water comes out of a hose with a nozzle at greater velocity. It is important here to disambiguate "pressure": we say that the fluid comes out with more external pressure, meaning that the ordered motion of the fluid as a bulk in one direction is faster, but that is because the internal pressure, meaning the random motion of its individual molecules in all directions, is lower at the narrowing than at the wider precedent area.
- Shielding: a smaller boat passes by a bigger one, which shields it from the pressure of that side of the ocean; so the mass on the other side outweighs the mass on the side of the passing big boat and pushes towards it the water and together with it the small boat.
The other type is situations where directly the fluid is pushed and accelerated by some factor, thus reducing the numbers in that part:
- You blow between two pieces of paper, so as the air is passing between them more quickly, the pressure is lower there than outside, hence the pieces of paper are drawn to each other.
