Recently, my daughter asked me during bathtime, “Why does my arm feel like it wants to float up when it’s underneath the water?”
The explanation I gave her seemed to resonate in a sort of good way, both for me and her, and I wanted to take the time to write it down, and support it visually. I was playing with
paper.js for supporting presentations last night and thought it was a good subject to try it on — actually I was also just looking for a fun thing to write during Covid-19 quarantine.

The reason why things float up is one of the things that confused me terribly as a child, and even as a young adult. It took a long while before I was able to explain it properly.
High-schools teach about these physically fundamental mechanisms with very difficult to understand concepts such as “density” and “forces”, and I think I am not the only one who could be terribly confused by these things, which felt like magical laws that needed to be memorised rather than fully understood.

Explaining it using only simple concepts may ultimately have its own share of problems, but it might at least feel novel to you, depending on your familiarity with the subject.

The Body in the Water

Sebastien the sea bass

We do not care how it got to be fully immersed into water either: someone put it there and then swam away quickly, without leaving a trace.

Actually, this is quite a complex philosophical topic: when do we say several parts constitute “one object”?
Is a partially melted sugar cube, or an oil slick, still “one object”, or are they partially part of the water around them?

Is a sea urchin, with all its spikes and microscopic follicles and pockets of air and organs pumping blood cells round, each cell an intricate little organic machine in its own right, one object or multiple “objects”? What does it mean to be something? Is anything even something? Aah!?
These are good questions, but we do not worry about this for now, we’ll deal with it later! Procrastinating complexity is doing science!

Now, when you’re inside water, you are actually inside an extremely busy mesh of water molecules. Molecules are groups of atoms, but you don’t need to remember that.
These water molecules all look the same and you can largely look at them like little triangular magnets repelling and attracting each other. But more importantly, they wriggle around endlessly. This is called “Brownian motion”, a terminology that you can forget for now.

The foremost thing you need to really really understand about these molecules, is that there are a lot of them.
Here’s our fish in the water, with this new insight.

100000 molecules are on top of each other, in the above image. The result is pretty much the same, except for a wriggly structure, and that’s the point!
Just to get some understanding of the size of an actual real-life molecule, this is the amount of molecules in one small drop of water:

One drop = 2000000000000000000000 molecules!

A royalty-free drop of water

Molecules are tiny, but numerous up to a point that you simply cannot fathom. In a way this is to be expected since there is nothing “else” that makes up “stuff”, so they have to be packed pretty tightly; the number and ubiquity of them is still very hard to comprehend.

You are at this very moment surrounded by a dense mesh of air molecules, bouncing around you and hitting you in every way you can imagine. You are also made out of molecules, doing the same inside your body.
Actually, the reason you are not exploding from the molecules inside your body doing the same against your skin from the inside out, is because you are surrounded by these air molecules constantly hitting you and keeping your skin somewhat firmly shrink wrapped around your body, which is really trying to push it outwards. There is an equilibrium here that would be fairly easily broken by removing all of the air.

If you would now be in space, without these air molecules, a place that is called “the vacuum”, then softer “free” molecules in your body would try to escape, ultimately succeeding in holes through your skin, and you would partially boil and explode, with only the more rigidly connected molecules remaining clogged together — your bones and teeth, for instance.
The reason this doesn’t happen for anyone on Earth’s surface is because there’s a lot of stuff in the Earth’s atmosphere moving, and the amount of that movement per volume is called “atmospheric pressure”. Our bodies and everything here have evolved in that pressure, and their structural integrity is attuned to it.

“That scene” in Total Recall

Note the reason I said “boil” is because the concept of temperature is nothing more than the amount and force of the bounces of these molecules, which is what is measured when measuring heat.
If they are shaking very violently, trying to wriggle free, they might even break free from each other’s attractive force, turning into something steamy instead of something more liquid.
If they’re moving slowly, they will end up in a more crystal-like configuration that is determined by their geometry, like ice.
Your body would actually do both things at the same time left exposed to space, for reasons fairly easy to understand if you remove the effect of the surrounding air. It would not be a good experience for you, not without a suit that creates an artificial atmosphere with bouncing molecules inside it to counteract yours.

Let’s go back to our shape and look at the molecules around it and draw the current “action” for some of them. You could call this action “momentum” but really I’m just drawing where the molecules might be heading at this moment.

The molecules shake around against each other, affecting each other’s motion continuously, and because they have been interacting with each other in such a great number, for such a long time, we can only look at their current movement as being mostly random by now.
Random is actually a fairly misleading word: they bounce against each other and against the object in ways that you can fairly accurately relate to the way billiard balls might bounce against each other.

This simple model of motion is called “classical mechanics”, and consists of simple rules that you are entirely familiar with since it’s what you see around you daily and have been using since you were a kid throwing toys around.
Based on a situation, to get to the next step, you look at the velocities, the masses, and determine the motion.
If you’re more advanced, you can predict complex systems by solving differential equations, or can start making approximations (f.i. Navier-Stokes), but all of that is out of topic of our simple explanation.
You would also be surprised at how little you can really calculate in a real physical sense: even predicting how the interaction between three molecules will end up turns out to be impossible, let alone doing this for trillions and trillions of them — something that is known as the many-body problem. It’s not possible to calculate things accurately in physics because there is too much stuff.

I also said the motion is represented “fairly accurately” with these classical mechanics, because the little bits that make up the molecules, the vibrating atoms and their components, behave in a much weirder way that has no visual classical counterpart, and nevertheless they also affect the way the molecules behave when they rub and interact with each other, albeit in a much smaller degree.
This is called “quantum mechanics” but this little nuance is so small, it has no real bearing on our explanation.

It also is important to note that “classical mechanics” emerges from quantum mechanics because of all the incredible amount of stuff that’s going on gets kind of averaged out through a further weird quantum physical process; there’s no reason to assume there’s two separate regimes at play that you need to remember but the classical one is a lot easier to “see” visually.
Just remember, molecules are billiard balls that have no idea what they’re doing, they just bounce into each other and everything creating random motion because there’s such an insane amount of them.

Now let’s look at the average momentum coming into the fish from all sides.

With this picture in mind it seems that our body gets hammered from all directions with molecules in roughly the same amount and thus, it should stay in place.
To be more accurate, the most it should do is some of that good ol’ Brownian motion itself, twitching around in microscopic amounts, not observable by the human eye. It certainly should not be floating up!
This would be actually be the case if we were in deep space. But we’re not in deep space, are we? We are leaving out an important other phenomenon.

How gravity makes things go up

A different way of saying this, is that the water on the bottom feels the weight of all the water on top of it. To be honest there is some difficulty with that image so I prefer imagining the molecules being pulled down and becoming more densely populated at the bottom, with their interconnected mesh that is a lot stronger than gravity making sure they’re not torn apart from each other.

Why does this happen, why does gravity happen? Ultimately, nobody knows.

It is possible to go in a long amount of detail in how gravity can be reinterpreted by several models, but ultimately, the gist of it is: we do not know, and it doesn’t matter for our explanation. It is the only part of this explanation that you will just have to take for granted. Gravity is.

It seems like the way things float up can easily be described by taking gravity as a thing that just happens. It also seems — I haven’t tried — that if you try to do the opposite, and take it for granted that things float up, it becomes a lot harder to derive gravity and other phenomena out of that. You can do it but the maths would become more complex and require new concepts.
From a mere language perspective then, it’s easiest to treat gravity as something that is just there. It is one of four fundamental phenomena that create everything, four phenomena which will all hopefully end up being explained by one thing, one day.

Look at what happens if we add the effect that gravity has on the water!

Can you see it? The water is being pushed down, and on the bottom of the fish, there is now a lot more interaction with molecules going on than at the top. So, the bottom of the shape sees a lot more bouncing than the top. This business is called “pressure” — there is more “pressure” on the bottom.

Let’s color all the “upwards” moving molecule action just to make a point.

As you can see, there are way more upwards-moving molecules hitting the bottom of the fish than there are downwards hitting molecules hitting the top.

Ultimately, this will cause two things; for one, the fish will be hit with more “upwards” momentum than “downwards” momentum, and also, the mesh of molecules around the fish will be strengthened together to move “upwards”, since the fabric of the water stays together.

The net effect of these two will cause the body to go up, and up, …
One thing stopping it from going up freely is the friction by the water above it, so there is a subtle equilibrium involved there — if things end up floating up very quickly, they will be slowed down more by the conflicting momentum of all the water molecules above it.

There, we are done!

Not quite, of course.
We forgot about some very important things!
Let’s look at least three more important observations before bringing it all together.

Gravity affects everything

Actually, is this so easy to understand? What does this mean in the case of my daughter’s arm for instance, and the little hairs on her arm and the immense complexity inside it? How does gravity know what part of everything is part of the object and what part is part of the water, or something else?

What really happens is that gravity affects everything, in just the same way, but things we identify as “objects” constitute of molecules that are more or less interconnected with each other, so they kind of stay together or at least oriented in a predictable manner relative to each other, and thus keep being the same “object” when they are being pulled or pushed apart.
A drop of tar falling from a height will actually deform from gravity as it breaks free, but ultimately its molecules will stay together: even when they splatter flat somewhere, they are still somewhat interconnected with their magnetic force.

Thus, gravity affects the shape in the exact same way as the water, but the shape itself can be considered “one object” because it has some kind of structure that stays the same or deforms predictably. That answers our debate about the sea urchin.

The molecules in my daughters arm are hugely complex, with an extreme diversity. You will probably be surprised however that 99% of my daughter’s molecules are water molecules.
Most people know that 72% of a body’s weight is water, but less people know that that means that 99% of the molecule diversity is water — simply because the other types of molecules are heavier!

So, gravity pulls at all the molecules that make up our shape, and they stick together because they have strong connections, so we can still keep talking about the same “shape”. Gravity pulls down the fish, and it pulls it down stronger based on how much mass the fish has.

Naturally, as you can see, the object will only move “up” when the up arrow is bigger than the down one. So what matters for that to happen?

Size matters — in a way

Ultimately, the surface does not matter, and it is easy to figure out why.
The thing is, while the molecules need to affect the surface for them to affect the object, for each piece of surface they hit, it is sort of the height of the object above that small surface that determines how big the pressure difference is when the movement ripples through the object.
To get this total pressure difference, and thus the total net upwards effect, you need to average this out for all surfaces. This is what in mathematics is called an integral and it is quite easy to visualise.

Taking a small part from the fish’s belly, we can try to calculate how much the pressure difference is for any vertical movement propagating just through that part. If you then do this for infinitely small surfaces, with a chosen direction of motion, you are effectively calculating an integral across the fish.

Well, the thing is, the result of this integral is nothing else than the volume of the fish. The maths work out in such a way that the surface is completely cancelled out in the equations and only the volume is important for the motion of the object.
So, the up arrow grows larger as the volume becomes larger, and is also determined by how much gravity squeezes the water.

Now, as you might understand more easily, the “down” arrow grows larger as the “mass” of the object grows, because gravity pulls on all of the parts of the object “harder” if their combined mass is larger. Actually, mass is something very complex that ends up being a mostly theoretical concept, but since it is the main component affected by gravity you can easily understand it by that effect, an effect we call “weight”.

Therefore, size matters for the Up arrow, and actually we’ve established three things matter:

- heavy objects (large mass) will cause a big Down arrow — because of gravity
- large objects (large volume) will cause a big Up arrow — because of the influence by motion and the pressure difference
- the way that gravity affects the liquid water, the amount that it can “squeeze” it, also determines the size of the Up arrow.

Now, the buoyancy of the water as I’ve stated above it largely relates to the weight of the water itself — how much one volume of water weighs, in terms of mass. This simplifies the entire thing to “the Up arrow is bigger than the Down arrow when the object is less dense than the water”.
I myself am not fully satisfied with that, since ultimately some other factors can end up affecting the influence that gravity can have on the molecular motion, and just comparing the weight of the water with that of the object, is not really a good description of the entire effect. But if you read about “buoyancy” it will usually refer to that shortcut concept.

So now you know that things float when the Up arrow wins from the Down arrow.
You end up with a fairly accurate explanation of why a bowling ball sinks, because the Down arrow wins from the Up arrow, and why a light, large object might have a winning Up arrow, and thus float — a piece of ice is a fairly good example of the latter.
Also, some liquids cause objects to float that would sink in other liquids, because of the third point!

When you calculate everything out, and stick to simple conditions, some stuff cancels out and ultimately you will end up with Archimedes’ law. However, this law breaks down in a lot of dynamic situations, so it’s better to understand it from its core concepts, in my opinion. I also always had a hard time understanding this law as a child, although this may be a personal thing.

Actually, this is an ok explanation of flotation, but there are still a couple of things to understand for a full picture.

Does shape matter?

Well, if the parts don’t really disturb the molecular motion — so if the arms do not move, then it doesn’t matter if the shape is a squid or a cube or a sphere. You could melt it down to something else, and it will just float in the same way.
Ultimately, the volume is all that matters. This may come as somewhat of a surprise, because you can imagine a strange creature with millions of tiny arms — looking like small hairs — and it wouldn’t matter!

The shape however, may balance itself as it is floating upwards — just because of the molecular movements finding an equilibrium — and in doing so, it may affect the molecular path and cause a delay in its flotation. Different mechanisms can be created as such, that speed or slow down flotation just because of how they affect the molecular pressures around them as they float up. Similarly, the octopus may suddenly decide to swim, which creates an entirely different complexity.

‘Helicopter’ seeds that have structures to affect the air around them as they float

While shape does not matter, if you are creating air pockets, meaning holes that capture air, things do matter, because then you have effectively changed the volume vs weight of the object and you are affecting the Up/Down arrow sizes. This is obviously why a football would float up in most liquids, because its volume is quite high for its weight, so it gets a lot of that molecular movement without gravity being able to win against it.

I would also like to add that if your shape ends up with very very thin structures, you can abuse some of the quantum mechanical effects — quantum electrodynamics, notably, and break the statement that shape does not really matter.
These would very thin structures, and it would probably be impossible to speak of them as belonging to an object as there might no longer be a persistent connection to them and the octopus, or fish.

When the water moves

Suppose the fish is surrounded by a somewhat more “organised” mesh of water molecules:

Someone might have jumped into the water, and caused a disturbance that effects the mesh of molecules in such a way that they follow a more organised path of motions — and they reinforce this in the mesh of each other’s attractions, obviously, so it’s kind of a viral effect of motion.

Now, just note that it is impossible to do this perfectly. Meaning, you will never be able to make all the molecules behave in exactly an organised manner. To do so you would need to create such a large speed that the water would probably turn into a ball of plasma and you would not be able to keep it together — not without doing another thing that is impossible. There will always still be some of that Brownian random motion lingering in each molecule.
I guess that is a deep difference between physics and maths: things are never fully clean.

Obviously, when the water moves, the shape of your body will become important. You will enter the terrain of hydrodynamics, (this is basically aerodynamics, but with gravity still affecting the medium in a significant manner), and the nose of the fish will tend to go “up” simply of the way that its surface tension — the molecular rigidity of its skin, bones, etc — capture the bounces of the molecules and convert it into an organised motion (for instance “slightly up”). So yes, now shape does matter, obviously, and the octopus knows this very well.

But there is a more subtle and amazingly simple fact that emerges as well: if a molecule follows a certain direction, well, then it can’t follow a different direction at the same time, can it?
Meaning, for molecules that are going decidedly “left”, they cannot at the same time go “up” as well can they?

Well, this is precisely how an airplane stays in the air!

By shaping the wing in a highly specific manner, you create areas where the air is very “organised” and small vacuum pockets where the air is “non-organised”. The “organised” air cannot push “down” on the wing anymore because it is too pre-occupied with going in a certain direction. But the unorganised air will push “up”, simply because it will still have some molecules with that direction. So, by tricking the air in such a way, you can end up with a larger “up” vector, and the airplane wing stays in the air. The only thing it requires is for the air to hit it with a certain speed to develop the effect — and also, for there to be air molecules in the first place!
And, you need a wing, built according to exact specifications — so you need a lot of order to convert the inherent disorder into something useful!

One last thing about things “moving” up

This actually makes things extremely more complex because objects may be stretched and squeezed in all sorts of ways while they are floating “up”, in a kind of chaotic slinky pattern.
So, if you really want to know how things float, you’re not quite there yet as you would need to agree on what exactly it means to “move” something.

There, we’re at the end of our story. This is to a large extent, why things float up.



Dad, nerd, runner.

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