Because they’re more or less dense than water. QED. Any more questions?

Yeah, sure… of course that’s true, but why? Why do denser things sink while less dense things float?

That’s a harder question. Let’s start with a common explanation (that’s pretty much correct, by the way) and then try to poke some holes in it.

## Common explanation

Consider a vertical column of water. Now consider a horizontal plane $$p_1$$ cutting through the column of water at a depth of $$h_1$$. What are the forces acting on this horizontal plane?

Well, all the water sitting above the plane is pushing down on it. With what force? The force is just the weight of the water, so $$F = mg$$ where $$m$$ is the mass of all the water above the plane.

Are there any opposing forces? Yes. The water below the plane is pushing back (up) with an equal and opposite force. In a sense, it’s holding the water up. This would be more obvious if we were talking about solids, but it’s still true for liquids.

Ok, now consider the exact same scenario except position the horizontal plane at a greater depth $$h_2 > h_1$$. Let’s call this (deeper) plane $$p_2$$. What are the forces on $$p_2$$?

It’s the same story as before - the force pushing down is $$F = mg$$. The only difference is that $$m$$ is greater since there is more water (with more mass) above the plane.

Similarly, the water below the plane is pushing back up with the same, greater force.

Now imagine the water that’s sitting between $$p_1$$ and $$p_2$$. We’ve just established that there’s a greater force pushing up on $$p_2$$ (at a depth of $$h_2$$) than pushing down on $$p_1$$ (at a depth of $$h_1$$). So if there’s a net force up on that region of water, why isn’t it moving up? Because the net force up perfectly balances with the force of gravity pulling that region/volume of water down. A net force of zero means it doesn’t move up or down.

For the last step of this explanation, let’s imagine replacing all the water in the column between $$p_1$$ and $$p_2$$ with an empty box with zero mass. Now what happens? There is still a net force up on that region of water but - since the box has no mass - there is no balancing gravitational force. In this case there really is a net force pushing up and so the box starting floating up towards the surface.

This argument doesn’t rely on the box having literally zero mass. Anything with a mass that is less than the net force pushing up will float up. Since the net force up exactly equals the gravitation force on that volume of water, anything that has less mass per unit volume than water will float up. In other worse, anything that’s less dense than water will float. QED.

## What’s the problem?

The problem is that I can make the exact same argument about solids. Instead of imagining a column of water with three discrete regions (the water above $$p_1$$, between $$p_1$$ and $$p_2$$, and below $$p_2$$), imagine three blocks of iron stacked on top of each other. Let’s go through the argument again.

$$p_1$$ is the plane that separates the top block $$b_1$$ from the second block $$b_2$$. What forces are acting on that plane? Same as before - the block $$b_1$$ is pushing down with a gravitational force $$F = m_{b_1}g$$. The block $$b_2$$ is pushing back up with an equal and opposite force.

What are the forces on $$p_2$$? The force pushing down is the gravitational force of block $$b_1$$ and $$b_2$$ combined: $$F = (m_{b_1} + m_{b_2})g$$. The block $$b_3$$ must be pushing back up with an equal and opposite force.

If $$b_3$$ is pushing up on $$b_2$$ with a force of $$(m_{b_1} + m_{b_2})g$$ and $$b_1$$ is pushing down on $$b_2$$ with a force of $$m_{b_1}g$$, then there is a net force up of $$m_{b_2}g$$. Why isn’t $$b_2$$ moving up then? Because the net force is exactly equal to the gravitation force on $$b_2$$ itself: $$F = m_{b_2}g$$.

Now replace block $$b_2$$ with a equally sides block of wood that weighs less than iron. Let’s call this block $$w_2$$ ($$w$$ for wood). The force pushing up on the wood block is greater than the force pushing down on it. Furthermore, the net force up is greater than the gravitation force on the wood block ($$m_{b_2}g > m_{w_2}g$$). So what will happen? The wood block will start floating up!

Oh wait… except it won’t. Common sense tells us that if you stack three solid blocks on top of each other, the middle block will definitely not start floating in the air - even if the middle block weights less than the other two blocks.

## So what went wrong?

The argument does not rely on the ability of the medium to flow. But clearly this is an important property when it comes to whether something will actually be able to sink or float in real life. Things can float or sink in liquids and gases, but not solids. Liquids and gases can flow; solids cannot.

We can also approach “what went wrong” from a different angle. In the case with three solids, one step of the argument is just false. When you replace the second block of iron with a wood block, the force pushing up on the wood block from below will decrease. Why? The force pushing up on the wood block is exactly equal to the gravitation force of the top two blocks pushing down. When you replace the iron middle block $$b_2$$ with a wooden one, the total mass of the top two blocks decreases, which in turn decrease the force with which they push down, which in turn decreases the force with which the bottom block pushes up. Everything changes to quickly reach a new equilibrium where all forces balance and nothing moves.

So why doesn’t that happen with water? I think in some cases it can, but usually it doesn’t. Why not?

Honestly I’m not 100% sure. Any argument I put forth will very likely have flaws at least as serious as the argument I’m currently poking holes in. But if you want me to give it a hand-wavy attempt, here goes:

Liquids have a tendency to equalize pressure in a way that solids don’t. I think this relates to their ability to flow.

Consider two stacks of solid blocks side by side. One stack contains iron blocks and the other wood blocks. The pressure in the bottom iron block will be much greater than the pressure in the bottom wooden block. There is no transfer of pressure between the solid blocks.

Now consider two columns of water side by side (and water can flow between these two columns). I think it would be difficult to create a situation where the pressure at the bottom of one column was very different than the pressure at the bottom of the other columns. Water would quickly flow from the high pressure columns to the low pressure column to equalize the pressures.

## Fixing the argument

Again, I’d like to caveat this section by saying that I’m not sure how to fix the argument. This is just my best guess.

Let’s say we start with a large pool of water in equilibrium. Then, we magically snap our fingers and replace some small region of water with an empty massless box. What happens?

The water below the box was previously at whatever pressure was necessary to hold up all the water above it. Now, however, it has less mass weighing down on it than before. So, at time zero, it’s pushing back “too hard” and it will move the column of water up just a tiny bit as it expands (and reduces its internal pressure).

However, now we’re in a strange state where there is a region of water (under the empty box) which has a lower pressure than the water directly to the left and right of it. Remember, the water directly to the left and right is still holding up just as much water as before we snapped our fingers and added the empty box. So what happens?

Since the water in the region below the box has lower pressure it’s pushing less hard than it used to on the water to its left and right. The water to the left and right is pushing just as hard as before. Therefore, the water to the left and right will expand just enough to equalize the pressure. But now it has a lower pressure than the water around it!

And this process continues on and on until everything is in rough equilibrium again and… long story short, the water below the empty box is pushing up on it roughly as hard as it was before, causing a net up force and moving the empty box up.

## Reflections

We started with an argument that made a decent amount of sense, but then realized that it must be wrong because it implied that solid objects would float. One way to realize that something was wrong with the argument was to see that it didn’t rely - in any way - on the substance’s ability to flow, and we know that’s required.

So am I trying to say the starting argument is “bad”? No, actually. It’s just simplified. It explained a lot, but it ignored some aspects of the problem. But what explanation doesn’t do that? Even after adding a reliance on the ability of a liquid to flow, I’m sure there’s still something wrong or unrealistic with the new explanation.

Really, I just thought this was an interesting problem to being with. I honestly did want to understand what made something float or sink. Furthermore, I thought the process of realizing that something must be wrong with the initial explanation by realizing that it applied perfectly well to solids was also interesting. It reminded me of the way proofs work in math. You have some argument, but then realize it implies things that are false, and you need to either abandon or update the argument.

Hope you found it interesting too!