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This question was originally inspired by this one:

Why does the air pressure at the surface of the earth exactly equal the weight of the entire air column above it

In it, the author mentions a really tall solid column (extending all the way up into space) whose base has an area of $1$ square inch, and which weighs $14.7$ pounds, which is equal to the pressure near the surface of the earth.

My question is this: would such a column...float?

After all, the weight of the column is $14.7$ pounds, its base has an area of $1$ square inch, and the pressure from the air at its base is $14.7$ pounds per square inch.

So, would the pressure from the air at its base balance out the weight of the column?

I'm assuming the pressure doesn't change once we put a solid column on top of some air. I don't know much about fluids yet, but this assumption may be incorrect...

Thanks!

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  • $\begingroup$ by the way, this argument allows you to estimate the total weight of the Earth's atmosphere, which is equal to (14.7 pounds/square inch) x (surface area of the Earth modeled as a smooth sphere, in square inches). $\endgroup$
    – niels nielsen
    Jul 4, 2019 at 18:02

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Yes, it should float.

Finding a solid that tall with such a low weight would be difficult (presumably not possible); but we already have a great example of something gaseous that does this: the air.

The air at any given point on the surface has developed it's air pressure due to the weight of all the air above it that it has to "hold" in place. Essentially, the pressure of the air has just enough force to keep the air above it "held up". If you had a solid object that weighed just as much as the air that it displaced, it would float due to buoyancy; which is exactly what the air itself does.

If the pressure on the bottom of the object were equal to the atmospheric pressure, and the pressure at the top is 0 (as we have in the case of something extending up out of the atmosphere) it is said to be neutrally buoyant, and thus would float.

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  • $\begingroup$ Thanks, JMac! +1! I was thinking the same thing, but the problem with the air was that it's in constant motion - it's not really being "held" in place, is it? It's continuously moving in and out of that "column", creating tiny differences in pressure that balance out almost immediately as air comes in to replace it. So it's hard for me to picture what's happening at the base of that column when it's made out of air. It's not like the air under the column is really "supporting it," since the column itself is made out of air...but I do see that if some of the air "in" the column were to... $\endgroup$
    – joshuaronis
    Jul 4, 2019 at 17:35
  • $\begingroup$ ...fall further down and not be pushed back up by air molecules under it almost immediately, there would be a difference in pressure...? Actually, I'm really confused. Can you help me better picture how the pressure is due to the column of air on top of it? Thanks! $\endgroup$
    – joshuaronis
    Jul 4, 2019 at 17:37
  • $\begingroup$ @JoshuaRonis Yes, it's obviously a bit more complicated in the dynamic system of Earth, where the surface is heated, so the low air tries to expand due to the ideal gas law, and then that pressurizes some areas less than others, which generates a pressure gradient, which makes wind; which propagates, etc. But consider the perhaps easier to picture when you think of hydrostatics. It's really just a force balance. If the air on the bottom didn't push back with enough force, the air above would accelerate down due to gravity. $\endgroup$
    – JMac
    Jul 4, 2019 at 17:39
  • $\begingroup$ @JoshuaRonis This develops pressure, because obviously the molecules can't all be in the same place, and start to push back. $\endgroup$
    – JMac
    Jul 4, 2019 at 17:41
  • $\begingroup$ Thanks again, but I'm still confused...even if there were no wind, and no heating...etc etc, but just a column of air and the gravity of the earth, wouldn't air molecules still be moving in and out randomly from that column? It's not like the column is really being "held up" by the air molecules below it...this relates to another question I just asked, but how would the air particles at the base of this "column of air" be moving? Would they just be moving up and down, into and out of the column, keeping the pressure mostly constant? $\endgroup$
    – joshuaronis
    Jul 4, 2019 at 17:45
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If you want to know if something floats, you have to compare the densities of the medium which the thing is floating in (here, Earth's atmosphere) and the thing that is floating (the solid column of air). The pressure and the weight are different things and are not necessarily related.

So to answer your question, while freezing this column of air, you would not change the mass but you would decrease the volume. Thus you would increase the density because density is mass divided by volume. So the density of the frozen air is heavier than the density of normal air and your column would sink.

If you could somehow freeze the column of air without changing the density (I don't see how you could do this), then it would be neutrally buoyant because the densities are equal. In other words, the buoyancy force and the gravity forces would exactly cancel and there would be no net force on the cylinder (excluding things like friction). Technically, neutral buoyancy is not the same thing as floating, since if you let go of your column at the bottom of the atmosphere (sea level), it would stay there, not float upwards.

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Yes, it would float because air is much lighter than water, but a miracle would be required to float it. The air at the bottom of the column has a pressure of 14.7 lb per sq in because that is the weight of the air pressing down on it from above. I can't think of any way you could encapsulate this column and float it in the sea. If you laid the column horizontally to float it, the 14.7 lb per sq in pressing down on it would not be from the column itself, but from the external atmosphere. I know thought experiments can have scenarios which are impossible in the real world, but they usually do it to illustrate a principle. This one doesn't.

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  • $\begingroup$ No no, I wasn't talking about the sea, I mean would it float on the air itself? $\endgroup$
    – joshuaronis
    Jul 4, 2019 at 17:47
  • $\begingroup$ It depends what you mean by float. In the air it would have neutral buoyancy, assuming that the temperature was the same as the rest of the atmosphere. It's like asking would a column of seawater float in the sea? Yes, for a while if the temperature was much higher than the surrounding seawater, but when the temperature equalised it would have neutral buoyancy.. $\endgroup$
    – Michael Walsby
    Jul 4, 2019 at 17:57

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