# Archimedes' principle for gases

These two pages say that Archimedes' Principle is valid for gases.

If that is so then: Doing a quick internet search tells me that density of air is about $$1.2kg/m^3$$. I have an object of density less than that of air. Why don't I see it flying, accelerating up to start floating at the top of the air surface, as predicted by the law of floatation:

$$\frac{Volume Of Object Immersed}{Entire Volume Of Object} = \frac{Density Of Solid}{ Density Of Fluid}$$

Since density of object is less than that of air, RHS of the equation is less than 1 and therefore volume immersed must be less than the volume not immersed, that is, object floats.

Is the Archimedes' principle valid for fluids of incompressible category, and thus it is not valid for gases?

The book that I am using to learn Archimedes' Principle doesn't mention anything about the fluid being ideal ,non ideal, non Compressible etc. It just talks about a fluid.

What's the problem here?

I have an object of density less than that of air. Why don't I see it flying, accelerating up to start floating at the top of the air surface

According to a wikipedia article the least dense solid known to exist is graphene aerogel with a density of 1.6 kg/m$$^3$$. So it seems there are no solids that will float in air of density 1.2 kg/m$$^3$$. The only thing that can rise in air is another gas of lower density than air.

But you generally can't "see" another gas rising in air unless that gas is enclosed in some visible solid, such that the density of the combination of the visible solid and gas is less than air. That's the case of a balloon filled with a gas lighter than air (e.g., helium).

Hope this helps.

• So that means, if I have an object(if possible) less dense than air, it rises up. Wow!!!!! And archimedes principle is valid on fluids of any sort. Commented Nov 18, 2023 at 17:03
• @ApoorvaShukla Yes, but no such solid object exists. It would have to be another gas of density less than air. And if its a gas, you will not "see" it rise up. Commented Nov 18, 2023 at 17:05
• I understood. Thank you Commented Nov 18, 2023 at 17:07

Here's the problem is about logic or comprehension, not physics.

Archimedes' principle holds for fluids. The question in both sites asks if it's not valid for gases. The right answer is False, because the principle holds for fluids, gases included.

The equilibrium of a floating object is established by the buoyancy (Archimedes) force. Forces acting on the body are:

• its weight pointing downwards, $$- m g \mathbf{\hat{z}}$$
• Archimedes force, equal to "the weight of the fluid displaced", pointing upwards, $$\mathbf{F} = m_{f,displ} g \mathbf{\hat{z}}$$.

At the equilibrium, sum of forces is zero, and thus $$-m g + m_{f,displ} g = 0$$

• I think that's what my question is. If it is valid for gases why don't I see my object accelerating up... Commented Nov 18, 2023 at 15:22
• edited. Archimedes' force pointing upwards but object weight pointing downwards, with the same intensity, and thus equilibrium Commented Nov 18, 2023 at 15:29
• The Archimedes principle already accounts for the buoyant and weight of the object. When equilibrium is attained the object has already reached the top of the surface and thus is said to float, if it has less density that fluid. But as pointed out in the other answer there is no such object existing which has smaller density than air, but if it exists it should float, like a balloon. Commented Nov 18, 2023 at 17:16