- #1

Amaterasu21

- 64

- 17

I understand where Archimedes' Principle comes from in liquids:

If we imagine a cylinder immersed in a liquid of density ρ whose cross-sectional area is A and whose top is at depth h

_{1}and whose bottom is at depth h

_{2}:

Force

_{(top of cylinder)}F

_{T}= ρgh

_{1}A

Force

_{(bottom of cylinder)}F

_{B}= ρgh

_{2}A

Buoyant force = ρgA(h

_{2}-h

_{1})

= ρA(h

_{2}-h

_{1})g = ρV

_{cylinder}g = m

_{displaced liquid}g = weight of displaced liquid

therefore buoyant force = weight of displaced liquid.

I also understand that this works even if the object immersed in liquid isn't a cylinder because pressure at any depth is constant, so the horizontal components of pressure on any surface of the object cancel out, and the vertical components produce the same buoyant force as the cylinder.

But what I don't understand is why this is true for gases as well. I've been told that the buoyant force in gases is equal to the weight of displaced gas. However, gases unlike liquids are compressible and a large mass of gas in a gravitational field (e.g. the atmosphere) will be far denser at the bottom than at the top. Therefore while ρ is constant during the derivation for liquids, it would certainly

**NOT**be constant across the height of the cylinder in a gas!

Force

_{(top of cylinder)}F

_{T}= ρ

_{1}gh

_{1}A

Force

_{(bottom of cylinder)}F

_{B}= ρ

_{2}gh

_{2}A

Buoyant force = gA(ρ

_{2}h

_{2}- ρ

_{1}h

_{1})

...and I don't see how we get from here to "= weight of displaced gas!"

Any help would be appreciated!