Buoyant Force and what densities to consider?

[Ian Baughman]

So I know

F_B = ρ_DF × g ×V_DF
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when we submerge an object in a fluid such as water but what if we were talking about a hot air balloon? In this case would it be correct to use

F_B = (ρ_in - ρ_out) × g ×V_object?​

Where

ρ_in = density inside balloon and ρ_out = density outside balloon.
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If this is the case how do we know when to differentiate between the two?

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[Simon Bridge]

Your two equations appear to be using different definitions for "buoyant force"...

The rule is that the fluid-pressure exerts a net force equal to the weight of the fluid displaced ... if this is bigger than the weight of the object, it rises.

That rule works for all situations... don't bother memorising any specific equation, you already know all the ones you need.

The weight of fluid displaced is $\rho V g$ (V is the volume displaced, and $\rho$ is the density of the fluid)
The weight of the object is $mg$
... so the net upwards force is $F=\rho g V - mg$

If all the object is in the fluid, then I can write $m = \rho_mV$ since the volume displaced is the same as the volume of the object
... which gives: $F = (\rho-\rho_m)gV$
... which only works for the case that the entire object is immersed in the fluid.