Why is hot air lighter than cold air?

So that slight increase of density in the lower air is enough to exert an upward force on the balloon. Yes..that makes perfect sense. Thanks!

 

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Correct, and the denser air has a higher pressure so there is a higher force below the center of the balloon than above the center of the balloon so there is a net upward force. So it is up up and away...

 

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Is this pressure the same as barometric pressure?

 

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Yes, pressure is force per unit area. So barometric pressure is the pressure from the air. In a cyclone (low pressure area) you will have low barometric pressure for instance.

 

That's the core of the physical principle expressed glibly as "Hot air doesn't rise, it's the cold air that falls."

That question is about the mechanism of buoyancy which is answered by the phenomena of the pressure exerted by a fluid under gravity.

Fluidic pressure is equally transmitted in all directions so if we have a surface, the pressure is a force per unit area perpendicular to the surface.
Real fluids have density, ρ, and a fluid in hydrostatic equilibrium has to hold up the weight of all the fluid about it. But Newton's third principle means the pressure of the liquid is what holds up that weight. Thus as you descend a distance, Δz, into a fluid of density ρ, under constant gravitational acceleration g, the pressure increases, ΔP = gρΔz.

Now imagine a cylinder in that fluid of radius r and height Δz. Because of symmetry there will no net force in the horizontal directions, so all we have to do is consider the forces in the vertical direction. On the top surface of the cylinder there will be a pressure P₀ in the down direction which means a force of πr²P₀, but on the bottom surface there is a force of πr²(P₀ + ΔP) in the up direction for a net force of buoyancy of πr²(P₀ + ΔP) – πr²P₀ = πr²ΔP = gρπr²Δz = gρV where V = πr²Δz is the volume of the cylinder. This is the principle of displacement:

The net upward force of buoyancy on any object is equal to the weight of fluid the object displaces.​

It works for cylinders, cones, spheres and any shape. It works on boats.

Now if the cylinder has mass M then it has a net force upward if gM < gρV (or, stated another way, if M/V < ρ). If M/V > ρ then the cylinder sinks because it feels a net force of gM – gρV > 0 in the down direction.

https://en.wikipedia.org/wiki/Archimedes'_principle
https://en.wikipedia.org/wiki/Displacement_(fluid)
https://en.wikipedia.org/wiki/Displacement_(ship)

// Edit -- I see that Fraggle Rocker in post #20 gave a similar answer. Thank you.

 

On a related note, what makes helium rise? Is it less dense than our air, or does it actually have less mass?

 

Um, both?

Air is mostly nitrogen molecules, with a molecular weight of 28. Oxygen, Carbon Dioxide and Argon are components of dry air with a molecular masses larger than nitrogen. So the molecular mass of dry air averages about 29 per particle. For helium, it is only 4. So at the same temperature and pressure a given volume of helium masses less than 1/7 of that of air.