I understand the archimedes principle as it relates to water, but am having trouble reconciling a thesis related to a hot air balloon. If theoretically, I could increase the temperature of air contained in a fixed pocket/container, would the increased pressure exerted upward lift the container? The archimedes principle would argue against the prospect of lifting the container due to the fact that the container remains the same size and therefore displaces the same amount of air outside. However, I find it difficult to understand why increased upward pressure on the container would not lift the container. Can you explain?
I case of an hot air balloon, the volume of the balloon remains unchanged when the air is heated. But because the heated air expands, part of it escapes from the balloon since it is not closed (the bottom is open). Therefore the weight of the air inside the balloon is reduced. The hot air in the balloon has a lower density that the cool air around it. Therefore the balloon lifts.
Would a closed balloon of unstretchable fabric be used, then the pressure in the balloon would rise but the volume of the balloon would remain unchanged. Therefore the density of the air inside the balloon and the buoyancy would remain the same and the balloon would not lift off.
When you increase the pressure in a closed volume, the increase pressure will act on all parts of the surface. Since both the upward-facing and the downward-facing area is the same, those pressures cancel each other.
If you want to reduce the area on the lower end of the volume, you need to cut a hole in it. Now the upper area is bigger by the area of the hole, and the volume will be forced upwards by a force which is the product of the pressure difference to outside pressure and the area difference, which is the area of the hole. This kind of lift, however, is not buoyancy, but rocket thrust. Unfortunately, as soon as you open that hole, gas will flow out and the pressure difference will vanish, unless you pump more gas into the volume.