# Basic buoyancy question

If I have a cup of water filled with air at the bottom of a pool, then when the cup is "upside down" the air doesn't leave because the water pressure is pushing it up against the top of the container.

But then when I flip the cup, such that it is no longer upside down, the air rushes upwards. But why on earth does this happen? I mean the water's pressure is pushing this air downwards against the bottom of the container. If I were to draw an FBD which force is causing this air bubble to rush upwards?

Thanks

I think I see your question, if the surface of the air bubble were perfectly flat, and the air + cup didn't float, then the surface would have an equal pressure across it and it would not move. The system, however, is in an unstable equilibrium, the slightest perturbation will cause the bubble to rise out of the cup.

Consider this, since the pressure of the water is proportional to the height of the water above it, a bump up in the surface of the bubble would have less water pressure above it. The air pressure is essentially constant pressure throughout the bubble, so it would push into the bump and expand it. The bigger it gets the more the air can expand it. The water in the areas around the bump move down, increasing the pressure and so increasing the rate that they move down.

In addition, the water adheres to the cup walls, creating a meniscus along the edge of the bubble, drawing the water down and starting the rest of the air bubble to rise.

When the cup is tilted up, the water wants to flow into the cup. That is what water does - it attempts to flow downstream. In doing so it displaces the air. Now the air experiences the force of the water (pressure below bubble > pressure above)

• But how is that the water can flow into the cup and not simply compress the air inside? Suppose I submerge my cup with a sealing on top at first, and then rapidly remove it why isn't the air inside simply squished ?
– DLV
Jul 2, 2014 at 1:40
• Water on top of air is in an unstable equilibrium - the situation you describe cannot hold (although it depends on surface tension - when surface tension is a significant factor it can bee bought to make the bubble stable. See bubbles on inside of beer glass... When they get too big they disconnect. Jul 2, 2014 at 15:54

Like you said, the water is pushing UP. It is always pushing the air up (it is the definition of buoyancy). Therefore, when you flip the cup, the buoyancy force continues to push the air upwards. Before you flipped the cup, the cup is exerting the downward force that counteracts the buoyancy force.

Therefore in the FBD, the buoyancy force (pointed upwards) is what causes the air bubble to rush upwards. There is also a downward force of gravity, but it is negligibly small compared to the buoyancy force.

• But the buoyant force is the sum of all forces in contact with the air. The water is in contact with the air just on top. How can the buoyant be pointing upwards if the water isn't in contact with the air on the bottom?
– DLV
Jul 2, 2014 at 1:33
• The air bubble is in contact with the water all around. The bottom portion of the bubble receives greater force due to greater pressure (pressure increases with depth). Jul 2, 2014 at 1:37
• Suppose I submerge my cup with a sealing on top at first. Then I remove it, in that situation the water is clearly not in contact with the air bubble all around.
– DLV
Jul 2, 2014 at 1:38
• I see. In this case, the water will displace the air because it is unlikely that the air inside the cup has a high enough pressure to counteract the downward force of the water. Jul 2, 2014 at 1:43
• But I mean, if I do the experiment the air bubble does go upwards right? It doesn't just get compressed.
– DLV
Jul 2, 2014 at 1:46

It is due to Archimedes' principle. The total force on the 'air bubble' under the water is the force of gravity on it (downwards) plus the force of gravity on the water that the 'air bubble' displaces. Since the same volume of water that replaces the air is more massive (because of higher density) there is a net upward force on the bubble and causes it to move up. The actual equations of motion of the bubbles in fluids are quite complicated to calculate however.

Lastly when the cup is upside down as you wrote, the air bubble stays inside because the normal force provided by the bottom of the cup.