Simple question about buoyant force

[Signifier]

Say I have some volume V of some gas with a pensity P less than the density of air. How can I calculate how much a balloon (of volume V) of this gas could lift?

Is this called the "buoyant" force, or is this something else? What I'd like to do is find the find the lifting "power" (...) of some volume of helium, or the upward force in the fluid of air that a volume of helium could exert. How much of a gas would I need to pack into a container in order to lift some mass up some distance, etc.

Is there a branch of physics these sorts of questions neatly fit under, BTW, that would help my search for information? I remember a strange attribute of my first three physics class, which were supposed to provide a solid introduction to all physics (minus quantum) was the complete lack of discussion on things like hydrostatics and buoyancy...

Thank you, sorry for the simplicity of these questions!

 

[pseudovector]

use archimedes's equation: g * V * dp
where g - gravitational acceleration, V - volume, and dp the difference between the densities of the matter inside and outside the volume V.

 

[Signifier]

Really, that's it? Are there any other equations that are used to find the amount of upward force some volume of some uniformly dense gas can exert?

 

[Doc Al]

That's really all there is to it. The buoyant force on an object (a balloon filled with gas, or whatever) is equal to the weight of the fluid it displaces. (That's Archimedes' principle.) The buoyant force is the same regardless of the contents of the balloon. Of course, to use the balloon to lift something else that buoyant force must be greater than the weight of the balloon (and its contents). That difference will tell you how much additional weight the balloon can lift.

 

[Signifier]

So let me get this straight (sorry!): if I have a balloon filled with gas of density p, enclosed in a volume V, carrying a payload of mass m, then in order for the balloon to lift upward I must have

gV|dp| > gm

(where dp is the difference in densities between the gas inside and the atmosphere)? Yes, have I got it? Or is it:

gV|dp| > g(M+m)

where M is the mass of the balloon with all of its gas and m is the mass of the payload. This last expression would reduce to:

V(Patm - 2Pgas) > m

where Patm is the density of the atmosphere and Pgas is the density of the gas inside the balloon...

?

 

[Doc Al]

So let me get this straight (sorry!): if I have a balloon filled with gas of density p, enclosed in a volume V, carrying a payload of mass m, then in order for the balloon to lift upward I must have

gV|dp| > gm

(where dp is the difference in densities between the gas inside and the atmosphere)? Yes, have I got it?

That's it. Realize that "m" also must include the mass of the balloon material as well as the load. Further realize that we are ignoring the (probably small) buoyant force of the payload.

Or is it:

gV|dp| > g(M+m)

where M is the mass of the balloon with all of its gas and m is the mass of the payload.

You've already included the mass of the gas within the balloon in the term V|dp|; don't count it twice.

This last expression would reduce to:

V(Patm - 2Pgas) > m

See? You're counting the mass of the gas twice.

Things with be much clearer if you work out the formula from the basic principles: The bouyant force must support the weight of everything.

Buoyant force = g P(air) V
Weight of everything = g(mass of balloon material + gas + payload)

Of course, mass of gas = P(gas) V.

So:
g (Pair - Pgas)V > g(mass of balloon material + payload)

Or,
(Pair - Pgas)V > (mass of balloon material + payload)

 

[rcgldr]

Here's a thought question, say you have a model balloon, complete with deflated balloon and helium tank inside a large sealed container filled with air (plus the model). The container (with the model in it) rests on a scale and the scale indicates a total weight of 100lbs. Then, via remote control, the helium in the model is transferred from the model's tank to the model's balloon filling it, so that the model balloon now hovers inside the container. What will the scale indicate for the weight of this closed (sealed) system with the model balloon hovering? Support your answer.

 

[OlderDan]

Here's a thought question, say you have a model balloon, complete with deflated balloon and helium tank inside a large sealed container filled with air (plus the model). The container (with the model in it) rests on a scale and the scale indicates a total weight of 100lbs. Then, via remote control, the helium in the model is transferred from the model's tank to the model's balloon filling it, so that the model balloon now hovers inside the container. What will the scale indicate for the weight of this closed (sealed) system with the model balloon hovering? Support your answer.

100lbs. The weight of the system, including the container, does not change when you redistribute the contents of the container. All you are doing when you blow up the balloon is move some helium out of the tank into the balloon and displace air molecules that were originally above the balloon to other parts of the container. Nothing has been added or taken away.

Now as long as we are changing the question from "simple" to more complex, what if the scale is the lower wall of the container, with the tank and the balloon both resting on the scale?

I've been pondering a somewhat similar buoyancy problem related to bubbling water. As seen on TV, some have suggested that bubbling water is less buoyant because of its lower density, speculating that ships have been lost in the Bermuda triangle because they sank in bubbling water. Archimedes principle is quoted as justification for this conclusion. No doubt the principle is well established for fluids of uniform density, but I doubt it applies in such a simple way to bubbling water. I'll start another thread sometime soon to expand on this.

 

[rcgldr]

Here's a thought question, say you have a model balloon, complete with deflated balloon and helium tank inside a large sealed container filled with air (plus the model). The container (with the model in it) rests on a scale and the scale indicates a total weight of 100lbs. Then, via remote control, the helium in the model is transferred from the model's tank to the model's balloon filling it, so that the model balloon now hovers inside the container. What will the scale indicate for the weight of this closed (sealed) system with the model balloon hovering? Support your answer.

100lbs. The weight of the system, including the container, does not change when you redistribute the contents of the container.

Say the container box weighs 50lbs, the air 49lbs, and the model 1lb. How does the air generate a 49lb downwards force on the box (model resting at bottom), and 50lbs downwards when the model is hovering? (I'm ignoring the small boyancy factor of the air against the model when it's deflated, although the answer below takes care of this as well).

Highlight below to see answer (hold down mouse button and pan pointer across the text):

The pressure of the air in the container decreases with altitude, less at the top, more at the bottom, and this pressure differential results in a net downwards force equal to the weight of the air. When the model is hovering, it displaces the air, increasing the pressure of the air, and also increasing the rate of pressure decrease versus altitude so the pressure differential generates a downforce equal to the weight of the air and the model. If the balloon were expanded further, it float to the top and would exert an upwards force on the container, and the pressure differential of the air would generate a downwards force equal to the sum of weight of the air, the model, and the upwards force on the top of the container.