Register to reply

Buoyant Force on a Helium Balloon

by e(ho0n3
Tags: balloon, buoyant, force, helium
Share this thread:
e(ho0n3
#1
Jun17-07, 01:23 AM
P: 1,367
1. The problem statement, all variables and given/known data
A helium balloon has volume V0 and temperature T0 at sea level where the pressure is P0 and the air density is [itex]\rho_0[/itex]. The balloon is allowed to float up in the air to altitude y where the temperature is T. (a) Show that the volume occupied by the balloon is then V = V0(T/T0)ecy where [itex]c = \rho_0g/P_0[/itex]. (b) Show that the buoyant force does not depend on altitude y. Assume that the skin of the balloon maintains the helium pressure at a constant factore of 1.05 times greater than the outside pressure. [Hint: Assume that the pressure change with altitude is P = P0e-cy].


2. Relevant equations
The ideal gas law and the buoyant force equation.


3. The attempt at a solution
(a) is pretty simple since the hint gives it away. What concerns me is (b). Since the buoyant force on the balloon is equal to the weight of the volume of air displaced by the balloon, and since the volume depends on altitude, then it seems logical that the buoyant force depends on altitude. What gives?
Phys.Org News Partner Science news on Phys.org
What lit up the universe?
Sheepdogs use just two simple rules to round up large herds of sheep
Animals first flex their muscles
Doc Al
#2
Jun17-07, 07:11 AM
Mentor
Doc Al's Avatar
P: 41,453
But the density of the air decreases with altitude. What matters is how the product density*volume--the weight of the displaced air--varies with altitude.
e(ho0n3
#3
Jun17-07, 12:51 PM
P: 1,367
The mass of air inside the balloon remains the same regardless of its change in volume. Thus, the buoyant force is constant right? (Actually, it wouldn't since the force of gravity decreases with altitude.)

Doc Al
#4
Jun17-07, 02:22 PM
Mentor
Doc Al's Avatar
P: 41,453
Buoyant Force on a Helium Balloon

Quote Quote by e(ho0n3 View Post
The mass of air inside the balloon remains the same regardless of its change in volume.
Yes, the mass of the helium remains the same, but the buoyant force equals the weight of the displaced air, not the balloon contents.
e(ho0n3
#5
Jun17-07, 03:49 PM
P: 1,367
Ah, OK. I was getting confused. Let [itex]\rho = m/V[/itex] be the density of air at altitude y; m is the mass of air displaced by the balloon whose volume V is as given in the problem statement. The buoyant force is [itex]\rho V g = mg[/itex]. Hmm...so now how do I know that m is always the same?
Doc Al
#6
Jun17-07, 06:42 PM
Mentor
Doc Al's Avatar
P: 41,453
How does the density of the air depend on pressure and temperature?
e(ho0n3
#7
Jun18-07, 09:37 AM
P: 1,367
Applying the ideal gas law to air, then

[tex]PV = P \, \frac{m}{\rho} = nRT [/tex]

Rearranging yields

[tex]\rho = P \, \frac{m}{nRT}[/tex]

The fraction m/n is surely constant as it equals air's molar mass. Now what about P/T? If they both decrease at the same rate, then it will definitely be constant. I don't have an expression for T though.
Doc Al
#8
Jun18-07, 09:52 AM
Mentor
Doc Al's Avatar
P: 41,453
Keep going. Now find the buoyant force, which is:
[tex]\rho g V[/tex]
e(ho0n3
#9
Jun18-07, 12:45 PM
P: 1,367
If I substitute and simplify, I get that the buoyant force is mg which is what I had derived in post #5. Again, I don't know if m is constant though.
Doc Al
#10
Jun18-07, 02:07 PM
Mentor
Doc Al's Avatar
P: 41,453
What matters is how the density of the air varies. From the ideal gas law, the density of air will be proportional to:
[tex]\frac{nR}{V} = \frac{P}{T}[/tex]

Plug that into the expression for buoyant force.
e(ho0n3
#11
Jun18-07, 05:21 PM
P: 1,367
I see. You're suggesting that the buoyant force is proportional to nRg right? However, like m, doesn't n depend on altitude as well?
Doc Al
#12
Jun19-07, 08:04 AM
Mentor
Doc Al's Avatar
P: 41,453
All we are concerned with at this point in the solution (as in post #10) is what the air density depends on. I'm not concerned with any particular mass of air or number of moles.

It will turn out that the mass of air displaced by the balloon is independent of altitude, but no need to assume that at this point.


Register to reply

Related Discussions
Helium Balloon... Introductory Physics Homework 10
Helium Balloon Introductory Physics Homework 1
Helium balloon Engineering, Comp Sci, & Technology Homework 1
Buoyant force on balloon Introductory Physics Homework 1
Helium balloon Introductory Physics Homework 11