Calculating Balloon Volume Change with Altitude

[physicsss]

A helium balloon has volume V0 and temperature T0 at sea level where the pressure is P0 and the air density is p. The balloon is allowed to float up in air to altitude y where the temperature is T1.

a) Show that the colume occupised by the balloon is then V = V0(T1/T0)e^(+cy) where c = pg/P0 = 1.25 X 10^4 m^-1.

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 factor of 1.05 times greater than the outside pressure. [Hint] Assume that the pressure change with altitude is P = (P0)e^(+cy).

I was able to use the hint to do a), but I'm confused by b). Why does the book say to assume he skin of the balloon maintains the helium pressure at a constant factor of 1.05 times greater than the outside pressure?

 

[member 553083]

How does this relate to the buoyant force? a) The volume occupied by the balloon is related to the temperature, pressure and density of the air. The equation for the volume change due to a change in temperature is given by V = V0(T1/T0). The equation for the volume change due to a change in pressure is given by V = V0e^(-cy), where c = pg/P0. Combining these two equations gives us the total volume change due to both pressure and temperature changes: V = V0(T1/T0)e^(-cy). b) The buoyant force is equal to the weight of the displaced air, which is equal to the mass of the displaced air times the gravitational acceleration. The mass of the displaced air is equal to the volume of the balloon times the density of the air. The density of the air is related to the pressure by the equation p = P/RT, where R is the gas constant and T is the temperature. Thus, the buoyant force can be written as Fb = Vpg/RT. Since the pressure changes with altitude (P = P0e^(+cy)), the density of the air (p = P/RT) also changes with altitude. However, if the skin of the balloon maintains the helium pressure at a constant factor of 1.05 times greater than the outside pressure, then the helium pressure inside the balloon will remain constant, thus eliminating the effect of pressure change on the buoyant force. Therefore, the buoyant force does not depend on altitude y.