A helium-filled balloon at atmospheric pressure is tied to a 3.0 m long, 0.070 kg string.?

Question

A helium-filled balloon at atmospheric pressure is tied to a 3.0 m long, 0.070 kg string. The balloon is spherical with a radius of 0.40 m. When released, it lifts a length (h) of the string and then remains in equilibrium as in Figure P9.78. Determine the value of h. When deflated, the balloon has a mass of 0.25 kg. (Hint: Only that part of the string above the floor contributes to the load being held up by the balloon.)

http://www.webassign.net/sf/p9_78.gif < figure

Answer 1

Air contains about 78% N2, 21% O2, 1% Ar, and small amount of H2O vapor and CO2, and very small amount of others. Assume Air just contains 78% N2, 21% O2 and 1% Ar, its "averaged molar mass" is:
78%28.0+21%32.0+1%39.9 = 28.96 g/mol
Assume the temperature is 25ºC. The density of air is:
n*28.96/V = P*28.96/RT = 28.96/(0.08206*298.2) = 1.183g/L = 1.183 kg/m^3.
The density of helium is:
n*4.003/V = P*4.003/RT = 4.003/(0.08206*298.2) = 0.1636g/L = 0.1636 kg/m^3
Buoyancy of the helium in the balloon is:
(1.183-0.1636)*(4/3)*π*0.40^3 = 0.2733 (kg)
Hence the value of h is: (0.2733-0.25)*3.0/0.070 = 0.999(m) (answer)