physics fluid mechanics question. uh ohh.?

Question

A child's rubber balloon of mass 2.5g is filled with helium gas of density .33kg/m^3. The balloon is spherical, with a radius of 12cm. A long cotton string with a mass of 2.0g per meter hangs loosely from the bottom of the balloon. Initially, the string lies loosely on the floor, but when the balloon ascends, it pulls the string upward and straightens it out. At what height will the balloon stop ascending, having reached equilibrium with the hanging portion of the string? Assume that the surrounding air has the density 1.29 kg/m^3

Answer 1

Do a force balance on the object, setting the total force equal to zero (there's no net force, the object is at equilibrium):

F = m*g - buoyant force
F = (rubber mass + gas mass + string mass)*g - buoyant force = 0
F = (rubber mass + gas mass + string mass)*g - (displaced air mass)*g = 0
F/g = rubber mass + gas mass + string mass - displaced air mass = 0
F/g = (2.5 g) + (volume)*(helium density) + (2 g/m)*(balloon height) - (volume)*(air density) = 0

volume = 4/3*pi*r^3 = 4/3*pi*(12 cm)^3 = 0.007238 m^3

F/g = (2.5 g) + (0.007238 m^3)*(0.33 kg/m^3) + (2 g/m)*(balloon height) - (0.007238 m^3)*(1.29 kg/m^3) = 0
F/g = (2.5 g) + (2.389 g) + (2 g/m)*(balloon height) - (9.337 g) = 0

combining terms and rearranging, we get:

4.448 g = (2 g/m)*(balloon height)
2.224 m = balloon height

So, the balloon rests, balanced with 2.224 meters of string hanging in the air.