Fluid Mechanics of helium-filled balloon

[vmind]

Hi,

I would really appreciate help in solving the following problem.

A helium-filled balloon is tied to a 2.00 m long, 0.050 kg uniform

string. The balloon is spherical with a radius of 0.400 m. When

released, the balloon lifts a length h of string and then remains in

equilibrium. The envelope of the balloon has a mass of 0.250 kg.

What is the value of h?

 

[HallsofIvy]

Since 2 meters of the uniform string as mass 0.05 kg, h meters of the string will have mass (h/2)(0.05)= 0.025h kg and so weight 0.025 gh= 0.245h Newtons. The balloon itself has mass 0.25 kg and so
weight 0.25*g= 2.45 Newtons so the balloon is lifting 2.45+ 0.245h Newtons. You will need to use the bouyancy of helium (the balloon contains (4/3)(pi)(0.4³= 0.268 m³ of helium) to determine how much weight the balloon can lift and so what h is.

 

[vmind]

Hello again,

Is the following formula the one I should use to find the buyoancy of helium?

B_helium = p_helium * V_helium * g

Also will the combined weight of string and balloon equal the buyoancy of helium at height h?

I really appreciate your help.

 

[Doc Al]

Your formula will give you the weight of the helium.

For equilibrium, the forces must balance. The forces acting down are the weights of the balloon envelope, the helium, and the length (h) of string. These must be balanced by the buoyant force acting upward, which equals the weight of the displaced air.