How can I find the equilibrium length of the string for a helium-filled balloon?

[physics214]

Homework Statement

A helium-filled balloon at atmospheric pressure is tied to a 3.4 m long, 0.060 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.

Homework Equations

Bernoulli's equation
A1V1=A2V2
P = Po + (density)gh

 

[physics214]

Here's my futile attempt at the problem:

There's weight and pressure acting downward...mg and atmospheric pressure.
There's also the same pressure acting upward...so wouldn't the two pressures cancel out?

Basically, the only thing that is stopping the balloon from drifting off is the weight of the string, but i don't know what formulas to use.

 

[alphysicist]

Hi physics214,

Here's my futile attempt at the problem:

There's weight and pressure acting downward...mg and atmospheric pressure.
There's also the same pressure acting upward...so wouldn't the two pressures cancel out?

They won't cancel completely. The idea is that the force from air pressure pushing down on the balloon is less than the force from air pressure pushing up on the balloon, so overall the effect of the air is to give a buoyant force upwards. So you'll need to take into account the buoyant force from the air.

(Your equation $$P = P_o +\rho g h$$ shows why the pressure is greater at the bottom of the balloon.)

Basically, the only thing that is stopping the balloon from drifting off is the weight of the string, but i don't know what formulas to use.

Draw a force diagram, since you know the balloon is is equilibrium. What has to be true of all the forces?

(There's not enough details in your last post for me to be sure, but it's possible that you might be missing some of the forces, when you say that only the weight of the string keeps the balloon from drifiting off.)