Solving the Equation: Finding h in a He-filled Balloon

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In summary, the problem is to determine the value of h, the length of string lifted by a helium-filled balloon at atmospheric pressure. Using the equations P = mg/A and P2 = P1 + dg(h), the pressure of the balloon is found to be 4.874 Pa. However, the pressure is not relevant to finding h. The net lift of the balloon, taking into account the weight of the balloon fabric, is calculated to be 300.15 kg. This is equal to the weight of the air displaced by the balloon, which can only lift the amount of rope with a weight equal to the net lift of the balloon. Using B=pVg and considering all the forces acting on the balloon, h is
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Homework Statement



A helium-filled balloon at atmospheric pressure is tied to a 3.1 m long, 0.100 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.)


Homework Equations



P2 = P1 + dg(y1 - y2)
d = density



The Attempt at a Solution



First I determine the pressure of the balloon by using P = mg/A, and I got 4.874 Pa
Then I use the equation P2 = P1 + dg(h)
4.874 = (0) + (1.29)(9.8)(h). The 1.29 is density of air.
I have no idea why I set p1 equal 0.
The answer I got is wrong and I am stuck. Help please.
 

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  • #2
I don't think the pressure is relevant, but I could be wrong.

How much does a sphere of air weigh when r=0.4m?
How much does a sphere of helium weigh at the same radius?

After adding in the weight of the fabric of the balloon, what is the net lift of the balloon?
 
  • #3
density of air is 1.29 and density of He is 0.179

mass of air = 2161.4
mass of He = 299.9

299.9 + 0.25 = 300.15 the mass of the balloon + He

Im still not sure what to do here. How is the net lift relative to finding h? (I thought it was a pressure problem because it came from a chapter with Bernoulli equation and fluid flow).
 
  • #4
It's good to get in the habit of including units in all of your equations and work (and check to see that the units agree all along through your calcs).

The lift of the balloon is what lifts some length of the rope up. It can only lift up the amount of rope that has a weight equal to the net lift of the balloon, right? You are given enough info to calculate the linear density of the rope...
 
  • #5
ooo thanks for your help, I managed to solve it using B=pVg. I found all the forces acting on the balloon and set that equal to the weight of air. Thanks
 

1. How do you determine the value of h in a He-filled balloon?

The value of h can be determined by using the ideal gas law equation, PV = nRT, where P is the pressure of the gas, V is the volume of the balloon, n is the number of moles of gas, R is the ideal gas constant, and T is the temperature in Kelvin. By rearranging the equation to solve for h, we can calculate the height of the balloon.

2. What is the purpose of using a He-filled balloon to solve for h?

Using a He-filled balloon allows us to create a controlled environment with a known gas and volume. This makes it easier to calculate the value of h and also ensures that the results are more accurate.

3. Are there any limitations to using this method to find h?

Yes, there are some limitations. The ideal gas law assumes that the gas behaves perfectly, which is not always the case in real-life situations. Additionally, other factors such as atmospheric pressure and air resistance may affect the results.

4. Can this method be used for any type of gas or only He?

This method can be used for any type of gas as long as the ideal gas law is applicable. However, using a lighter gas like helium (He) is preferred as it helps to reduce any potential errors and makes the calculations easier.

5. How is this method useful in scientific research?

Solving for h in a He-filled balloon can be used in various scientific experiments and studies to determine the height of different objects or structures. This method is also used in weather forecasting to measure the height of the atmosphere and in studying the properties of gases in different conditions.

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