Balloon Buoyancy Homework: Find Tension in Line

In summary, to find the tension in the line of a filled balloon fastened to a vertical line, we can use the formula T = B - wh - wb. The volume of the balloon should be calculated using the radius of the empty balloon. The weight of helium should be calculated using the density of air, as we are finding the weight of the displaced air. By making these corrections, we arrive at the correct solution of 4.52537 N.
  • #1
gmorrill
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Homework Statement


An empty rubber balloon has a mass of 0.0122 kg. The balloon is filled with helium at a density of 0.179 kg/m3. At this density the balloon has a radius of 0.467 m. The acceleration of gravity is 9.8m/s2. If the filled balloon is fastened to a vertical line, what is the tension in the line? Answer in units of N.

Homework Equations


V = (4/3) * pi * r3
Buoyant force B: rhoair * V * g
Weight of helium wh: rhohelium * V * g
Weight of balloon wb: m * g

The Attempt at a Solution


I think I understand all the concepts, but I'm not coming up with the solution in the book. Book's solution is 4.52537 N.

I drew a free body diagram to determine the sum of forces:
B - wh - wb - T = 0

So:
T = B - wh - wb

The density of air isn't given, so I went with what the chart in the book listed: 1.20 kg/m3

V = 0.42662

B = 1.20 * 0.42662 * 9.8 = 5.01703

wh = 0.179 * 0.42662 * 9.8 = 0.74838

wb = 0.0122 * 9.8 = 0.11956

So I end up with: 4.14909 N.

So where am I going wrong? Thanks :)
 
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  • #2


Hi there,

Thank you for posting your attempt at a solution. Your approach is correct, but there are a few small calculations errors that may have led to your incorrect answer.

First, for the volume of the balloon, you used the radius of 0.467 m, which is the radius of the balloon when it is filled with helium. However, the initial problem states that the balloon is empty, so the correct radius to use for V = (4/3) * pi * r^3 is 0 meters.

Second, for the weight of helium (wh), you used the density of helium (0.179 kg/m^3) instead of the density of air (1.20 kg/m^3). This is because we are calculating the weight of the displaced air, which is equal to the buoyant force (B).

Making these two corrections, your final answer should be: 5.01703 - 1.20*0.179*9.8 - 0.0122*9.8 = 4.52537 N, which matches the solution in the book.

I hope this helps clarify any confusion and good luck with your studies!
 
  • #3


Your calculations are correct, but you made a mistake in converting the density of helium from kg/m3 to kg/m^3. The correct conversion would give a density of 0.179 kg/m^3, not 0.179 kg/m. This small error leads to a difference in the final answer.
Corrected calculation:
wh = 0.179 * 0.42662 * 9.8 = 0.74838

Corrected final answer:
T = 5.01703 - 0.74838 - 0.11956 = 4.14909 N

So your solution is actually correct, just a small error in the conversion. Good job!
 

FAQ: Balloon Buoyancy Homework: Find Tension in Line

1. What is balloon buoyancy?

Balloon buoyancy is the upward force exerted on an object immersed in a fluid, such as air, due to the difference in density between the object and the fluid. In simpler terms, it is the ability of a balloon to float in the air.

2. How is buoyancy related to tension in a line attached to a balloon?

The tension in a line attached to a balloon is directly related to the balloon's buoyancy. As the balloon rises, the tension in the line increases because the balloon is experiencing a greater upward force from the surrounding air. This tension helps keep the balloon from floating away too quickly.

3. What factors affect balloon buoyancy?

The two main factors that affect balloon buoyancy are the density of the balloon and the density of the surrounding air. A balloon filled with helium, which is less dense than air, will have greater buoyancy and rise higher. Additionally, the temperature and pressure of the air also play a role in balloon buoyancy.

4. How can the tension in a line attached to a balloon be calculated?

The tension in a line can be calculated using the equation: T = mg + (ρVg), where T is the tension, m is the mass of the balloon, g is the acceleration due to gravity, ρ is the density of the surrounding air, and V is the volume of the balloon. This equation takes into account both the weight of the balloon and the buoyant force acting on it.

5. What is the purpose of finding the tension in a line attached to a balloon?

Calculating the tension in a line attached to a balloon is important for understanding the stability and movement of the balloon. It can also help determine if the balloon is able to lift a certain weight or if it is reaching its maximum height. Additionally, knowing the tension in the line can help prevent the balloon from floating away or being pulled down too quickly.

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