Calculate Net Force on Hot-Air Balloon Using Ideal Gas Law | Homework Solution

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SUMMARY

The net force on a hot-air balloon with a volume of 1.5m³, where the internal air temperature is 75°C and the external air temperature is 24°C, can be calculated using the ideal gas law and Archimedes' principle. The density of the air inside the balloon is approximately 1 kg/m³, while the density of the external air is 1.19 kg/m³. The assumption that the pressure inside the balloon is equal to the pressure outside is valid due to the open bottom of the balloon. The net force can be determined by calculating the weight difference of the displaced external air versus the internal air.

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  • Understanding of the ideal gas law (PV=nRT)
  • Knowledge of Archimedes' principle
  • Basic concepts of pressure and density
  • Familiarity with temperature conversions (Celsius to Kelvin)
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  • Study the ideal gas law applications in real-world scenarios
  • Learn about Archimedes' principle and its implications in fluid mechanics
  • Explore calculations involving buoyancy and net forces on objects in fluids
  • Investigate the effects of temperature on gas density and pressure
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Homework Statement


A hot-air balloon has a volume of 1.5m^3 and is open at the bottom. If the air inside the balloon is at temperature of 75C, while the temperature of the air outside the balloon is 24C, at a pressure of about 1 atm, what is the net force on the balloon and its contents? (Neglect the Weight of the balloon itself).



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The Attempt at a Solution


I know that this question is about the ideal-gas law, but for some reason I just can't think how to do it.
 
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Force is pressure times area. This is all about the pressure of the gas inside the balloon.. That should be enough to get you going.
 
Consider Archimedes' principle. How does the density of the air inside the balloon compare to that outside?
 
So to find the density, I presume I first find the number of moles in the balloon using the ideal gas law. But to do so, can I assume that the pressure inside the balloon is the same as the pressure outside?

When I do this, I calculate 35 moles per m^3. This leads me to a density of just over 1kg/m^3. This is compared to the density of the air outside of 1.19kg/m^3

Oh.. ok I got it. So now I just figure out how much more gravity would pull down 1.5m^3 of the normal density air, and this is the force on the balloon.

Initially my problem was just that I wasn't sure if I could assume pressure inside to be equal to outside - because I thought the material of the balloon would add its own elasticity to the forces making the balloon smaller - but I guess because the bottom of the balloon is open this never becomes an issue.

Anyway, thanks for the help!
 

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