Calculating the Radius of a Hot Air Balloon to Withstand a Load of 300 kg

In summary, the conversation discusses the calculation of the radius of a hot air balloon that can withstand a total load of 300 kg. The relevant equations are Archimedes' law and the buoyant force. The formula used is r=m/[(density of air-density of hot air)*(4/3)*pi], with the density of hot air being calculated using the equation rho=(p*M)/(R*T). The conversation also mentions the importance of adding the weight of the heated air to the total load. Ultimately, the calculated radius for the balloon is 8.0 m.
  • #1
Lenoshka
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Homework Statement
The air in the hot air balloon is heated to 70 ° C. At the start, the surrounding atmospheric pressure is 98 kPa and the temperature is 28 ° C, the balloon is open from below, assuming a spherical shape and Mr air is 29.
What must be its radius to withstand a total load of 300 kg (not including the air)? [12,2 m]
Relevant Equations
Archimedes law, buoyant force
The result is supposed to be 12,2 m but every time I get 8,016 m... I used for example this formula >r=m/[(density of air-density of hot air)*(4/3)*pi]
For density I used > rho=(p*M)/(R*T)

Am I forgetting something? Thanks in advance.
 
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  • #2
The buoyant force comes from the weight/mass of the air at ambient temperature that is displaced, but don't forget to add the weight of the heated air to the load of 300 kg.
 
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  • #3
Lenoshka said:
Homework Statement:: The air in the hot air balloon is heated to 70 ° C. At the start, the surrounding atmospheric pressure is 98 kPa and the temperature is 28 ° C, the balloon is open from below, assuming a spherical shape and Mr air is 29.
What must be its radius to withstand a total load of 300 kg (not including the air)? [12,2 m]
Relevant Equations:: Archimedes law, buoyant force

r=m/[(density of air-density of hot air)*(4/3)*pi]
You forgot a power of 1/3 here, but otherwise it looks correct to me.

Charles Link said:
The buoyant force comes from the weight/mass of the air at ambient temperature that is displaced, but don't forget to add the weight of the heated air to the load of 300 kg.
That is where the density of the hot air comes from in the (corrected) formula above:
$$
r = \left(\frac{3m}{4\pi (\rho_{\rm air} - \rho_{\rm hotair})}\right)^{1/3}
$$
If he just took the buoyant force only the density of the cold air would be in the formula.

Plugging the numbers into Octave, I get 8.0 m (the input data really does not support using more significant digits).
 
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FAQ: Calculating the Radius of a Hot Air Balloon to Withstand a Load of 300 kg

What is the radius of a hot-air balloon?

The radius of a hot-air balloon refers to the distance from the center of the balloon to its outer edge. It is typically measured in meters or feet.

How is the radius of a hot-air balloon determined?

The radius of a hot-air balloon is determined by the size and shape of the balloon's envelope, which is the fabric that contains the hot air. It is also affected by the amount of air that is heated and the weight of the basket and passengers.

Why is the radius of a hot-air balloon important?

The radius of a hot-air balloon is important because it affects the balloon's lift and stability. A larger radius means a larger surface area and more lift, while a smaller radius can make the balloon more maneuverable.

Can the radius of a hot-air balloon be changed?

Yes, the radius of a hot-air balloon can be changed by adjusting the amount of hot air in the envelope. Adding more hot air will increase the radius and lift, while releasing hot air will decrease the radius and lift.

How does the radius of a hot-air balloon affect its flight?

The radius of a hot-air balloon plays a crucial role in its flight. A larger radius means more lift and stability, allowing the balloon to stay in the air longer. It also affects the balloon's maneuverability, as a smaller radius allows for quicker turns and changes in direction.

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