Question

Hot air balloons float in the air because of the difference in density between cold and hot air. Consider a balloon in which the mass of the pilot basket together with the mass of the balloon fabric and other equipment is mb. The volume of the hot air inside the balloon is V1 and the volume of the basket, fabric, and other equipment is V2. The absolute temperature of the cold air outside the balloon is Tc and its density is ρc. The absolute temperature of the hot air at the bottom of the balloon is Th (where Th>Tc). The balloon is open at the bottom, so that the pressure inside and outside the balloon is the same here. Assume that we can treat air as an ideal gas. Use g for the magnitude of the acceleration due to gravity

What is the density ρhρhrho_h of hot air inside the balloon? Assume that this density is uniform throughout the balloon?

What is the total weight WWW of the balloon plus the hot air inside it?

What is the magnitude of the buoyant force FBFBF_B on the balloon?

For the balloon to float, what is the minimum temperature TminTminT_min of the hot air inside it?

Answer 1

What is the density ρhρhrho_h of hot air inside the balloon

just compare the density and temperature

_h = _cT_c / T_h

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What is the total weight W of the balloon plus the hot air inside it

W is the total weight of gas ( balloon) inside

so,

W = m_bg + _h V₁ g

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What is the magnitude of the buoyant force FB on the balloon?

F_b = Vg

so,

F_B = _c (V₁ + V₂ )g

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For the balloon to float, what is the minimum temperature Tmin of the hot air inside it

The balloon will just begin to float when the magnitude of the buoyant force is equal to the magnitude of the force of gravity.

so,

m_bg + _h V₁ g = _c (V₁ + V₂ )g

T_min =