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A hot-air balloon consists of a basket hanging beneath a large envelope filled with hot air. A typical hot-air balloon has a total mass of 539 kg, including passengers in its basket, and holds 2.62 ✕ 103 m3 of hot air in its envelope. If the ambient air density is 1.27 kg/m3, determine the density (in kg/m3) of hot air inside the envelope when the balloon is neutrally buoyant. Neglect the volume of air displaced by the basket and passengers. HINT kg/m3

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Answer:

1.05045 kg/m³

Explanation:

[tex]\rho_a[/tex] = Density of air = 1.26 kg/m³

[tex]\rho_{ha}[/tex] = Density of hot air

[tex]m_a[/tex] = Mass of balloon = 539 kg

g = Acceleration due to gravity = 9.81 m/s²

v = Volume of air in balloon = [tex]2.62\times 10^3\ m^3[/tex]

The net force on the balloon will be

[tex]F_n=(\rho_a-\rho_{ha})vg[/tex]

Also

[tex]F_n=m_ag[/tex]

[tex]\\\Rightarrow 549g=(\rho_a-\rho_{ha})vg\\\Rightarrow 549=(1.26-\rho_{ha})2.62\times 10^3\\\Rightarrow -\rho_{ha}=\frac{549}{2.62\times 10^3}-1.26\\\Rightarrow \rho_{ha}=1.05045\ kg/m^3[/tex]

The density of hot air inside the envelope is 1.05045 kg/m³