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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 534 kg, including passengers in its basket, and holds 2.77 ✕ 103 m3 of hot air in its envelope. If the ambient air density is 1.23 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/m

Answer :

Answer:

 [tex]\rho_{int}[/tex] = 1,037 kg/m³

Explanation:

This is a problem with the Archimedes principle that states that the thrust on a body is equal to the weight of the dislodged fluid.

Let's write Newton's second law for this balance

     B- W- [tex]W_{air}[/tex] = 0

     B = W + [tex]W_{air}[/tex]  

Where W is the weight of the basket and the passengers and Wair is the weight of the air inside the balloon

The thrust is

     B = [tex]\rho_{ext}[/tex] g V

We replace

   [tex]\rho_{ext}[/tex] g V = mg + [tex]\rho_{int}[/tex] g V

   [tex]\rho_{int}[/tex] = ([tex]\rho_{ext}[/tex] V - m) / V

   [tex]\rho_{int}[/tex] = [tex]\rho_{ext}[/tex] - m / V

   [tex]\rho_{int}[/tex] = [tex]\rho_{ext}[/tex] -m / V

  [tex]\rho_{int}[/tex] = 1.23 - 534 /2.77 103

  [tex]\rho_{int}[/tex] = 1,037 kg/m³

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