(a) It is known that density r of air decreases with height y as ρ = ρₒ e ¯ _ y/yₒ where ρₒ = 1.25 kg m¯³is the density at sea level, and yₒ is a constant. This density variation is called the law of atmospheres. Obtain this law assuming that the temperature of atmosphere remains a constant (isothermal conditions). Also assume that the value of g remains constant. (b) A large He balloon of volume 1425 m³ is used to lift a payload of 400 kg. Assume that the balloon maintains constant radius as it rises. How high does it rise ? [Take yₒ = 8000 m and ρ _He = 0.18 kg m¯³].

(a) It is known that density r of air decreases with height y as ρ =

1.	(a) It is known that density r of air decreases with height y as ρ = ρₒ e ¯ _ y/yₒ where ρₒ = 1.25 kg m¯³is the density at sea level, and yₒ is a constant. This density variation is called the law of atmospheres. Obtain this law assuming that the temperature of atmosphere remains a constant (isothermal conditions). Also assume that the value of g remains constant. (b) A large He balloon of volume 1425 m³ is used to lift a payload of 400 kg. Assume that the balloon maintains constant radius as it rises. How high does it rise ? [Take yₒ = 8000 m and ρ _He = 0.18 kg m¯³].
Answer» (a) It is known that density r of air decreases with height y as ρ = ρₒ e ¯ _ y/yₒ where ρₒ = 1.25 kg m¯³is the density at sea level, and yₒ is a constant. This density variation is called the law of atmospheres. Obtain this law assuming that the temperature of atmosphere remains a constant (isothermal conditions). Also assume that the value of g remains constant. (b) A large He balloon of volume 1425 m³ is used to lift a payload of 400 kg. Assume that the balloon maintains constant radius as it rises. How high does it rise ? [Take yₒ = 8000 m and ρ _He = 0.18 kg m¯³].

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