Mechanical Properties of Fluids
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Question : 31
Total: 31
(a) It is known that density r of air decreases with height y as
ρ = ρ 0 e −
wherer 0 = 1.25 k g m – 3 y 0
(b) A large He balloon of volume1425 m 3 y 0 = 8000 m ρ H e = 0.18 k g m – 3 ] .
where
(b) A large He balloon of volume
Solution:
(a) We know that the rate of decrease of density ρ ρ −
∝ ρ
= − K ρ
where K is a constant of proportionality. Here –ve sign shows thatρ y
∴
= − K d y
Integrating it within the conditions, asy y , ρ 0 to ρ
= −
K d y
[ log e ρ ] ρ 0 ρ = − K y
orlog e ρ − log e ρ 0 = − K y
orlog
= − K y
or
= e − K y
orρ = ρ 0 e − K y
HereK y 0 K =
ρ = ρ 0 e −
(b) The balloon will rise to a height, where its density becomes equal to the air at that height.
Density of balloon,
ρ =
=
=
=
k g ∕ m 3
A s , ρ = ρ 0 e − y ∕ y 0
∴
= 1.25 e − y ∕ 8000
ore −
=
ore
=
= 2.7132
Taking log on both the sides
= log e 2.7132
= 2.3026 log 10 2.7132
= 2.3026 × 0.4335 ≈ 1
y = 8000 × 1 = 8000 m
= 8 k m .
where K is a constant of proportionality. Here –ve sign shows that
Integrating it within the conditions, as
or
or
or
or
Here
(b) The balloon will rise to a height, where its density becomes equal to the air at that height.
Density of balloon,
or
or
Taking log on both the sides
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