The pressure p , and density, ρ , of the atmosphere at a height y above the earth’s surface are related by d p = − g ρ d y . Assuming that p and ρ satisfy the adiabatic equation of state p = p o ( ρ ρ o ) γ , where γ ≠ 1 is a constant and p o and ρ o denote the pressure and density at the earth’s surface, respectively, show that p = p o [ ( 1 − γ − 1 γ ) ( ρ o g y p o ) ] γ / ( γ − 1 ) .
The pressure p , and density, ρ , of the atmosphere at a height y above the earth’s surface are related by d p = − g ρ d y . Assuming that p and ρ satisfy the adiabatic equation of state p = p o ( ρ ρ o ) γ , where γ ≠ 1 is a constant and p o and ρ o denote the pressure and density at the earth’s surface, respectively, show that p = p o [ ( 1 − γ − 1 γ ) ( ρ o g y p o ) ] γ / ( γ − 1 ) .
Solution Summary: The author explains the adiabatic equation of the state.
The pressure
p
, and density,
ρ
, of the atmosphere at a height
y
above the earth’s surface are related by
d
p
=
−
g
ρ
d
y
.
Assuming that
p
and
ρ
satisfy the adiabatic equation of state
p
=
p
o
(
ρ
ρ
o
)
γ
, where
γ
≠
1
is a constant and
p
o
and
ρ
o
denote the pressure and density at the earth’s surface, respectively, show that
p
=
p
o
[
(
1
−
γ
−
1
γ
)
(
ρ
o
g
y
p
o
)
]
γ
/
(
γ
−
1
)
.
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