9. What would be the height of the atmosphere if the air density is uniform? Assume that at sea level the air pressure is 10 1300 p a and the air density is 1.3 kg per metre cube
Answers
Explanation:
(a)We use the expression for the variation of pressure with height in an incompressible fluid: p
2
=p
1
−ρg(y
2
−y1.
We take y
1
to be at the surface of Earth, where the pressure is p
1
=1.01×10
5
Pa, and y
2
to be at the top of the atmosphere, where the pressure is p
2
=0. For this calculation, we take the density to be uniformly 1.3kg/m
3
. Then, y
2
−y
1
=
ρg
p
1
=
(1.3kg/m
3
)(9.8m/s
2
)
1.01×10
5
Pa
=7.9×10
3
m=7.9km.
(b)Let h be the height of the atmosphere. Now, since the density varies with altitude, we integrate p
2
=p
1
−∫
0
h
ρgdy.
Assuming ρ=ρ
∘
(1−y/h), where ρ
∘
is the density at Earth's surface and g=9.8m/s
2
for 0≤y≤h, the integral becomes
p
2
=p
1
−∫
0
h
ρ
∘
g(1−
h
y
)dy=p
1
−
2
1
ρ
∘
gh.
Since p
2
=0, this implies h=
ρ
∘
g
2p
1
=
2(1.01×10
5
Pa)
=16×10
3
m=16km