Velocity of sound in air depends on pressure P, and density d.
Find the relation among them using dimensional analysis.
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The velocity v of sound in a gas depends upon its pressure P and density ρ, shown by using dimensional analysis, show that ν∝
ρ
P
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V∝P
a
ρ
b
From the relation, using dimensional analysis,
Dimension of LHS= Dimension of RHS
∴LT
−1
=[ML
−1
T
−2
]
a
[ML
−3
]
b
⇒LT
−1
=M
a+b
L
−a−3b
T
−2a
Comparing LHS and RHS,
a+b=0
−a−3b=1
−2a=−1
On solving the equations,
a=
2
1
& b=
2
1
∴V∝P
2
1
ρ
−
2
1
⇒V∝
ρ
p
)(V−b)= constant. The unit of a is :
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Hubble's law is expressed as (here, v= speed of recession, r= distance of galaxy, H= Hubble constant).
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