Question

Speed of sound in air depends upon pressure and density...by using the method of dimensional analysis, establish a formula for this speed. Take the constant proportionality K =1

Answer 1

Answer:

Formula for speed is $v=\sqrt{\frac{P}{\rho} }$

Explanation:

Speed of sound in air depends upon pressure P and density ρ.

$v=KP^a \rho ^b$

where constant proportionality K =1

$v=KP^a \rho ^b\\[LT^{-1}]=1\times [ML^{-1}T^{-2}]^a [ML^{-3}]^b$

by comparing the powers, we get

for L, $-a-3b=1.......(1)$

for T,

$-2a=-1\\a=\frac{1}{2}$

For M,

$a+b=0\\b=\frac{-1}{2}$

by substituting a, b

$v=KP^\frac{1}{2} \rho^\frac{-1}{2} \\v=1\times \sqrt{\frac{P}{\rho} } \\v=\sqrt{\frac{P}{\rho} } m/s$