Question

The velocity $v$ of sound in a gas depends upon its pressure $P$ and density $\rho$, shown by using dimensional analysis, show that $\nu \propto \sqrt{\frac{P}{\rho }}$

Answer 1

$V\propto P^a{\rho }^b$

From the relation, using dimensional analysis,

Dimension of LHS= Dimension of RHS

$\therefore L{T}^{-1}={\left[M{L}^{-1}{T}^{-2}\right]}^a{\left[M{L}^{-3}\right]}^b\Rightarrow L{T}^{-1}=M^{a+b}{L}^{-a-3b}{T}^{-2a}$

Comparing LHS and RHS,

$a+b=0$

$-a-3b=1$
$-2a=-1$

On solving the equations,

$a=\frac{1}{2}\&b=\frac{1}{2}$

$\therefore V\propto P^{\frac{1}{2}}{\rho }^{-\frac{1}{2}}$

$\Rightarrow V\propto \sqrt{\frac{p}{\rho }}$