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

According to given

$V=k{p}^a{\rho }^b$ $\left\{p=pressure,\rho =density\right\}$

Applying dimensional formula on both sides

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

On comparing dimension of $M,L$ and $T$

$a+b=0-\left(1\right)$

$-a-2b=0-\left(2\right)$

$a=\frac{1}{2}-\left(3\right)$

From equation(3)

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

From equation(i) and (ii)

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

Hence,

$v=k{p}^{\frac{1}{2}}{\rho }^{-\frac{1}{2}}$

$v=\sqrt{\frac{p}{\rho }}$