Question

Dimensional analysis of the equation velocity ${}^x=$ pressure difference ${}^{3/2}\times$ density ${}^{-3/2}$, gives the value of $x$ as ______

Answer 1

Answer: 3

$(\text{ velocity }{)}^x=P^{3/2}\times {\rho }^{-3/2}={\left(\frac{P}{\rho }\right)}^2$
$={\frac{\left[{\text{M L}}^{-1}{\text{T}}^{-2}\right]}{\left[{\text{M L}}^{-3}\right]}}^{\frac{3}{2}}$
$={\left[L^2{T}^{-2}\right]}^{3/2}$
$=\left[L^3{T}^{-3}\right]$
$=[L^1{T}^{-1}{]}^3$
But $\left[L^1{T}^{-1}\right]$ is the dimension of velocity.
$\Rightarrow x=3$