Question

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

Answer 1

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