Question

The hydrostatic pressure 'P' of a liquid column depends upon the density 'd',height 'h' of liquid column and also an accleration 'g' due to gravity.using dimensional analysis.derive formula for pressure P.

Answer 1

$\square$ $p$ is the hydrostatic pressure (Pa= 1 kg/(m·s²)),

$\square$ $d$ is the fluid density (kg/m³),

$\square$ $g$ is gravitational acceleration (m/s²),

h is height of liquid column (m).

$$p = d^{\alpha} g^{\beta} h^{\gamma} \gg k g * m^{-1} * s^{-2} = k g^{\alpha} m^{-3\alpha} m^{\beta} s^{-2\beta} m^{\gamma} \gg$$ $$\gg \alpha = 1, -3\alpha + \beta + \gamma = -1, -2\beta = -2 \gg \alpha = 1, \beta = 1, \gamma = 1$$

That’s why:

$$p = d g h$$