Question

by using dimensional analysis derive an expression for the height h to which a liquid of density rho and surface tension as will rise in a capillary tube of radius r given the acceleration due to gravity is g and h is inversely proportional to r

Answer 1

Given that,

Height = h

Radius = r

Density = ρ

Surface tension = T

We know that,

The height of capillary tube is defined as,

$h=\dfrac{2T\cos\theta}{\rho g r}$....(I)

Where, T = surface tension of liquid

g = acceleration due to gravity

r = radius of tube

ρ = density of liquid

We need to prove h is inversely proportional to r

Using equation (I)

$h=\dfrac{2T\cos\theta}{\rho g r}$

Here, 2 and cosθ are constant.

Put the dimension formula of all element

$h\propto\dfrac{[MT^{-2}}{[ML^{-3}]\times[LT^{-2}]\times[L]}$

Here, M = mass

T = time

L = height or distance

$L\propto\dfrac{1}{[L]}$

Here, L shows the height of capillary tube in left side and L shows the radius of capillary tube in right side.

Hence, This proved, h is inversely proportional to r.