Question

When a capillary tube of radius r is immersed in a liquid of density rho the liquid rises to a height h in it. if m is the mass of the liquid in the capillary tube, the potential energy of this mass of the liquid in the tube is?

Answer 1

radius of capillary tube = r

density of liquid in which capillary tube is immersed = $\rho$

height of liquid rises = h

mass of liquid in capillary tube = m

potential energy of liquid in the tube is ?

We know the formula for the gravitational potential energy in terms of the height of the liquid but it is spread over the entire length and not focused at the top. Thus we shall consider the concept of center of mass and we can assume that the entire mass is concentrated at the center of mass of the liquid which is at geometrically half of the total height.

so, U = mgh/2

here, m = volume × density

m = πr²h × $\rho$

and h = $\frac{2Tcos\theta}{\rho rg}$

where T is surface tension and $\theta$ angle of contact of liquid.e.g., 0°

so, h = 2T/$\rho$rg......(1)

now, potential energy, U = πr²$\rho$h²g/2

from equation (1),

U = πr²$\rho$g/2 × {4T²/$\rho^2$r²g²}

= 2πT²/$\rho$g