Question

(a) How high will water rise in a glass capillary tube with a 0.500-mm

radius?

(b) How much gravitational potential energy does the water gain?

(c) Discuss possible sources of this energy.

Short answer

(a) The water will rise in glass capillary tube by: $h=2.978cm$.

(b) The gravitational potential energy is obtained as:$E_P=3.4\times 10^{-6}J$.

(c) Surface tension provides this energy.

Step-by-step solution

Conceptual Introduction

Capillary action is the mechanism by which a liquid moves through a small opening defying external forces like gravity, or even working against them.

Given Data

The radius of the capillary tube is$r=0.500\times 10^{-3}m$

How high will the water rise?

(a) The height to which water rises in a glass capillary tube is given by:

$h=\frac{2\gamma cos\theta }{\rho gr}$

Here,

θ is contact angle = $0^0$ for water-glass surface

$\gamma$isthe surface tension of water at $20^{0}C=0.07275J/m^2$ .

ρ is the density of the water is$997kg/m^3$

substituting these values in the above equation, we get

$\begin{array}{l}h=\frac{2\ast 0.07275\ast cos0}{997\ast 9.8\ast 0.5\times 10^{-3}}\\ h=\frac{0.1455}{4.8853}\\ h=0.02978m\\ h=2.978cm\end{array}$

Therefore, the water will rise: $h=2.978cm$.

Evaluating gravitational potential energy

(b) Gain in gravitational potential energy by water is given by:

$E_P=mgh$

Here, $g$and ‘$h$’ is known.

The mass of the water in the cylindrical capillary tube is calculated as:

$\begin{array}{l}m=\rho \times V\\ m=\rho \times \pi r^{2}h\\ m=997\times 3.14\times (0.5\times 10^{-3}{)}^2\times 0.02978\\ m=2.3307\times 10^{-5}kg\end{array}$

The water molecules inside the capillary tube are at different heights, hence, we are going to consider their average height.

Therefore, the gain in gravitational potential energy is given by:

$\begin{array}{l}{E}_P=mg(h/2)\\ E_P=2.3307\times 10^{-5}\ast 9.8\ast (0.02978/2)\\ E_P=3.4\times 10^{-6}J\end{array}$

Hence, the gravitational potential energy is:$E_P=2.17\times 10^{-6}J$.

What is the possible source of energy?

(c) The surface tension of the water is providing the necessary force to increase the height inside the capillary tube.

Therefore, energy is provided by surface tension.