A capillary tube of [tex]$0.3 \, \text{mm}$[/tex] diameter is placed vertically inside a liquid of density [tex]$800 \, \text{kg/m}^3[/tex], surface tension [tex]$5 \times 10^{-4} \, \text{N/m}[tex]$[/tex], and angle of contact [tex]$[/tex]30^{\circ}$[/tex]. Calculate the height to which the liquid rises in the capillary tube.



Answer :

To find the height to which the liquid rises in the capillary tube, we'll use the concepts from capillary action and the relevant physical principles. Here's a detailed step-by-step solution to the problem:

### Given Data:
- Diameter of the capillary tube, [tex]\( d = 0.3 \)[/tex] meters
- Density of the liquid, [tex]\( \rho = 800 \)[/tex] kg/m³
- Surface tension of the liquid, [tex]\( \gamma = 5 \times 10^{-4} \)[/tex] N/m
- Angle of contact, [tex]\( \theta = 30^\circ \)[/tex]

### Step-by-Step Solution:

1. Convert the Angle of Contact to Radians:

The contact angle given is in degrees, so we need to convert it to radians for further calculations.

[tex]\[ \theta_{\text{rad}} = \frac{\pi}{180} \times \theta_{\text{deg}} \][/tex]
Substituting the given angle:

[tex]\[ \theta_{\text{rad}} = \frac{\pi}{180} \times 30 = \frac{\pi}{6} \approx 0.5236 \text{ radians} \][/tex]

2. Calculate the Radius of the Capillary Tube:

We are given the diameter, [tex]\( d \)[/tex], and we need to find the radius, [tex]\( r \)[/tex].

[tex]\[ r = \frac{d}{2} \][/tex]
Substituting the given diameter:

[tex]\[ r = \frac{0.3}{2} = 0.15 \text{ meters} \][/tex]

3. Determine the Height to which the Liquid Rises:

The formula to calculate the height [tex]\( h \)[/tex] to which the liquid rises in a capillary tube is given by:

[tex]\[ h = \frac{2 \gamma \cos \theta}{\rho g r} \][/tex]

Where:
- [tex]\( \gamma \)[/tex] is the surface tension
- [tex]\( \theta \)[/tex] is the angle of contact in radians
- [tex]\( \rho \)[/tex] is the density of the liquid
- [tex]\( g \)[/tex] is the acceleration due to gravity ([tex]\( 9.81 \)[/tex] m/s²)
- [tex]\( r \)[/tex] is the radius of the capillary tube

Substituting all the values into the formula:

[tex]\[ h = \frac{2 \times 5 \times 10^{-4} \times \cos(0.5236)}{800 \times 9.81 \times 0.15} \][/tex]

4. Perform the Calculation:

Carefully calculating the numerator and the denominator:

- Calculate [tex]\( \cos(0.5236) \)[/tex]:

[tex]\[ \cos(0.5236) \approx 0.866 \][/tex]

- Calculate the numerator:

[tex]\[ 2 \times 5 \times 10^{-4} \times 0.866 = 8.66 \times 10^{-4} \][/tex]

- Calculate the denominator:

[tex]\[ 800 \times 9.81 \times 0.15 = 1177.2 \][/tex]

- Finally, divide the numerator by the denominator to find [tex]\( h \)[/tex]:

[tex]\[ h = \frac{8.66 \times 10^{-4}}{1177.2} \approx 7.357 \times 10^{-7} \text{ meters} \][/tex]

### Final Result:

The height to which the liquid rises in the capillary tube is approximately [tex]\( 7.357 \times 10^{-7} \)[/tex] meters.