Answer :
Capillary action refers to the ability of a liquid to flow through a narrow space, such as a thin tube, without the assistance of external forces and sometimes even in opposition to gravity. This phenomenon occurs due to the interplay between the cohesive forces within the liquid and the adhesive forces between the liquid and the walls of the tube.
When considering the effect of changing the radius of the capillary tube:
- If the radius of the tube is increased, the height to which the liquid rises will decrease. This is because the weight of the liquid column becomes larger relative to the adhesive force pulling it up.
- If the radius of the tube is decreased, the height to which the liquid rises will increase, as the adhesive forces have a more significant effect relative to the weight of the liquid column.
Now, let's solve the problem with the given values:
1. Known Variables:
- Radius of the tube, [tex]\( r = 2 \, \text{mm} = 0.002 \, \text{m} \)[/tex]
- Contact angle, [tex]\( \theta = 35^\circ \)[/tex]
- Density of water, [tex]\( \rho = 1000 \, \text{kg/m}^3 \)[/tex]
- Surface tension of water, [tex]\( \gamma = 0.072 \, \text{N/m} \)[/tex]
- Acceleration due to gravity, [tex]\( g = 9.8 \, \text{m/s}^2 \)[/tex]
2. Convert Contact Angle to Radians:
The contact angle needs to be in radians for calculations:
[tex]\[ \theta_{\text{rad}} = 35^\circ \times \frac{\pi}{180} \approx 0.610865 \, \text{radians} \][/tex]
3. Calculate the Height of Water Rise ([tex]\( h \)[/tex]):
The height to which water rises in a capillary tube can be calculated using the formula:
[tex]\[
h = \frac{2 \gamma \cos(\theta_{\text{rad}})}{\rho g r}
\][/tex]
4. Substitute the Values:
[tex]\[
h = \frac{2 \times 0.072 \times \cos(0.610865)}{1000 \times 9.8 \times 0.002}
\][/tex]
5. Result:
After performing the calculations, the height ([tex]\( h \)[/tex]) is approximately 0.006018 meters or 6.018 millimeters. This indicates how far the water will rise in the capillary tube with the given conditions.
When considering the effect of changing the radius of the capillary tube:
- If the radius of the tube is increased, the height to which the liquid rises will decrease. This is because the weight of the liquid column becomes larger relative to the adhesive force pulling it up.
- If the radius of the tube is decreased, the height to which the liquid rises will increase, as the adhesive forces have a more significant effect relative to the weight of the liquid column.
Now, let's solve the problem with the given values:
1. Known Variables:
- Radius of the tube, [tex]\( r = 2 \, \text{mm} = 0.002 \, \text{m} \)[/tex]
- Contact angle, [tex]\( \theta = 35^\circ \)[/tex]
- Density of water, [tex]\( \rho = 1000 \, \text{kg/m}^3 \)[/tex]
- Surface tension of water, [tex]\( \gamma = 0.072 \, \text{N/m} \)[/tex]
- Acceleration due to gravity, [tex]\( g = 9.8 \, \text{m/s}^2 \)[/tex]
2. Convert Contact Angle to Radians:
The contact angle needs to be in radians for calculations:
[tex]\[ \theta_{\text{rad}} = 35^\circ \times \frac{\pi}{180} \approx 0.610865 \, \text{radians} \][/tex]
3. Calculate the Height of Water Rise ([tex]\( h \)[/tex]):
The height to which water rises in a capillary tube can be calculated using the formula:
[tex]\[
h = \frac{2 \gamma \cos(\theta_{\text{rad}})}{\rho g r}
\][/tex]
4. Substitute the Values:
[tex]\[
h = \frac{2 \times 0.072 \times \cos(0.610865)}{1000 \times 9.8 \times 0.002}
\][/tex]
5. Result:
After performing the calculations, the height ([tex]\( h \)[/tex]) is approximately 0.006018 meters or 6.018 millimeters. This indicates how far the water will rise in the capillary tube with the given conditions.
