At Westonci.ca, we provide reliable answers to your questions from a community of experts. Start exploring today! Get immediate and reliable solutions to your questions from a community of experienced experts on our Q&A platform. Get immediate and reliable solutions to your questions from a community of experienced professionals on our platform.
Sagot :
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.
### 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.
Thank you for choosing our platform. We're dedicated to providing the best answers for all your questions. Visit us again. We hope you found this helpful. Feel free to come back anytime for more accurate answers and updated information. Thank you for using Westonci.ca. Come back for more in-depth answers to all your queries.
