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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.
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