Answer :
The diameter of the capillary tube is approximately 0.175 mm.
To calculate the diameter of the capillary tube, we can use the principles of capillary action and the concept of the capillary rise equation, which states:
[tex]\[ h = \frac{{2\gamma}}{{\rho g r}} \][/tex]
Where:
- h is the height of the liquid rise (17 cm in this case).
- [tex]\( \gamma \)[/tex] is the surface tension of the liquid (water in this case).
- [tex]\( \rho \)[/tex] is the density of the liquid (water's density is approximately 1000 kg/m³).
- g is the acceleration due to gravity (approximately [tex]\( 9.81 \, \text{m/s}^2 \))[/tex].
- r is the radius of the capillary tube.
First, we need to convert the height from centimeters to meters (17 cm = 0.17 m). The surface tension of water at room temperature is about 0.0728 N/m. Plugging in the known values, we get:
[tex]\[ 0.17 \, \text{m} = \frac{{2 \times 0.0728}}{{1000 \times 9.81 \times r}} \][/tex]
Now, solving for r:
[tex]\[ r = \frac{{2 \times 0.0728}}{{0.17 \times 1000 \times 9.81}} \][/tex]
[tex]\[ r \approx \frac{{0.1456}}{{1663.7}} \][/tex]
[tex]\[ r \approx 8.75 \times 10^{-5} \, \text{m} \][/tex]
Finally, converting from meters to millimeters (1 m = 1000 mm):
[tex]\[ r \approx 8.75 \times 10^{-5} \, \text{m} \times 1000 \, \text{mm/m} \][/tex]
[tex]\[ r \approx 0.0875 \, \text{mm} \][/tex]
So, the diameter of the capillary tube is twice the radius, which is approximately 2 x 0.0875mm = 0.175mm.
Complete question :- Water rises in a glass capillary tube to a height of 17 cm. What is the diameter of the capillary tube?
Final answer:
In Physics, especially in fluid dynamics, capillary action describes how water can defy gravity in a narrow tube, with the height of ascent inversely related to the tube's diameter. Given the height of water rise doubles from the provided example, the tube's diameter likely halves, making option (a) 0.034 cm the most plausible answer.
Explanation:
The question revolves around the phenomena of capillary action, a physical principle pivotal in the study of Physics, specifically within the realm of fluid dynamics. This principle elucidates how water can climb upward in a narrow tube against gravity, a capability influenced by the tube's diameter, among other factors. When considering the provided example and applying the concept that the height to which water rises inversely correlates with the diameter of the tube, doubling the height from the reference (8.4 cm to 17 cm) suggests that the inner diameter of the capillary tube decreases in proportion. Given the initial example where water rises to a height of 8.4 cm with a tube diameter of 0.36 mm, an increase in height suggests a reduction in diameter; hence option (a) 0.034 cm seems plausible as it represents a halving, aligning with the theoretical understanding that inversely relates capillary rise to tube diameter.
