If 'm' is the mass of water that rises in a capillary tube of radius 'r', then what is the mass of water that will rise in a capillary tube of radius '3r'?

Question

30-04-2025
Physics

High School

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If 'm' is the mass of water that rises in a capillary tube of radius 'r', then what is the mass of water that will rise in a capillary tube of radius '3r'?

Answer :

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MoodyErica MoodyErica · Answer 1 · 2024-01-29T11:45:35-05:00

Final answer:

The mass of water that will rise in a capillary tube with a radius of '3r' is 9 times the mass of water that rises in a tube with radius 'r', assuming that the only factor changed is the radius of the tube and that the contact angle remains at 0°.

Explanation:

The mass of water that rises in a capillary tube due to capillary action depends on the radius of the tube among other factors. According to the formula for capillary rise, h = (2T cos θ) / (rpg), where h is the height the liquid rises, T is the surface tension, θ is the contact angle, r is the radius of the tube, p is the density of the liquid, and g is the acceleration due to gravity.

Since the density of water (ρ) is 1.0 g/cm³, and considering that water wets the glass leading to a contact angle of 0°, we can rewrite the height equation simply as h ≈ (2T) / (rp) because cos(0) equals 1.

Looking at the relationship between the height of water and the radius of the tube, we can see that height is inversely proportional to the radius. Therefore, if the radius triples, the height of water rise in the capillary will decrease to one-third. After finding the new height, we can determine the mass of the water that rises in the tube with radius 3r.

Since mass (m) is equal to the volume of water multiplied by the density (pdV), and the volume is proportional to the square of the radius times the height (πr²h), tripling the radius while the height reduces to one-third results in the mass of water rising by a factor of three (3² / 3 = 3).