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

Question

27-03-2025
Physics

High School

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

Answer :

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ConnorRobert ConnorRobert · Answer 1 · 2024-02-01T04:51:35-05:00

Final answer:

The mass of water rising in a capillary tube with radius '3r' will be three times the original mass 'M', given the relationship between the height of water rise and the radius of the tube is inversely proportional.

Explanation:

The mass of water that rises in a capillary tube is determined by the height to which the water rises and the volume of the tube, which relies on its radius. According to the capillary rise formula 2T cos θ h = rpg, where T is the surface tension of water, θ is the contact angle, h is the height of water rise, r is the radius of the capillary tube, p is the density of water, and g is the acceleration due to gravity. This equation indicates that the height of water rise, h, is inversely proportional to the radius, r. If the radius is tripled, the height of the water rise would become one-third, assuming the surface tension, contact angle, density, and gravity remain constant. Since volume is related to the square of the radius and height of the liquid column, if the radius is tripled, the volume and thus the mass of the water will increase by a factor of 3² times the new height, which is 1/3 of the original height. This leads to a total mass increase by a factor of 3. Therefore, for a capillary tube with a radius of '3r', the mass of water that would rise is 3M.