A cappilary of radius R is immersed in water and water rises in it to a height H.Mass of water in the cappilary tube is M.if the radius of the tube is doubled mass of water that willl rise in capillary tubbe will now be

To understand how the mass of water in a capillary tube changes when the radius is altered, we need to delve into the principles of capillarity and how they relate to the geometry of the tube. When a capillary tube is immersed in water, the height to which the water rises is influenced by the tube's radius, the surface tension of the liquid, and the density of the liquid.

Capillary Action Basics

Capillary action occurs due to the adhesive forces between the liquid and the walls of the tube, as well as the cohesive forces within the liquid itself. The height \( H \) to which the liquid rises in a capillary tube can be described by the formula:

H = (2γ cos θ) / (ρgR)

• γ = surface tension of the liquid
• θ = contact angle
• ρ = density of the liquid
• g = acceleration due to gravity
• R = radius of the tube

Effect of Doubling the Radius

Now, if we double the radius of the capillary tube (let's call the new radius \( R' = 2R \)), we can analyze how this affects the height of the water column. Plugging \( R' \) into the formula gives:

H' = (2γ cos θ) / (ρg(2R)) = H / 2

This shows that when the radius is doubled, the height \( H' \) of the water column is halved. This is a crucial point because it directly impacts the volume of water that can be held in the tube.

Calculating the Mass of Water

The volume \( V \) of water in the capillary tube can be calculated using the formula for the volume of a cylinder:

V = πR²H

Initially, the volume of water in the tube with radius \( R \) and height \( H \) is:

V = πR²H

When the radius is doubled, the new volume \( V' \) becomes:

V' = π(2R)²(H/2) = π(4R²)(H/2) = 2πR²H

This indicates that the volume of water in the tube with the doubled radius is twice that of the original volume.

Mass of Water in the Capillary Tube

The mass \( M \) of the water can be calculated using the density \( ρ \) of the water:

M = ρV

For the original tube, the mass is:

M = ρ(πR²H)

For the new tube with the doubled radius, the mass becomes:

M' = ρ(2πR²H) = 2M

Final Thoughts

In summary, when the radius of the capillary tube is doubled, the mass of water that will rise in the tube also doubles. Therefore, if the original mass of water was \( M \), the new mass of water in the capillary tube will be:

M' = 2M

This relationship highlights the interplay between geometry and fluid dynamics in capillary action, illustrating how changes in dimensions can significantly affect the behavior of liquids in confined spaces.