When a capillary tube is immersed in water, the height to which the water rises is influenced by the tube's radius due to capillary action. The relationship between the height of the liquid column and the radius of the tube can be described by the formula:
Capillary Rise Formula
The height \( h \) of the liquid column in a capillary tube is given by:
h = (2γcosθ) / (ρgr)
Where:
- 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
If the radius \( r \) is doubled (to \( 2r \)), the new height \( h' \) can be calculated as:
h' = (2γcosθ) / (ρg(2r)) = h / 2
This shows that the height of the water column is halved when the radius is doubled.
Mass of Water in the Tube
The mass of water \( m \) in the tube is related to the volume of water and its density:
m = ρV
The volume \( V \) of water in the tube is given by:
V = πr²h
When the radius is doubled, the new volume \( V' \) becomes:
V' = π(2r)²(h/2) = 2πr²h = 2V
Thus, the new mass of water \( m' \) is:
m' = ρV' = ρ(2V) = 2m
Final Answer
Therefore, if the radius of the tube is doubled, the mass of water that will rise in the capillary tube will now be 2m.
