Question

Calculate the rise of water inside a clean glass capillary tube of radius 0.1 mm, when immersed in water of surface tension $7×{10^{−2}} N/m$. The angle of contact between water and glass is zero, the density of water = $1000kg/{m^3}$ , g=$9.8m/{s^2}$ .

Answer 1

Hint: In this question, we will use the relation between the displaced water, surface tension, density and gravity, substituting the given values in the equation will give us the required result. Further, we will study about the Archimedes principle, which is related to buoyant force, when an object having some mass is immersed in the liquid.
Formula used:
$H = \dfrac{{2T\cos \theta }}{{\rho gr}}$

Complete answer:
When a capillary tube of radius r is dipped in a liquid having some density and surface tension T, the liquid rises or falls through a distance, given by:
$H = \dfrac{{2T\cos \theta }}{{\rho gr}}$
Substituting the given values in the above equation, we get:
$\eqalign{& \Rightarrow H = \dfrac{{2 \times 7 \times {{10}^{ - 2}} \times \cos \theta }}{{1000 \times 9.8 \times 0.1 \times {{10}^{ - 3}}}} \cr
  & \therefore H = 0.142m \cr} $
Therefore, we get the required result of rise of water inside the capillary tube, which is given by the above result.

Additional information:
As we know, Archimedes principle states that an object immersed in a fluid experiences some buoyant force that is equal in magnitude to the force of gravity on the displaced fluid. It is also known as the law of buoyancy. This principle was discovered by the ancient Greek mathematician and inventor Archimedes.
The weight of the displaced fluid is equal to the subtraction of weight of object in vacuum and weight of object in fluid. The weight of the displaced fluid is directly proportional to the volume of the displaced fluid (if the surrounding fluid is of uniform density).
The buoyant force on a body floating in a liquid or gas is also equivalent in magnitude to the weight of the floating object and is opposite in direction. So, in this situation the object neither rises nor sinks.
We should also know about the buoyant force. This is an upward force exerted by a fluid which opposes the weight of a partially or fully immersed object in the fluid. Here, in the fluid, pressure increases with depth as a result of the weight of the overlying fluid. So, the pressure at the bottom of a column of fluid is greater than at the top of the column.

Note:
One should notice that the Archimedes principle is only valid for fluids, where buoyant force can be observed. Also, the displaced fluid is equal to the weight of the object immersed in the water. In the displaced fluid, gravity also plays an important role.