Question

A long capillary tube of mass π gm, radius 2mm and negligible thickness, is partially immersed in a liquid of surface tension 0.1Nm Take angle of contact zero and neglect buoyant force of liquid. The force required to hold the tube vertically, will be (g=10 \[m/{{s}^{2}}\])
                                                                

Answer 1

Hint: We will use the formula and concept of surface tension. The tension on the liquid surface and the capillary tube has been calculated. After getting the value of tension, we will find force by substituting the given values in the formula. Three forces act on the system, one on the liquid, the second on the surface of the test tube and the downward force due to liquid filled in the test tube.

Formula used:
Surface tension= \[=\dfrac{Force}{Length}\]

Complete answer:
We have a long capillary tube having mass \[m\] and radius \[R\] . The tube is partially immersed in a liquid. The surface tension of the liquid is given. We need to find the force which is required to hold the tube vertically.
Let \[F\] be the upward thrust on the tube and \[{{S}_{1}}\] be the tension on the surface of liquid. Surface tension on the surface of the capillary tube be \[{{S}_{2}}\] . \[mg\] be the force acting downward on the capillary tube.
Then the surface tension for the liquid surface and tube surface be
For liquid surface \[{{S}_{1}}=2\pi R(S)\]
For tube surface \[{{S}_{2}}=2\pi R(S)\]
So force can be calculated as = \[{{S}_{1}}+{{S}_{2}}+mg\]
                                                     =\[2\times 2\pi R(S)+mg\]……(1)
Given:
\[\begin{align}
  & m=\pi gm \\
 & R=2mm \\
 & g=10m/{{s}^{2}} \\
 & S=0.1N/m \\
\end{align}\]
Substituting the given values in equation (1)
\[\begin{align}
  & =2\pi (2\times 0.1)\times {{10}^{-3}}+\pi \times 10\times {{10}^{-3}} \\
 & =8\pi \times 0.1\times {{10}^{-3}}+10\pi \times {{10}^{-3}} \\
 & =10.8\pi mN \\
\end{align}\]
Therefore the total force acting will be \[10.8\pi mN\] .

Note:
Surface tension is a force exerted on the liquid surface. It has the property to occupy minimum possible surface area. Surface tension depends on the radius of the test tube, density of the liquid and height of the column. In our daily life the principle of surface tension can be seen in thermometer, tooth paste, the detergent we use etc.