Question

Three capillaries of same length but radii \[2r,3r,4r\] are connected in series and a liquid flows through them in streamline flow. If the pressure across the third capillary is \[16mm\] of Hg the pressure across the first capillary will be(in mm of Hg)-
A. \[25.6\]
B. \[2.56\]
C. \[256\]
D. \[2560\]

Answer 1

Hint:Here three capillary tubes are directly connected with each other and they are exerting pressure. By using the formula we can find the value of pressure. We can define capillary action as a phenomenon where ascension of liquids through a tube or cylinder takes place. This primarily occurs due to adhesive and cohesive forces.

Complete step by step answer:

Consider P,${P_1}$, ${P_2}$ be the pressure exerted at the starting, joining and the ending of the capillary tube. These two tubes are joined in series(as shown)
So,
\[\dfrac{{\pi \left( {P - {P_1}} \right){{\left( {2r} \right)}^4}}}{{8\eta l}} = \dfrac{{\pi \left( {{P_1} - {P_2}} \right){{\left( {4r} \right)}^4}}}{{8\eta l}}\]
After solving this equation, we get-
\[\left( {P - {P_1}} \right) = \left( {{P_1} - {P_2}} \right)16\] --- (1)
\[P - 17{P_1} = 16{P_2}\] --- (2)
According to question,
\[P - {P_2} = 16mm\]
After solving all the given equations, by substituting the given values of r and pressure. in the question. We get-
\[{P_1} = 256mm\] of Hg.

Hence, option C is correct.

Note:If any resistors or tubes are directly connected then it means they are in series. For finding the total of them, we can find the value of the equivalent. A capillary tube is a tube with a very fine bore. The phenomena of rise and fall of liquid inside these tubes is known as Capillary. As we see inside the tube, water rises, it is because of adhesion of water molecules to the wall of the tube. It causes an upward force on the liquid at the edges and we see that the meniscus turned upward. This capillary action comes into account when the adhesion to the walls is stronger than the cohesive forces between the liquid molecule.