Capillary Tube , Out of The World!

Classical Mechanics Level 5

There is a cylindrical capillary tube of sufficient radius R and length L with one end closed and the other end kept open. Initially the tube is placed in open atmosphere, with its open end facing down. It is then slowly dipped vertically down into a liquid, which has density \(\rho \), surface tension T and atmospheric pressure \({ P }_{ o }\). Also assume that the contact angle between the capillary tube and the liquid system is zero.

Up to what depth ( y ) can this capillary tube be dipped in the liquid such that no liquid rises in the capillary tube?

The answer can be expressed as :

\[{y = \cfrac { a }{ b } R +\cfrac { c }{ d } (\cfrac { TL }{ { P }_{ o }R } )}.\]

Find the value of \( a + b + c + d \).

\(\bullet\) Assume That \[\displaystyle{\cfrac { T }{ { P }_{ o }R } <<1}\]
\(\bullet\) Assume That whole atmosphere is made of an Ideal Mono atomic gas.
\(\bullet\) Assume That wall of Capillary Tube is Adiabatic (No heat Loss Takes Place).
\(\bullet\) \(a,b,c,d\) are Positive integers , and pair \((a,b)\) and \((c,d)\) are co-prime .

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