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 $a + b + c + d$ .

Assumptions
$\bullet$ 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 .

This is Original

Try more Deepanshu's Mixing of concepts

The answer is 16.

1 solution

Deepanshu Gupta
Jan 1, 2015

I Will Post full Solution Later If No-one Post it , But for now the Hints and answer are :

Hints:

1)- Initially Gas is Present in Tube and when we dipped it in Liquid Then gas is Compressed by Liquid So That an Hemispherical Curved Surface ( Meniscus ) is formed at the Top most Liquid Level .

2)- For No liquid Rise The Pressure inside Capilary tube (which is exerted by gas molecules) Should be Balanced by the Lower Side of Meniscus.

3)- Now Walls are adiabatic So $\displaystyle{P{ V }^{ \quad \gamma }\quad =\quad constant}$ Initial Pressure, Volume is Known and final Volume is can be calculate in terms of 'y' and from Here we calculate Pressure of gas finally

Note : Most Important Step of this Question is that Final Volume is equal to Vol of Tube + Vol. of Hemispherical Meniscus(Just Like an Test Tube) , Since gas is Not only contained in cylindrical Vessel !

4)- Now use ${ P }_{ g }\quad ={ \quad P }_{ o }\quad +\quad \cfrac { 2T }{ L-y }$ .

5)- Stablished Relation b/w all known quantity and Use ${ (1+x) }^{ n }\quad =\quad 1+nx\quad (if\quad x<<1)$ .

You will get The Answer :

$\displaystyle{\boxed { y\quad =\cfrac { 2R }{ 3 } +\cfrac { 6TL }{ 5{ P }_{ o }R } } }$ .

Deepanshu, thanks for posting such interesting and engaging problems.

Here are some guidelines which will make your problems much better. 1) Use proper punctuation.
There is no need to randomly capitalize every other word. That makes it harder to read the sentences, as it is extremely distracting.
Use commas and fullstops as necessary.
2) Use proper grammar. Add "the, a, an" where appropriate.
3) Form complete sentences.
Each sentence should focus on conveying one main message. Avoid long convoluted sentences and run-on sentences.
4) Avoid using \quad in your Latex unless you really need to force a space.
Random spaces in an equation makes it harder to read. Latex automatically adjusts the equations to a suitable spacing.
5) I personally feel that there is no need to place your variables in quotation marks and bold them, as that is not standard. I have removed the bold, but left the quotation marks in.

I have made edits to several of your questions to help improve the formatting and presentation. Please review them to learn how to improve your problems.

I look forward to working on many more of your problems, and I am sure that the community would appreciate the time that you spend to make your problems more understandable!

Calvin Lin Staff - 6 years, 5 months ago

Thanks for Editing Sir . I will Try to Follow Your Instructions accordingly.

Deepanshu Gupta - 6 years, 5 months ago

Bhaiya, why is rad. = (L-y)\ ? and not R?

Md Zuhair - 3 years, 3 months ago

can you please write equation for pressure at meniscus

sashank bonda - 4 years, 1 month ago

why is radius of miniscus equal to L-y? pls explain

Ashutosh Sharma - 3 years, 5 months ago

Same question, I did it with R, and correct answer came

Md Zuhair - 3 years, 3 months ago

@Deepanshu Gupta can pls post ur full solution now.i am not able to get the above one.pls

Ashutosh Sharma - 3 years, 4 months ago

Capillary Tube , Out of The World!

This is Original

Try more Deepanshu's Mixing of concepts

The answer is 16.

1 solution

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