Fluid Mechanics: Weight vs Pressure

[Prannoy Mehta]

Homework Statement

A thin uniform circular tube is kept in a vertical plane. Equal volumes of (The liquids subtend a right angle at the centre) two non miscible liquids whose densities are a and b respectively fill half of the tube as shown. (The diagram depicts a>b) In equilibrium the radius passing through the interface makes an angle of 30 degrees with the vertical. The ratio of densities a/b is equal to.

Homework Equations

The basic equations of fluid mechanics given in any introductory course.

The Attempt at a Solution

I tried equating the weights on both the side as it must be necessary condition for the equilibrium.
Doing so I have obtained the answer as 3. The answer given in the text is 3.732. Where is the flaw in my concept.

Thanks for all the help.

 

[Simon Bridge]

How can a tube be kept in a plane?
Is the tube lying on its side or standing on end?
I think we need to see the diagram.

 

[Prannoy Mehta]

I thought it was not required, sorry for the trouble..

 

[Chestermiller]

The flaw in your logic us using the weights. You ought to be using dp/dz=-ρg, and equating the pressures at the interface.

Chet

 

[Prannoy Mehta]

I still did not understand why that concept of weight is wrong, for equilibrium the weight of the liquid with density a, subtending 60 degrees at the centre, should be equal to that, of the a subtending 30 degrees, and the liquid b which is subtending 90 degrees. I do realize there is a flaw in that concept, just wanted to know what is the flaw..

PS: I got the answer using that method. Thank you :)

 

[Chestermiller]

I still did not understand why that concept of weight is wrong, for equilibrium the weight of the liquid with density a, subtending 60 degrees at the centre, should be equal to that, of the a subtending 30 degrees, and the liquid b which is subtending 90 degrees. I do realize there is a flaw in that concept, just wanted to know what is the flaw..

You are aware that the bottom of the curved tube wall supports part of the weight of each fluid, correct?

Chet

 

[Prannoy Mehta]

Yes, I got it now. I din't think of it that perspective.