Fluid mechanics (shape of free surface)

[Mandeep Deka]

1. The problem statement, all variables and given/known data
A tube of radius 'R' and height 'H' (placed vertically), is filled with a fluid till a height 'h' (<H). Now it is rotated with an angular velocity 'w' about the central axis. Prove that the shape of the free surface is parabolic.


2. Relevant equations



3. The attempt at a solution
The proof of this equation is simple, and i have derived it. Now along with this question a statement was mentioned, which i aint able to prove.

Is is said that the initial height of the fluid is the arithmetic mean of the maximum height (i.e the height of the fluid at the end of the tube) and the minimum height (i.e the height of the fluid at the center). I have tried, but failed to prove it. Is the proposition valid?
If yes, I would be grateful if someone could give me some hint as to how i prove it!!

[fzero]

You will probably want to draw the radial cross-section with the parabola and the original water level crossing it. The volume of water above the original water level at the edges should equal the volume "missing" below it in the center.

[Mandeep Deka]

exactly, but how do i do it?
That's what i am asking, How will i calculate the volume of water above and below the original level?

[sgd37]

well you integrate your equation for the shape of the free surface from +R/2 to -R/2 and it should give 0 if you have placed the origin of the graph at (R/2,h) in the cylinder frame