# Fluid mechanics (shape of free surface)

## Homework Statement

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.

## 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