Which Happens First: Water Spills or Bottom Exposed in a Cylindrical Centrifuge?

[Doogan1]

1. Homework Statement

A cylindrical centrifuge of raidus 1 m and height 2 m is filled with water to a depth of 1 meter. As the centrifuge accelerates, the water level rises along the wall and drops in the center; the crossection will be a parabola.

a) Find the equation of the parabola in the figure in terms of h, the depth of the water at its lowest point.

b) As the centrifuge rotates faster and faster, either water will be spilled out of the top, or the bottom of the centrifuge will be exposed. Which happens first?




2. Homework Equations

R=1
H=2m
depth of water= 1m

3. The Attempt at a Solution

I've been thinking this problem must have something to do with integrals, but i am lost in the fact that the parabola from the cross section is constantly growing. I just can't seem to find an equation for the parabola, but i have made a hypothesis that the bottom will be exposed first, based of course off of no mathmatical basis. Help desperately needed.

 

[tiny-tim]

Welcome to PF!

I've been thinking this problem must have something to do with integrals, but i am lost in the fact that the parabola from the cross section is constantly growing.

Hi Doogan1! Welcome to PF!

Don't panic … questions like this are carefully written so that you can ignore the change in rotation.

Just assume that the rotation is constant, and work out the surface shape on the assumption …
i] the surface just touches the top
ii] the surface just touches the bottom.

 

[Doogan1]

Thx for replying and trying to help. I figured the problem out with my teacher today and basically it turned out to be just a big ol' integral.