Centrifugal Force on a Bucket of Water

[BAnders1]

I had a test the other day and it doesn't sound like it will be graded for quite some time (our TA seems to be over his head in classwork). I wanted to see if anyone here could tell me if my answer to one of the questions is correct and justified. Prethanks.

Homework Statement

"A bucket of water is set spinning around its symmetry axis, determine the shape of the water in the bucket"

Homework Equations

F=-mω×(ω×r) (centrifugal force)

The Attempt at a Solution

I first found the centrifugal force on a small mass of water (F=mω²r), then I integrated this with respect to r to find the potential energy associated with the centrifugal force [U=-(mω²/2)r²]. Since the water is bound in the bucket, it cannot attain any kinetic energy, thus in equilibrium the potential energy due to the centrifugal force must cancel the gravitational potential energy U=mgz. Solving for z, I got z=(ω²/2g)r², thus the water forms a paraboloid.

Is my reasoning correct? Thank you in advance.

 

[Redbelly98]

Looks good.

In fact large, parabolic telescope mirrors are made by spinning molten glass until the glass has cooled and hardened.