In a spinning cylindrical glass of water, the water's surface is a surface of revolution of what curve?

Question

I'm interested in the curve as a function of the radius r. I have two guesses at this point. Either a parabola or a catenary. I'm not sure about it because I don't fully understand the magnitude of the force that causes the water to move upwards in the first place.

(Assume gravitational force on the water is constant throughout the glass, don't bother to figure in the slight variation from that)

Of course, assume constant rotational velocity and that the system has had time to stabilize.

Answer 1

It's a parabola. I believe the astronomer Roger Angell proposed a parabolic reflector using mercury in a spinning container. The paraboloid is the natural equilibrium shape of minimum potential energy. The liquid assumes a level consistent with the local acceleration, which is the sum of gravity and radial acceleration given by w^2*r, where w is the rotational rate in rad/sec and r is the radius. The resulting paraboloid surface is perpendicular to the acceleration at all points.