Newton's Bucket
This applet shows a 2D cross-section of a 3D bucket of water, when the water is rotating with angular velocity omega. The height H and radius R of the bucket are adjustable. The parameter gamma (gam) sets the initial level of the water in the bucket. See the attached PDF for the math details of this analysis.
The vectors shown represent the gravitational force downward, the centrifugal force outward, and the resultant of these, acting on a small mass of water at the surface. The negative of that resultant is normal to the parabolic shape of the water surface. See the references in the PDF for more info about this aspect of the analysis. The vector location can be adjusted, along the parabola, and the lengths of the vectors can be scaled to a convenient size for display.
As the angular velocity is changed, note how the water surface always passes through the same two points (actually points on a circle, in 3D). This intersection occurs at the height of the initial level of the water.
Also note that, if the intial water level "gam" is less than half the bucket height, the angular velocity is restricted so that the water does not expose the bottom of the bucket (the model fails if this happens). The angular velocity omega is also restricted so that water cannot fly out of the top of the bucket. See the PDF for details on these restrictions.
