Final answer:
The equilibrium surface formed by spinning a bucket of water is a parabola, resulting from the combination of gravitational and centrifugal forces, where faster spinning results in a broader parabolic shape.
Step-by-step explanation:
When a bucket of water is spun about its vertical axis with an angular velocity ω, the water will form a parabolic surface in equilibrium due to the balance of gravitational and centrifugal forces. This equilibrium surface is an equipotential, meaning that the potential energy due to gravity and the centrifugal potential energy are constant across the surface.
To show that the surface is a parabola, we consider the potential energy at any point in the rotating frame: it includes both the gravitational potential energy (mgh) and the centrifugal potential energy (½mω²r², where r is the distance from the axis of rotation). Setting the sum of these potential energies equal to a constant and rearranging, we can derive the equation of the surface which turns out to be a parabola of the form h = ½ω²/g * r², where h is the height of the water above the bottom of the bucket at a distance r from the axis.
Thus, the velocity of rotation helps determine the shape of the water surface, with faster spinning (ω) resulting in a broader parabolic shape.