If I have a spinning bowl of water...
A spinning liquid of any sort forms a paraboloid; a three dimensional parabolic shape. Therefore, you can apply the physics principles relating to centripetal motion to any mass, m, on the surface of the liquid. Therefore, you may arrive at the conclusion that the position of the surface of the water, f(x), is equal to (2*Π^2*x^2)/(T^2*g) + c; where g = 9.81 m/s^2, T = the period of the spinning liquid, and c = a negative number in meters which is designated to the distance between the vertex of the resultant paraboloid and the initial surface of the water, which will be a function of T; x is measured in meters.
How does one go about finding this value, c?
Here is what I have so far, but I am not sure how close I am to finding it:
There are two points on the x axis which f(x) = 0, where the initial surface level of the liquid is also zero. The water is also displaced, which means the area between the curve and the x axis (indefinite integral) on the positive side of the positive x intercept and the negative area between the curve and the x axis between the vertex and the positive x intercept are equal to each other, no matter what T is.
This is my science fair project for 2018, and I'd like a little bump in the right direction, so I wish not for a complete solution and explanation, simply some ideas. Please help!