Consider an ellipsoid described the equation in cartesian coordinates $(x,y,z)$ $$ \frac{x^2}{a^2} + \frac{y^2}{a^2} + \frac{z^2}{c^2} = 1, \quad \text{where} \quad a < c.$$ This ellipsoid is usually called the prolate spheroid.
Now we introduce rotation around the $z$ axis,the angular velocity $\omega$ at the point $x = a,y = a, z= 0$ is $\omega = v/a$. Can we find the formula for angular velocity at any point on the ellipsoid?
What my attempt was: I thought that the angular velocity should be dependent only a the distance from the axis $z$, which I denoted $r = \sqrt{x^2 + y^2}$. From this i would get $$\omega = \frac{v}{r} = \frac{v}{\sqrt{x^2 +y^2}},$$ where of course $x$, $y$ are points on the ellipsoid.
But this doesn't seem quite right with my intuition, since the angular velocity as a function of $r$ will be the same in a sphere of radius $a$. Did I make a mistake or is my intuition wrong?
