It comes from the chapter 5 execise of Mathematical Analysis 1 by Vladimir Zorich.
Below is the problem:
A glass of water is rotating about its axis at constant angular velociry $\omega$.
Let $y=f(x)$ denote the equation of the curve obtained by cutting the surface of the liquid with a plane passing through its axis of rotation.
a) Show that $f'(x)=\frac{w^2}gx$, where $g$ is the acceleration of free fall.
b) Choose a function $f(x)$ that satisfies the condition given in part a).
c) Does the condition on the function $f(x)$ given in part a) change if its axis of rotation does not coincide with the axis of the glass?
These are my question about this problem.
First, I'm not sure what "cutting the surface of the liquid with a plane passing through its axis of rotation." means.
I think that looks like one of these two pictures.
Which is the right "figure" which describes the situation of this probblem? and I can't imagine how the glass look likes, does glass means that rectangular shape one?
Second, I couldn't set up the equation. I guess it has to be related with $sin, cos$ functions and $\omega$ will be included in the argument of $sin, cos$.
My guess is $<x(t),y(t)>=<r(t)sin(\omega*t)$,$r(t)cos(\omega*t)$>.
where $r(t)=$ (the distance of the point and the center of circle). So I guess (A) picture is the problem situation.
But I can't give you why it should look like this. and this equation is based on the assumption that a glass looks like cylinder. Could you give me a reason or if it's false, what is the right equation?