A vertical-axis, rotational paraboloid-shaped tall container has its focus 0.05 meters above the axis point. We pour a little water into the container. At what angular velocity $\omega$ should we rotate the container around its axis so that the water flows out at the top of the container?
The surface of the fluid is a surface of rotation, so it is sufficient to examine a single cross-section passing through the axis. It is useful to visualize the motion in a coordinate system fixed to the container, rotating with angular velocity $\omega$, where the fluid is at rest. The surface of the fluid is perpendicular to the resultant force acting on a small patch of the surface.

$1987-03-129-1$. eps

Two forces act on the selected fluid element: the gravitational force and the centrifugal force. According to the diagram, the slope of the surface at a distance $x_{0}$ from the axis is: $\operatorname{tg} \alpha=\frac{\omega^{2} \cdot x_{0}}{g}$.

We know that the functions whose derivative with respect to $x$ is $\frac{\omega^{2} \cdot x}{g}$ can be given by the formula $y=\frac{\omega^{2}}{2 g} x^{2}+C$. These functions define a parabola, whose equation is $y=\frac{\omega^{2}}{2 g} \cdot x^{2}+C$. (The origin is fixed to the bottom of the container, $C=0$ can be taken if the amount of fluid is sufficiently small.)

The fluid will flow out of the container, regardless of its height, when its surface is parallel to the wall of the container's paraboloid, where the equation of the cross-section is $y=\frac{1}{4 \cdot 0.05 \mathrm{~m}} x^{2}$.

The condition for parallelism is $\frac{\omega^{2}}{2 g} \cdot x^{2}=\frac{1}{0.2 \mathrm{~m}} \cdot x^{2}$, from which the required minimum angular velocity is

$$
\omega=\sqrt{\frac{2 g}{0.2 \mathrm{~m}}}=9.9 \mathrm{~s}^{-1}
$$

Remarks. 1. Some solvers cleverly used the small amount of water in the solution. Some referred to the entire amount of water as a "point mass," thus turning the problem into one of equilibrium for a ball placed in the container. Another solver had the small amount of water condensing into a ring in a horizontal plane.

2. Some did not see that the water can flow out regardless of the height of the container, and thus believed that the outflow of water could be prevented with a sufficiently tall container.