# The Coriolis Fountain

The Coriolis Fountain is an experiment that can be used to demonstrate the effect of a special form of inertial force called the Coriolis force. It is also sometimes referred to as an illusory force.

This Coriolis force acts on any body whose motion is observed from a rotating reference frame. It was first described mathematically in 1835 by the French mathematician Gaspard Gustave de Coriolis. The Coriolis fountain in MATHEMATICS ADVENTURE LAND was given its name because it is possible to visualize the Coriolis effect through rotations by means of suitably generated water jets. When the Coriolis fountain is rotated, the following phenomena are observed: the water jets coming out of the nozzles located in the center (in the direction of the rim) seem to “bend” against the direction of the rotational movement. And the water jets emerging from the nozzles located at the edge seem to “bend” in the direction of the rotary motion. Both effects increase with greater angular velocity of the disk, i.e. the faster it is rotated by an experimenter. If the water jets of the nozzles from the center meet those from the edge, at this moment the jets are illuminated with blue light. This phenomenon of “bending” water jets can be explained by means of the following mental experiment: Imagine that the nozzles do not emit a continuous water jet, but eject small balls (or individual water droplets) in “rapid” succession. The speed of the spheres is that of the water jet. The paths of the spheres lined up mentally can be understood as discretizations of the water jets coming out of the nozzles. It is therefore sufficient to analyze the path of a single sphere.

#### And now … the mathematics of it:

Assume that a circular disk rotates with the constant (vectorial) angular velocity , which — as a vector — is perpendicular to the plane of rotation. Now, if a sphere is ejected from the center of the rotating disk with constant (vectorial) velocity , then using a system in cylindrical coordinates and the associated unit vectors , we obtain (see Figure 2 below)

for the Coriolis force the following formula:

Here denotes the mass of the sphere and for the determination of the direction of the so called cross or vector product of two vectors and the three finger rule applies on the right hand (cf. figure 3 below):

We will derive this in the following: So where does the above equation for the Coriolis force come from? Let us assume that we, as observers, stand in the center of the counterclockwise rotating disk and follow the sphere just ejected from there in radial direction. Without rotation (and thus Coriolis force) it would simply move straight out (i.e. it moves in time by the distance along the radius). Now, however, after the sphere has been ejected, the disk continues to rotate “to the left” under it (and does not influence it any more). For the rotating observer, this gives the impression that the sphere is striving “to the right” (and in time by the distance ; cf. figure 2 above). This corresponds exactly to a uniformly accelerated motion to the right, with the constant Coriolis acceleration . Therefore, it appears to the observer that a force acts on the said sphere according to the above formula.