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.

Figure 1: Snapshot during clockwise rotation

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 \vec{\omega}, 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 \vec{v}, then using a system in cylindrical coordinates (r,\varphi,z) and the associated unit vectors \vec{e}<em>r,\vec{e}</em>\varphi,\vec{e}_z, we obtain (see Figure 2 below)

Figure 2: Coriolis effect modeled mathematically

for the Coriolis force the following formula:

    \[\vec{F}<em>{\mathrm{C}}=2m(\vec{v}\times\vec{\omega})=2m\lvert\vec{v}\rvert\lvert\vec{\omega}\rvert\vec{e}</em>\varphi.\]

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

Figure 3: The three finger rule

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 t by the distance r=\lvert\vec{v}\rvert t 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 t by the distance s=r\lvert\vec{\omega}\rvert t=\lvert\vec{v}\rvert\lvert\vec{\omega}\rvert t^2; cf. figure 2 above). This corresponds exactly to a uniformly accelerated motion to the right, with the constant Coriolis acceleration \vec{a}<em>{\mathrm{C}}=\vec{F}</em>{\mathrm{C}}/m=2\vec{v}\times\vec{\omega}. Therefore, it appears to the observer that a force \vec{F}_{\mathrm{C}} acts on the said sphere according to the above formula.


Supplemental comments:

  1. All movements on the earth are exposed to the Coriolis force (along the equator minimally!), because the earth turns around its own axis and thus represents a rotating reference system. However, the Coriolis force (due to the Earth’s relatively low approximate angular velocity of 360°/24h) only plays a role where large-scale motions occur. For example, rivers in the northern hemisphere undermine more the right bank, in the southern hemisphere more the left bank (in the direction of flow).
  2. The Coriolis force is “responsible” for the fact that air masses around large-scale high-pressure areas in the northern hemisphere of the earth move clockwise, but low-pressure areas move counterclockwise. Thus, in a low-pressure area, the air moves inward because of the pressure gradient. This flow is deflected to the right in the northern hemisphere by the Coriolis force. This results in a counterclockwise rotational motion.
  3. The Coriolis force plays an important role in the Foucault pendulum (observable as an exhibit in MATHEMATICS ADVENTURE LAND). The oscillation plane of the pendulum only appears to move. In fact, it maintains its direction while one “circles” the pendulum on the globe.
  4. In rail traffic, the Coriolis force in the northern hemisphere causes that in straight lines the rail which is on the right in the direction of travel is slightly more loaded than the left rail. A train (e.g. an ICE 3 with a mass of 400t) traveling at a geographical latitude of 51° north (e.g. Cologne) at a speed of 250km/h experiences a force of 3,200N (Newton) to the right. This corresponds to approx. 0.1% of the weight force. If the train has e.g. eight cars with four axles each, each right wheel is pressed against the rail with a Coriolis force of approx. 100N (Newton) to the right. In comparison, at this speed, a curve radius of 3,000m results in a lateral force of 20,000N on each wheel, i.e. 200 times the value of the Coriolis force.

Literature

[1] Körner, W.: Physik — Fundament der Technik, 10. Auflage, Leipzig, 1989.

[2] Lindner, H.: Lehrbuch der Physik, 12. Auflage, Leipzig, 1989.

[3] Stommel, H.M. und Moore, D.W.: An introduction to the Coriolis force, New York, 1989.