The Marvelous Rotating Fish Tank: Deriving the Paraboloid
My dad used to say that water “seeks its own level.”
While the marvelous rotating fish tank appears to be an exception to that rule, I think that the old saying is actually a great place to start this derivation. The first thing to realize is that when we say level we don’t necessarily mean flat. After all, we’d all agree that (aside from a few waves and the odd tsunami) the oceans are level — and yet they curve all the way around the earth.
What we mean when we say that water is level is that it is perpendicular to direction of local gravity. When we think about it that way, then the water in the fish tank is level since each part of the water’s surface is perpendicular to its local gravity.
Albert Einstein’s brilliant Equivalence Principle helps us understand this. According to Einstein, we can “assume the complete physical equivalence of a gravitational field and a corresponding acceleration of the reference system.” (Einstein 1907). This means that we can treat the centripetal acceleration inward as equivalent to a gravitational force outward.
When the aquarium is rotating, the water experiences an “effective gravitational field” that is equal to the vector sum of the earth’s gravity (equal to g and pointing downward) and virtual gravity produced by rotation (equal to ω^2 *radius and pointing outward).
For the water to be level, its surface must be perpendicular to that local gravitational field. That means the slope of the water at any point must be the angular velocity squared times the radius divided by g as shown in the figure below.
Then to find the height of the surface at a point R simply integrate the slope from zero to R.
This is the equation for any cross section of the surface of the water in the rotating tank. Even the section we see on the glass walls of the aquarium follow this equation, although the distance from a point on the wall to the axis of rotation is not the same as the distance as measured from the center line of the glass wall.
Using this Activity for AP Physics Practice
The Pivot Interactives activity for this video is an excellent opportunity for AP Physics students to practice graphing, and for determining the meaning of the slope of a linearized data set. The activity has four parts:
Derive the shape of the paraboloid using Newton’s Second Law.
Collect data for the height vs radius, h(r), for the surface of the paraboloid.
Compare the slope of the linearized data to the equation from the derivation.
Explore a second derivation based on Einstein’s Equivalence Principle (not on the AP exam).
The activity has sample data and copious notes for the instructor.
Swimming in a Rotating Pool?
Viewing the water in the tank, one might wonder: what would it be like to swim in a rotating swimming pool? To find out, we put a few dozen goldfish in the tank and recorded how they swim while the tank is rotating. As expected, the fish experience the effect of local gravity. Just as they normally do, fish near the surface swim with their bodies parallel to the surface. As they swim, the changing direction of local gravity causes the fish to swim in a curved path.
Making this Video
While recording this phenomenon, we wanted students to be able to see the cross section that passes through the axis of rotation. This is the simplest cross section to analyze because the distance from the axis of rotation is easiest to measure. In order to make this cross section visible, we experimented with many different methods. First, we tried simply adding coloring to the water. This helped, but the under surface was still not clearly visible. We needed a thin, sharp layer coating the surface floating on the water. We tried glitter (only floats on the surface due to surface tension, then sinks), vegetable oil dyed with pigment (adheres to the glass, and does not form a clean edge where it contacts the water), tiny styrofoam beads (also clings to glass, and does not form a consistent smooth layer) sawdust (mucky mess). After days of experimentation, Greg Schmidt, a chemistry teacher walked past our studio-lab and asked what we were doing. After hearing all our failures, he said, “Why don’t you try hexane with some iodine dissolved in it?” About an hour later, we had our shot. Hats off to chemists!
When recording the motion of the fish in the rotating tank, we attached a 2-meter long cross bar to the rotating platform with the camera on one end, and a counterweight on the other end. A scaffold over top of the tank allowed us to light the tank from directly above. The scene of the scaffold, the cross bar, and the tank filled with fish whizzing around was captivating to passers-by. We were glad when the session was successfully concluded without incident.
This is another example of how we at Pivot Interactives provide opportunities for students to analyze phenomena they may otherwise be unable to access. This classroom-ready activity is sure to engage students, and prompt in-depth discussion.