Urbano and Houghton, 2006
From GeoMod
- Article describing the vpython model of the coriolis effect.
- Additional details and references/links discovered since the publication of this paper can be found in the coriolis links section of the model documentation web page.
An Interactive Computer Model for Coriolis Demonstrations
Urbano, L., and Houghton, J., 2006. An Interactive Computer Model for Coriolis Demonstrations, Journal of Geoscience Education, 54, no. 1 pg. 54-60.
- http://www.nagt.org/nagt/jge/abstracts/jan06.html#v54p54 - Link to article on the Journal of Geoscience Education website (free download).
Abstract
The coriolis effect can be a difficult concept for students to understand, particularly in large classes where the effectiveness of physical demonstrations is limited by visibility. We developed a fully interactive computer visualization aimed at introductory undergraduate and precollege students based on the physical demonstration of a marble rolling across a turntable. The marble's velocity, turntable angular velocity and direction, and friction between the marble and the surface can be controlled to allow significant instructional flexibility. Pre and post demonstration student surveys indicate that the application improved student understanding of the coriolis effect, and the use of this type of demonstration was favorably received by the students. This program is written in the free, open-source Python programming language, specifically with the VPython module, which makes three-dimensional, physically-based, real-time visualizations efficiently programmable for geoscience demonstrations by nonprofessional programmers.
Keywords: coriolis, computer visualization, interactive model, meteorology, oceanography
Contents |
Introduction
Wind speed and direction over the surface of the Earth are controlled by the balance among the pressure gradient force, the coriolis effect and friction with the ground. The interaction of these three forces is vital in describing atmospheric motion but can be difficult to convey both individually and in concert. The coriolis effect is the most complex and least intuitive of the three forces because it involves relative motions perceived between two reference frames resulting in a fictitious force. As a result, much effort has been devoted to simplified explanations (e.g., Vandegrift, 1995; Schmidt, 1985), and a number of physical demonstrations have been designed to show this effect in the classroom (such as O'Connell, 2000; Arne, 1982; Levine, 1978).
The most common demonstrations involve moving an object, such as a marble, in
a straight line across a rotating disk. Although the object moves in a relatively straight
line, its path across the turntable transcribes a curve as the turntable spins out from under
it. These demonstrations are, however, difficult to visually project to large lecture classes,
leading to the development of a number of variations to address this problem. Arne
(1982) suggested using a drop of colored water that leaves a trail as it drips vertically
across a tilted rotating disk, while Levine (1978) used a pen to trace straight-line
segments on a piece of paper as it is rotated over a discrete angle, creating a type of stop
motion animation. Even with these modifications the effectiveness of any physical model
is limited by its ability to be seen from the back of an auditorium.
For computer-based lectures, another alternative is to use an animation, a number
of which are freely available on the Internet. Dray (1999) provides an excellent directory
to many of these. These animations, however, tend to be small in size (eg., only filling
one-fourth of the projected screen at 1024 by 768 pixels resolution) and of low resolution
to accommodate the current bandwidth limitations of many personal Internet connections,
limiting their usefulness in lectures. In addition, separate animations have been developed
for each of the three component of atmospheric circulation, so that attempts to
demonstrate the force interactions during a single lecture can be difficult. Most of these
animations can only demonstrate simple scenarios and lack the flexibility and drama of a
model in which fundamental parameters can be manipulated interactively during a live
demonstration. Reddy (2005) has produced an excellent interactive Java Applet for
coriolis demonstrations but it too is restricted in size and to two dimensions. There have
also been some attempts to produce internet-based coriolis games. EOA Scientific
Systems (2005), for example, have produced a game using an interactive Flash program,
but it too suffers from a substantial degree of abstraction by using a fixed Mercator
reference map and a linear progression of challenges.
After attempts to explain the coriolis effect using globes and computer animations
available on the internet were met with a particularly unenthusiastic response during
lecture we sought an alternate method of conveying the concept. To combine the realism
and spontaneity of a physical demonstration with the visual clarity of projected images,
we opted to create a computer visualization that could be controlled interactively from the
podium to demonstrate the coriolis effect and its interactions with the pressure gradient
and friction forces. The program we created (Figure 1 shows a black and white rendering
of the interface) simulates a marble that rolls in a straight line across a rotating turntable
leaving a trail that shows its path across the turntable. The program is fully interactive
and has been used in large introductory geoscience lectures, in laboratory sections by
graduate teaching assistants and by student groups doing extra credit assignments.
Because the interface is intuitive and requires only a few seconds explanation, we expect
this program to be useful in other educational settings and levels as well. The software
was developed in the Python programming language (van Rossum, 2004) and is available
on the Internet (Urbano, 2004). The VPython module (Scherer et al., 2000) of the Python
programming language is a free, open-source tool, useful for developing vector-based
models with three-dimensional visualization.
Visualizations with VPython
Creating this program was possible because the VPython module of the Python programming language allows for easy programming of three-dimensional visualizations. VPython supplies an array of three dimensional primitives such as spheres, cylinders and lines that can be easily transformed in space and time by vector manipulation functions. Moreover, objects on the screen can be programmed to allow direct user interaction using the mouse. Python and VPython are open source, freeware, and of particular importance, come with their own interpreter (IDLE), a dedicated text editor that greatly facilitates programming by highlighting syntax and allowing interactive debugging.
The easy three dimensional programming in Python makes it a much better tool
than the most common Internet-based alternatives: Java (and Java 3D) (Hopkins, 2001)
and VRML, the Virtual Reality Markup Language (Sonstein, 1996). VRML, in particular,
as a more mature language, has a larger (and more reliable) set of three-dimensional
primitives and objects, but it lacks the interpreter and vector-based manipulation
functions that make it easy to design dynamic, interactive programs in Python. The syntax
and structure of Python, a high level language, are also simpler for non-professional
programmers, being similar to FORTRAN and BASIC rather than C, while still being
completely object-oriented.
While Python programs can be transferred easily over the Internet, they do not
function interactively within a web page. Python and VPython must be installed on the
local computer to execute the coriolis program. While these programs are freely available,
unlike Java they are not typically installed on most computers. Although I have provided
direct links on my class website, some students have had difficulty installing Python and
VPython on home computers. The primary impediment has been the need to deactivate
the graphics acceleration on some computers using the MS Windows operating system.
While making this change is relatively simple, the instructions for doing so vary with the
version of Windows so the solution can be difficult to convey. Once the Python programs
are installed, however, any Python program can be used. Since Python programs are
typically saved as text scripts, the programs themselves are extremely small and easy to
transmit; this coriolis program is only 11 kilobytes, for example. Versions of Python and
VPython are available at www.python.org and www.vpython.org respectively for all
common operating systems so the coriolis program is effectively platform independent.
The Model Interface
As the coriolis program was designed for instructional purposes in a variety of settings, it was necessary for it to be easy to use and intuitive. At the core of the program is a simple rotating turntable, crossed by two lines to enable visual tracking of the rotation, across which a marble can be fired. The angular velocity and direction of rotation (clockwise or counterclockwise) is controlled by a single slider located to the right of the turntable. As in the physical demonstration, the center of the turntable represents the north or south pole, and the circumference a line of latitude. Counterclockwise rotation is analogous to a view of the northern hemisphere. While it might have been preferable to show the outlines of continents on the surface of the disk to 6 provide a better analogy, doing so increased the number of objects to be rotated on the disk, which reduced the performance of the program and cluttered the scene.
In the program, the marble is fired from a “cannon,�? which the user can drag, by
its barrel, to any position in the scene. As the marble is fired, a line straight from the
“cannon�? appears, which provides visual proof of the marble's inertial (straight line)
trajectory. The program traces a trail behind the marble with a thicker curved line that
tracks the marble's motion on the rotating turntable to show the apparent deflection of the
coriolis effect. There is currently no control for rotating the “cannon�?, and although this
would be simple to integrate into the model's interface, we have so far found it to be
unnecessary. The marble's velocity can be changed by the user via a slider in the upper
right of the scene, while the degree of friction between the marble and the surface is set
by a slider on the lower right.
When dragged onto the turntable, the sphere from the upper left of the scene
labeled “Target�? rotates with the turntable, as if resting on its surface. This feature is
critical to the explanation of angular velocity versus actual linear velocity. By placing the
target at different distances from the center of the turntable (e.g., close to the pole versus
the equator), students can measure the time it takes to complete a rotation and estimate
the distance it moved. It then becomes obvious that there is a difference in the linear
velocity at different points on the surface of the Earth while the angular velocity at all
points is the same.
The final control is an unmarked button in the upper right of the scene that allows
the user to view the scene from the perspective of the marble as it crosses the turntable
(Figure 2). This view takes advantage of the three-dimensional model space to help
capture student interest. This button is unlabeled because, while useful, it is not essential
to the coriolis demonstrations, and is left there as a reward for a student (or faculty)
exploring the functionality of the program. The interactions controlled in the model result
in complex behavior similar to that of computer games with open simulation
environments, such as SimCity, where the users must often define and explore their own
objectives. The coriolis model was designed to be flexible enough to be used as such a
game. Unmarked and unannounced rewards, called “Easter eggs�? in the computer gaming
community, add interest and increase the playability of games.
The Physics in the Model
The coriolis effect is itself a matter of perspective since it depends on the observation of motion from two reference frames. Motion of an object seen from a nonrotating frame, call this the inertial frame, is dictated by inertia and appears to move in a straight line independent of the rotation of the surface below. However, when viewed from the rotating frame of reference, motion disconnected from that frame appears to be 8 deflected by a force perpendicular to the direction of motion, hence the synonymous terms coriolis force and coriolis acceleration. This apparent effect is intrinsic to any explanation of large-scale fluid motion over the surface of the Earth and was first described by G.G. Coriolis in the early 19th century. The magnitude of the coriolis force (Fc), as perceived from a fixed location on the surface of the Earth, for a moving object of unit mass is
F_c = 2 \omega sin(\theta) v
where \omega is the angular velocity of the rotating Earth (2\pi radians/day), \theta is the latitude, and v is the velocity of the moving object (Oliver and Hidmore, 2002). This formula indicates that the coriolis force is directly proportional to the angular rotation and the velocity of the moving object, and is maximum at the poles (q = 90°) and zero at the equator (q = 0°). A more complete, vector-based derivation of the corolis force is developed for use in undergraduate physics classes can be found in Vandegrift (1995). On a rotating disk, Bush et al. (1976) explicitly derive the equations in Cartesian coordinates for the special case where an object is moving at a constant velocity from the center of the disk.
This coriolis animation is achieved by rapidly showing a series of images in which
small changes have been made in the position of the marble and the disk based on the
simulated time between each frame, which is called the timestep, and the angular
velocity, marble velocity and friction set by the user. In each timestep the marble travels a
discrete distance in the x,y plane of the global Cartesian coordinate system (x,y,z) (the
inertial frame). The disk rotates about the z-axis of the inertial frame (representing the
north or south pole), and is referenced by its own Cartesian coordinate system (the
rotating frame) that is oriented by the normalized vector (xr, yr, zr). The only interaction
between the two coordinate systems is the marble's trail, which is drawn as a line of
points in the rotating frame. As the marble moves, each new position is translated from
the inertial frame to the rotating frame and added to the trail. This is accomplished by
determining the angle (a) between the axes, a = arctan(xr/yr), and rotating the marble's
position through that angle about the axis of the disk. As a result, the marble can be fired
from any position and in any direction, with its path accurately traced.
For each timestep the coriolis model uses the constant value set by the slider as
the velocity of the marble, which is an idealized representation of the marble/turntable
physical demonstration but not of the actual forces that determine atmospheric motion. A
more accurate representation would require that the velocity of the marble be controlled
by the interaction between the pressure gradient, coriolis and friction forces. In such a
model the user would set the pressure gradient force acting on the marble instead of its
velocity. This force acting alone in the absence of friction would continuously accelerate
the marble across the turntable. The pressure gradient force could easily have been used
in the program instead of velocity. However, we made the choice to model the physical
demonstration of a marble on a turntable rather than the actual forces, since depicting all
the forces might be too abstract for an introduction to coriolis.
Simplifying the pressure gradient force for the program leads to its own set of
limitations, primarily in that the frictional force needs to be represented in a way that is
commensurate with the pressure gradient approximation. As a result, friction in the model
does not decelerate the marble, but only allows the marble to be rotated by the turntable.
Thus the marble's velocity is not reduced by friction except where the direction of
rotation is counter to the direction of the marble motion. In fact, if the marble crosses a
region of the turntable that moves with a velocity component parallel to the marble's
direction, the marble can even be accelerated. This behavior can be used to explain
differences between upper and lower atmospheric circulation patterns, but in terms of the
demonstration it may prove an unnecessary complication. This is a difficulty we have
found even with the actual physical demonstration. Fortunately, both in the physical and
computer model demonstrations this behavior only becomes obvious when the marble
velocity is low. With this approximation for friction, the modeled interactions between
velocity and friction result in behavior similar to the real interactions between the
pressure gradient force and friction. Thus, a marble fired with a non-zero velocity and a
non-zero friction in the model will, correctly, show a reduced coriolis deflection (Figure
3). The model is not, however, able to simulate geostrophic winds, which occur when the
pressure gradient force is balanced by the coriolis force and the resultant motion of the
wind is perpendicular to the pressure gradient (parallel to the isobars).
Application in a large lecture class
Based on students' responses to a prototype of the coriolis model we devised five series of simulations to explain: 1) the coriolis effect in the two hemispheres, 2) the effect of friction on the coriolis deflection, 3) the difference in the coriolis effect between the equator and the pole, 4) the dimensions of atmospheric circulation cells, and 5) the importance of the frame of reference to the perception of the coriolis effect. Table 1 summarizes the conditions set for each simulation.
| ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
Table 1. Series of lecture demonstrations used to describe the coriolis effect. The five series show:
| ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
The Coriolis Effect
To show the coriolis effect, we first fire the marble from the cannon's default position at the bottom of the scene without rotation or friction. The marble moves in a straight, vertical line and leaves a straight trail behind it. We next set the turntable rotation to “maximum�? in a counterclockwise direction, simulating a view of the Earth centered on the north pole. Once the rotation of the disk returns the initial straight trail back to the cannon, a second marble is fired. This marble again moves in a straight line, but leaves a trail on the turntable that is curved to the right, as shown in Figure 1. These two simulations alone provided a convincing demonstration of the coriolis effect. However, to reinforce the message we move the cannon to different positions on the turntable and fire the marble repeatedly to leave a number of trails. We then clear the trails from the turntable with the “Reset All�? button, reverse its direction of rotation, and fire the marble again to show leftward coriolis deflection in the southern hemisphere.
Friction
A second set of simulations demonstrates the effect of friction on the coriolis deflection (Table 1, simulation 2), where the marble is fired at maximum velocity onto the rotating table, once without friction and once with full friction (Figure 3). The marble affected by friction no longer follows the inertial trajectory from the cannon and, as a result, the coriolis deflection is greatly reduced. With friction the track of the marble is partially a function of the marble velocity set by the slider, and partially a result of the turntable dragging the marble.
Difference in the coriolis effect between the equator and the pole
In order to better explain the difference in the coriolis effect between the equator and the pole, we then perform a pair of simulations with the marble starting at the center of the rotating turntable (the north pole) with varying marble velocities. We place the “Target�? on the turntable for reference so that all marbles can be fired in the same direction relative to the turntable. The results are shown in Figure 4. The first marble has maximum velocity and no friction (Figure 4a), similar to a previous simulation, while the second has low velocity (20%) and no friction (Figure 4b). These models resulted in curves described by Bush et al. (1976), with the lower velocity marble tracing out a spiral from the pole to the equator throughout a full rotation of the turntable. This low velocity simulation clearly shows the change in the curvature of the spiral trail, indicating that the coriolis deflection decreases as the marble moves equatorward in a way that is not apparent in the high velocity model. This is a visual representation of the fact that the rate of change of linear velocity is greatest near the pole and decreases away from it.
Atmospheric circulation cells
The development of atmospheric circulation cells, such as Hadley cells is a result of the heat imbalance between the pole and the equator, and the coriolis deflection of the resulting atmospheric motion. Coriolis deflection breaks up the single (pole to equator) heat transport system that would develop on a non-rotating globe by creating converging and diverging trade winds and westerlies approximately every 30 degrees of latitude on the Earth. One consequence of these interactions is that faster planetary rotation can result in an increase in shrinkage of the width of individual circulation cells and an increase in the total number of atmospheric circulation cells.
The coriolis model is used to illustrate the difference in sizes of the circulation
cells resulting from fast and slow rotations. First, a marble is fired with a low velocity
(20%), high friction (100%), and maximum rotation the marble transcribes a circular path
on the turntable (Figure 5 – simulation 4a). Next, the rotation rate of the turntable is
reduced (Figure 5 – simulation 4b), and the circle made by the marble expands illustrating
the larger potential circulation cell.
Frames of reference
To reinforce the importance of different frames of reference, we perform a final pair of simulations with zero marble velocity and counterclockwise rotation. The first simulation is with no friction and the second with full friction. In the scenario without friction, the marble is stationary but leaves a circular trail on the rotating turntable, while in the scenario with friction the marble rotates with the turntable and leaves no trail. Thus, we can explain the two frames of reference since we are watching the two simulations from outside the rotating frame of reference (from space looking down on the north pole). We see the lack of motion of the frictionless marble and the rotation of the marble dragged by friction with the turntable. However, if we were standing on the Earth (the rotating turntable) our perspective would be reversed, and we would see the marble with friction standing still and the marble without friction appear to move, even though in both cases the marbles are given zero velocity. A practical example of this is the illusion of motion of stationary objects in the sky (e.g. the sun and stars) because we, in fact, are moving.
Evaluation
To determine the efficacy of the demonstration we performed written (short answer) pre and post- demonstration surveys of students attending lecture in a large lecture hall (150 seats) where the primary presentation media was a single digital projection screen. The pre-survey was administered prior to the in-class explanation of coriolis to determine students' prior knowledge, while the post survey several lectures after the coriolis demonstration. Students were asked for a personal evaluation, specifically “Did you find the visualization of the coriolis effect to be useful in explaining the concept?�? and the following four questions to determine their degree of understanding of the concept:
- Q1: What causes the coriolis effect?
- Q2: Why is the coriolis effect opposite in the northern hemisphere compared to the southern hemisphere?
- Q3: If you are traveling in an airplane due south in the northern hemisphere, will the coriolis effect cause you to miss to the east or the west of your target?
- Q4: How is the coriolis effect different when you are close to the poles comparedto when you are near the equator?
The results, shown in Table 2, indicate that there was substantial improvement in
students' understanding of the coriolis effect between the pre and post tests. The fraction
of students with correct answers ranged from 24% (on Q2) to 65% (on Q1) for the entire
class. Of the subset of class that saw the demonstration, the improvement on Q1 is 88% were able to provide the correct answer.
In response to the question of the usefulness of the demonstration, the 25 survey
responses were unanimously favorable, and their comments fell in line with what we
expected. In particular, 8 comments mentioned the benefits of using a visual
demonstration specifically. As one student noted, the demonstrations were useful because
“visuals always are.�? The three-dimensional effects were favorably received; one student
explicitly stated that “3D animation made it more clear than 1D or 2D,�? and another
indicated that it was “Nice how it showed the effects of the Earth's rotation from the
'driver's seat'.�? However, a third student suggested that “more visual details might help –
more 3D movement?�?
Comments also suggested that the demonstration was very helpful in
understanding the coriolis effect when used as a supplement to the textbook. Comments
included: “before I saw it I had a lot of trouble grasping the concept�?, I “understand it so
much better�?, “it helped when studying at home later�?, and “helpful because it gave a
different point of view. Able to see why the deflection happens by seeing an object go
through a path.�?
The only negative comments (2 out of 25 responses) indicated that the
demonstration may have been performed too quickly.
Conclusions
This interactive computer model allows us to explain the coriolis effect in large lecture sections with a degree of detail not possible previously. By using a computer model we are able to program the three atmospheric forces that govern coriolis motion into the visualization. By creating an interactive model we permit the instructor to design the demonstration and provide the pacing necessary. The three-dimensional rendering of the visualization increases student interest and allows them to more easily immerse themselves in the simulation. The benefits of the demonstration are borne out by the improvement in student performance on survey questions, and student comments.
While we have only tested this program in a large lecture setting, the coriolis program has the potential for broader application in its current form or with slight modification. Correctly applied, the use of a complex system in demonstrations can help nurture critical thinking skills if, for example, students are asked to answer hypothetical questions involving the interaction of the various forces in the model before testing with the model. The simple controls of the program allow it to be used by graduate teaching assistants in laboratory sections with very little training. This has been attempted by one instructor so far, who attests to its effectiveness in this setting. Based on our experience with the program in lecture and laboratory and student interviews, we believe the use of this coriolis visualization has lead to a substantial improvement in our instruction regarding the coriolis effect, and we can suggest a number of insights into the effectiveness of this and similar programs.
First, for use in the classroom the demonstration programs need to be captivating.
Three dimensional rendering of the scene is one step toward this goal, even though a ball
rolling across a disk can easily be compressed into two dimensions. More importantly, the
ability to view the scene from an oblique angle aids in understanding and explaining the
coriolis effect. The ability of the program to show the scene from the perspective of the
marble greatly increased student interest and aids the students' understanding of the
different frames of reference necessary to create the coriolis effect.
Second, using a model with flexible controls that is based on the actual physics of
the processes demonstrated is much more powerful than pre-made animations since it
retains much of the spontaneity of a physical demonstration. Using the model has
improved our own understanding of coriolis much more than we anticipated when writing
the program, and we suspect that it would be similarly useful to students if they were able
to “play�? with it either as a homework assignment or in a laboratory exercise. Interactive
models such as this one are similar in structure to open simulation environment computer
games like the Sims, which are designed to be played non-linearly without specific
predefined objectives. The “driver's seat�? view was originally included in the model
specifically to enhance the “playability�? of the model, although it has proved to be
immensely popular in lecture demonstrations as well.
Third, the flexibility of an interactive model is both an opportunity and a
challenge for the instructor. The model is powerful in that can show a variety of complex
behaviors depending on the conditions set by the user, but representing the physics of a
globe on a disk produces subtle differences in behavior that can be easily missrepresented.
A good example can be seen in Simulation set 4, which show that the size of
atmospheric circulation cells depend on the rotation rate of the Earth. To achieve a
circular circulation cell in the model requires friction to be set to a high value. Friction,
however, is not necessary for the development of general circulation cells in the Earth's
atmosphere. To achieve the full potential of this model, instructors need to possess an
expert level of knowledge of the coriolis processes. Fortunately, they can acquire that
knowledge by experimentation with the model.
On a final note, an unanticipated but revealing experience with the program came
from an 8 year old relative who discovered me working on the program during winter
break. This individual, who is a very experienced user of computer, console and arcade
games, spent 20 consecutive minutes trying to hit the target with the marble. Interestingly,
he was only interested if the view was set to the marble's perspective. The educational
benefit of this is hard to evaluate, but it does suggest that a wide range of age groups may
find the program interesting, thus empowering the program to be a useful geoscience tool
in a broad range of educational settings and levels. The software, as published on the
Urbano (2004) website is available for modification and experimentation.
References
Arny, T. T., 1982. Coriolis force simulation, American Journal of Physics, v. 50, n. 4, p.381.
Bush, R.T., Digiambattista, D.J., and Coats, D.A., 1976. Coriolis deflection on the rotating platform, American Journal of Physics, v. 44, n. 9, p. 883.
Dray, T., 1999. Coriolis effects via "Earth Hockey", http://curvebank.calstatela.edu/earth/earth.htm (1999).
EOA Scientific Systems, 2005. Coriolis Effect, http://www.eoascientific.com/campus/earth/multimedia/coriolis/view_interactive (September 2005).
Hopkins, G., 2001, Java 3D tutorial, http://www.java3d.org/tutorial.doc (2001)
Levine, M. D., 1978. Coriolis Machine, or Demonstration of Inertial Motion from a Rotating Reference Frame, American Journal of Physics, v. 46, n. 7, p. 759.
O'Connell, S. B., 2000. Visualizing the coriolis effect, Journal of Geoscience Education, v. 48, p. 580.
Oliver, J.E. and Hidmore, J.J., 2002. Climatology, 2nd edition, Prentice Hall, p.89.
Reddy, B.S. 2005. General Physics Java Applets. http://www.surendranath.org/Applets.html (2005).
Schmidt, A., 1985. Coriolis acceleration and conservation of angular momentum, American Journal of Physics, v. 54, n. 9, p. 755.
Scherer, D., Dubois, P., and Sherwood, B., 2000, VPython: 3D interactive scientific graphics for students, Computing in Science and Engineering, Sept./Oct. 2000, p. 82-88.
Sonstein, J.,1996, Practical Applications for VRML 2.0, http://www.vrmlsite.com/aug96/spotlight/vrml2/vrml2.htm (1996).
Urbano, L., 2004. 3d demonstration models using VPython, http://des.memphis.edu/lurbano/vpython/VPython_models.html (December, 2004).
van Rossum, G., 2004, Python tutorial (release 2.4), Python Software Foundation, 114 p.
Vandegrift, G., 1995. On the derivation of the Coriolis and other noninertial accelerations, American Journal of Physics, v. 63, n. 7, p. 663.
Acknowledgements
We would like to thank Jennifer Bulmanski for her feedback concerning the use of the coriolis program in laboratory sections, and Pablo Ranlett for testing the program. We would also like to thank Dr. Robert MacKay for insightful comments on the manuscript.
Categories: References | GeoEd | Education | Article
