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# 27
Tevian Dray
Department of Mathematics
Oregon State University
Corvallis, OR 97331
tevian@math.orst.edu
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Coriolis Effects via "Earth Hockey"
Coriolis Acceleration: A Term from Physics
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In this section . . .
An Application
of
Vector Calculus
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For the student . . . .
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Dray has also provided animations
of a map of the Earth using vaious rotations about a fixed point or fixed
line. Please click on the map below. This animation shows
how to generate any rotation by using two successive 180 degree rotations
about different axes.
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Background . . . .
Briefly,
Coriolis acceleration is the apparent acceleration one sees when observing
particle motion from a rotating, rather an an inertial, reference frame.
For instance, the needle of a phonograph moves (nearly) in a straight line,
yet traces a spiral on the rotating record.
Mathematically, this is equivalent to considering the opposite
situation, such as an object moving along a radial line from the center of
a rotating disk. You may visualize this phenomenon by placing yourself
at the center of a merry-go-round and then walking toward the edge. [ Ignore
the horses or any other obstruction. ] This situation is analyzed in
the box below.
For surface geometry, study the globe on the left.
The black line gives the great circle path a frictionless hockey puck would
follow if the Earth were not rotating. The blue line gives the
great circle it actually follows. The red line shows the apparent path
as seen from the Earth which rotates underneath the blue line - much as a
phonograph rotates underneath the needle. Now click on the globe
to the left to select a variety of animations.
For the mathematical analysis we must use vector calculus.
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According to Harry F. Davis, writing in 1961, the Coriolis acceleration is "more complicated and is usually
not discussed in elementary physics textbooks." He suggests a careful
examination of the derivation will show the term applies "partly to the change
in direction of the radial component of velocity, and partly to the
fact that, as the radius changes, the transverse component of velocity changes,
even if the angular velocity is constant."
Professor Davis did not have the advantage of computer
animation. Tevian Dray has used the surface of the earth to illustrate
this phenomenon. Moreover, modern calculus texts with a chapter on
"Vector Functions" covering planetary motion may have problems on this subject.
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References
Harry F. Davis, Introduction to Vector Analysis,
Allyn and Bacon, Inc. 1961.
James Stewart, Calculus, 5th ed., THOMSON Brooks/Cole,
2003.
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Tevian Dray presented the
Vector Calculus Bridge Project at MathFest 2003. His minicourse emphasized
vectors and vector calculus.
In the program he wrote, "The key
to bridging the gap between mathematics and the physical sciences is geometric
reasoning."
This approach was pioneered by the
French mathematician Gaspard Gustave de Coriolis (1792 - 1843).
He showed that the laws of motion
could be used in a rotating system if an extra force, now called the Coriolis
acceleration, is added to the equations of motion.
In 1835 Coriolis wrote on a mathematical theory of
billiards. Similarly, Dray has chosen to call his investigations "earth
hockey."
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