Back to . . . . Curve Bank Home  NCB Deposit  # 27 Tevian Dray Department of Mathematics Oregon State University Corvallis, OR 97331 tevian@math.orst.edu Coriolis Effects via "Earth Hockey" Coriolis Acceleration: A Term from Physics In this section . . .   An Application           of Vector Calculus
For the student . . . .

 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.
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.

 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. References Harry F. Davis,  Introduction to Vector Analysis,  Allyn and Bacon, Inc.  1961. James Stewart,  Calculus, 5th ed., THOMSON Brooks/Cole, 2003.
 Additional Links For other animations showing that rotations do not necessarily commute, see        < http://www.physics.orst.edu/~tevian/Rotations/RotXY.gif  >        < http://www.physics.orst.edu/~tevian/Rotations/RotYX.gif  >. Windpower and the Coriolis effect  < http://www.windpower.org/en/tour/wres/coriolis.htm >. For a biography of Coriolis, see  < http://www-gap.dcs.st-and.ac.uk/~history/Mathematicians/Coriolis.html  >. For the Vector Calculus Bridge Project, see  <  http://www.physics.orst.edu/bridge>. For the Paradigms in Physics Project  ( NSF DUE Project #9653250 ), see  <  http://www.physics.orst.edu/paradigms>. The Earth Hockey animations were developed as part of the Paradigms Project. For more information, please see David McIntyre's article in  American Journal of Physics, 68,1097-1105 (2000).  Type  "Coriolis" in the "Keyword Quick Search."       McIntyre created these animations using Mathematica®. In addition, we recommend visiting his web site  < http://www.physics.orst.edu/~mcintyre/coriolis >.
 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."