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The Coriolis Force

by Vincent Mallette
Copyright © 1999 Inwit Publishing, Inc.

Strange to say, the Coriolis force gets into some very deep waters, some of which even have a philosophical or paradoxical content. The paradoxical aspect comes about from the fact that the Coriolis force is one of three important classes of forces in physics that are called "fictitious." Yet people experiencing these forces cannot tell that they are fictitious in any sense.

Indeed, the cornerstone of Einstein's general relativity theory is the equivalence principle, which tells us that "fictitious" forces cannot be distinguished in any way whatsoever from real forces by any "interior" experiment. For example, you are in an enclosed elevator in outer space that is accelerating upwards at 9.8 meters per second per second. Inside the elevator you feel what seems to be the familiar tug of gravity. You step on a bathroom scale and you weigh exactly what you weigh on earth. For all you know you are on earth. Einstein asserted that it is impossible by any experiment performed inside that elevator, to determine whether the force you feel is due to acceleration, or to a gravitational attraction. This is the equivalence principle.

Why, then, were the "fictitious" forces ever called fictitious? It is a technical term, somewhat similar to the way certain perfectly good numbers are called "imaginary" in mathematics. (Indeed, some books call fictitious forces "noninertial-frame" or "pseudo" forces.) The point is, fictitious forces are discussed separately from "real" forces because they are not rooted in matter. The four "real" forces we know in physics are all rooted, we believe, in interactions between matter particles. Gravity is an attraction between all matter particles; the electromagnetic force is mediated by photons; and the strong and weak nuclear forces are mediated by even more obscure particles (one of which is called the "gluon"). But all these four "real" forces are dependent on matter interactions. The fictitious forces are not. When you're accelerating upwards in your sealed elevator, the force you feel holding you to the floor has nothing whatsoever to do with matter.1 There needn't be any matter in the universe at all, except for you and your elevator, for you to feel that force. It is not a "matter" force; it is a fictitious force. However, you only know that from outside the elevator.

There are two other important classes of fictitious, or noninertial-frame, forces. One of these is the apparent force popularly called centrifugal. You round a corner at high speed in a car and are thrown against the door. You whirl a bucket full of water over your head and don't spill a drop. You're squashed, and the water is held in place, by the fictitious centrifugal force. But consider: if the door of the car were open, you wouldn't feel any force — you'd just go flying into space. You feel a force from the door, because your body is being restrained from going straight, which it would like to do (Newton's first law). That force you feel is not any kind of attraction; the door has not suddenly developed a "super-gravity" of its own. The apparent force is a consequence of your motion in an accelerating reference frame. This brings us at last to the Coriolis force.2

Like the elevator force, and the car-rounding-a-bend force, the Coriolis force is due to an accelerating reference frame. That accelerating reference frame is the familiar 24-hour rotation of the earth.3 (Circular motion like this amounts to a continuous, constant acceleration.) Moving bodies on earth behave as though they are being acted upon by a sidewise force — the Coriolis force.4

I'm going to give you several examples of the Coriolis force, but first I would like you to consider the following: Imagine that a large artillery shell is fired exactly south,5 say from Atlanta to Florida.6 The earth is turning from west to east: counterclockwise as viewed from above the north pole. The speed of the surface of the earth7 depends on your latitude; it is zero at the poles, and about 1000 miles an hour at the equator. The shell leaves Atlanta with the west-to-east speed due to the earth's rotation at that latitude. When it reaches Florida the scrub pines of Florida are moving faster beneath it than the scrub pines of Georgia when it left Atlanta. Let's say I fired the gun from the Fernbank Science Center in Atlanta at a longitude of 84 degrees, 19 minutes west of London. But when the shell reaches Florida the 84 degrees 19' meridian has sped away to the east, because of the earth's greater surface speed at that latitude. So the shell will be west of 84 degrees 19'; that is, it will have been "deflected" to the right as though a force (the Coriolis force) had acted on it. This is a general rule for projectiles fired in the northern hemisphere; they bend off to the right of the plane of firing. The same is true for so-called ballistic missiles, those that are not guided after they leave the launch pad.

Here is a real-life mishap caused by a failure to correctly allow for the Coriolis force:

"During the naval engagement near the Falkland Islands which occurred early in World War I, the British gunners were surprised to see their accurately aimed salvos falling 100 yards to the left of the German ships. The designers of the sighting mechanisms were well aware of the Coriolis deflection and had carefully taken this into account, but they apparently were under the impression that all sea battles took place near 50 degrees N latitude and never near 50 degrees S latitude. The British shots, therefore, fell at a distance from the targets equal to twice the Coriolis deflection." (Classical Dynamics of Particles and Systems, Second Edition — by Jerry B. Marion, Academic Press, Inc., 1970, p.346 fn.)

Quoting now from a book by Desloge:8

"Rivers and railroad trains experience a slight force to their right in the Northern Hemisphere and to their left in the Southern Hemisphere. Whether there are any noticeable effects of this force however is questionable."

Actually it has been shown that rivers in the northern hemisphere "scour their right banks more severely than the left," in particular the Mississippi and the Yukon.9 As to the putative effects on trains and cars I have nothing definite to report at this time. The Coriolis force on a 2-ton racing car going due north at 350 mph at a latitude of 45 degrees N is about 3.5 pounds.10 There is talk about the flywheels of race cars wearing differently when they are run on northern and southern hemisphere tracks, but I have been unable to verify this.

Back to Desloge:11

"If a simple pendulum, consisting of a heavy bob suspended vertically by a very long string from an essentially frictionless pivot, is set in motion in a vertical plane, the plane of motion is observed to rotate gradually about a vertical axis. The rate of rotation of the plane of motion will depend on the angular velocity of the earth and the latitude at which the pendulum is located. Such a pendulum is called a Foucault pendulum. The rotation of the plane of motion is due to the Coriolis force."

There is nothing speculative about this. Foucault pendulums hang in many science museums. I spent half an hour watching the Foucault pendulum at the Smithsonian, and saw its motion change during that time (it knocks over wooden cones to show the change). The Coriolis force due to the earth's rotation continuously nudges the pendulum to swing in a new plane. In 1851 Leon Foucault (1819-1868) strung a "very large iron ball" on a 200-foot steel wire from the center of the great dome of the Pantheon in Paris. Fine sand was sprinkled on the floor to mark the ball's motion. With great care the iron ball was drawn to one side and secured with a thread. A flame was applied to the thread; it burned through and launched the ball. With bated breath the audience waited. In one hour the plane of vibration of the ball had turned just over 11 degrees — exactly the calculated amount. For the first time in the history of the world the earth's rotation had been proved.12 (Contrary to popular belief, neither Copernicus nor Galileo demonstrated the rotation of the earth13 — Foucault, with the aid of the Coriolis effect, was the first. The year after his big pendulum demonstration, Foucault invented the gyroscope which provides "an even more striking visual proof of the earth's rotation."14 After Foucault's one-two punch, there was really no way to deny the rotation of the earth.)

"Vincenzo Viviani..., a pupil of Galileo, had noticed as early as about 1650 that a pendulum undergoes a slow rotation, but there is no evidence that he correctly interpreted the phenomenon."15

Isaac Newton predicted the eastward deflection of a falling body in 1679,16 and hoped to prove the rotation of the earth by observing the effect, from a tower for example.17 Robert Hooke (1635-1703) actually gave it a try,18 but this is not an easy experiment to do in practice. A marble dropped from a tower 328 feet (100 meters) tall at latitude 45 degrees will be deflected a little over half an inch.19 This is useless, because turbulence in the flight path, not to mention wind, will obscure this tiny deviation. Besides, there were few buildings in Hooke's and Newton's time that were even 328 feet tall — the Lighthouse of Alexandria may have been 400 feet high, but it was destroyed by earthquake in 1375 A.D.20 True, the inside height of the famous dome of St. Peter's is 393 feet,21 but I shudder to think what would have happened to Hooke and Newton if they had tried dropping cannonballs from the dome of St. Peter's. A fellow named Reich dropped pellets down a 188-meter mine shaft in 1831; he claimed deflections of 28 mm. on average.22 This is less than the theoretical amount, not surprising in view of air effects.

We come now to the largest-scale effect of the Coriolis force — its influence on the planet's air and water. I will quote Desloge again:23

"The Coriolis force plays a large role in the great mass movements of the oceans and the atmosphere. In particular it plays a significant part in the formation of cyclones and the trade winds."

Far and away the best manifestation of the atmospheric effect of the Coriolis force — and one which confronts us almost daily on TV news reports in the fall — is the famous rotation of hurricanes. In the northern hemisphere the rotation is counterclockwise as viewed from above, and clockwise in the southern hemisphere. But wait — didn't we just say that a rocket in the northern hemisphere pulls to the right? If you go right...and right...and right, you are going clockwise, not counterclockwise. What's wrong here. The matter is so confusing that even one advanced physics text has it wrong, affirming that hurricanes and cyclonic storms in the N. H. go clockwise. Yet the time lapse pictures on TV clearly show the N. H. storms cranking to the left. Are we all watching our TV's in mirrors, as you sometimes have to do in sports bars? No, the explanation is rather subtle. Imagine a mass of high-pressure air moving toward a large low-pressure area. Now imagine another mass of h-p air 90 compass degrees from the first one, moving toward that same low. Now box the compass and imagine two more hunks of high pressure coming toward that same low patch. So we have this low-pressure area being invaded by high-pressure masses from north, south, east, and west. Each of these h-p masses skews rightward as it dives into the big low. The multiple rightward pushes drive the low into counterclockwise rotation. Think about it — if you punch me on the chin with a rightward-directed blow, my head will crank counterclockwise (if you ever meet me, don't bother demonstrating). You can find these concepts laid out in more detail in standard physics texts.24

This brings us to a subject that fascinates the public — does your toothpaste go down the drain ccw in the northern hemisphere? Does the toilet swirl cw in the southern hemisphere? It may — and it may not. In any case it has little to do with Coriolis. The earth's Coriolis force is far too feeble to act on such small, slow systems.25 Even phenomena as big as whirlpools and tornadoes don't give Coriolis much respect. It's true, scientists have constructed special ponds, about as big as wading pools, in which the water does go down the drain, Coriolis-correct. It's been done in both the northern and southern hemispheres, and the water follows its marching orders. But incredible precautions have to be taken. The pools must be constructed with a neutral geometry around the drain, and the water must remain undisturbed for hours, lest the water will go down the tubes however it went down the last time as though it had a memory (it's really what they call residual angular momentum). This brings me to a fictional radio drama that I heard years ago. A famous industrialist had been kidnapped and taken to an undisclosed location. The kidnappers did, however, let the poor fellow brush his teeth, and he noted that the dregs went down the drain clockwise. Aha, says he — I'm in the southern hemisphere. He got this message to his company, and that was enough of a clue for him to be rescued. (I said it was fiction). The main point, though, is that hardly any string-and-sealing-wax experiment performed in a room of reasonable size could tell him what hemisphere he was in. In principle, you could set up a Foucault pendulum if the ceiling were high enough and if you had a cannonball, a steel wire, and a precision astatic bearing. Then over a period of several hours, you could make measurements that would establish your hemisphere and even your latitude to within a few degrees. The same thing could be accomplished if you had a precision gyrocompass — you should never leave home without one. (The best scheme for a kidnappee would be to get a glimpse of the sky — the Pole Star, or an area near the Southern Cross. That would give you a rough latitude, and would certainly determine your hemisphere).

A familiar physics demonstration actually lets us feel a Coriolis force. This is the demonstration in which we draw in weights toward our body while spinning on a piano stool. "The reaction of the weights against our hands in the direction to increase our angular velocity is the Coriolis force."26 Of course, this is the Coriolis force due to the spinning of the stool, not the spinning of the earth — you will never feel the latter, unless you let yourself be bonked by a Foucault pendulum!

Let's return now to what we started with: fictitious forces. You saw how a change in reference frames could make an apparent force, the centrifugal force, disappear. It turns out that even the "real" forces of nature (gravity, electromagnetic, nuclear) can be made to disappear through a similar maneuver: the Abelian and non-Abelian forces of the universe will disappear if an appropriate transformation, called the gauge transformation, is applied.27 So here we sit, anchored and held together by forces that don't exist! At least they wouldn't exist if we found the right transformation. But like opening the door of the speeding car, perhaps it's something we shouldn't do until we're fully prepared!


* "Any level bubble, which is being carried on a ship or plane, will be deflected [by Coriolis] from its normal position. The deflection will be perpendicular to the direction of motion of the ship or plane. The correction for this effect may amount to several miles in the determination of a position of the ship or plane by methods of celestial navigation if the bubble octant is used in the necessary observations." Van Nostrand's Scientific Encyclopedia, Fifth Edition — ed. by Douglas M. Considine, Van Nostrand Reinhold Company, 1976, article "Coriolis Effect," p. 685

* Example of Coriolis effect on ocean currents: "...the Gulf Stream veers toward the east as it moves northward [because] the earth is turning, and the eastward velocity of the earth at the equator (about 1000 mi/hr) is greater than the eastward velocity (about 800 mi/hr) at the latitude of the target, Labrador.... Any water leaving the West Indies [i.e., the equator] has a greater eastward component of velocity than does the target, so the water 'gets ahead of' the target, and strikes the British Isles rather than Labrador." College Physics — by Franklin Miller, Jr. (New York: Harcourt, Brace & World, Inc., 1959), p. 62

* An approximation to the Coriolis force for cases in which the rotating frame turns through an angle of no more than 3 degrees during the time the "projectile" moves from the point of origination to the "target":

FCoriolis = 2mv(2pf)

where v is the "muzzle velocity" of the "gun" and 2pf is the angular velocity of the rotating frame. Modern College Physics, Sixth Edition — by Harvey E. White (New York: Van Nostrand Reinhold Company, 1972), pp. 711-712

* What would happen if a body were shot out of a gun, directly upward, at the equator? Would it fall back into the barrel? Be deflected eastward? Westward? (Of course, we ignore air resistance and any influence that makes the situation less than ideal.) I'll leave this as a problem for the reader. The answer is found in Physics: Basic Principles — by Solomon Gartenhaus (New York: Holt, Rinehart and Winston, Inc., 1975), vol. 1, p. 306.

* Nine out of 10 physics books derive the Coriolis force from an application of Newton's laws to a rotating reference frame. However, it is also possible to derive Coriolis from the principle of conservation of angular momentum. This is done, for example, in Physics: Basic Principles — by Solomon Gartenhaus (New York: Holt, Rinehart and Winston, Inc., 1975), vol. 1, pp. 305-306. The same book also shows how Coriolis deflections can be viewed as an attempt by a particle to pursue a (very truncated) elliptical orbit around the earth (pp. 306-307).

* A caution about fictitious forces: "They can be useful in describing motion in a rotating system, but they are often incorrectly used to describe rotating particles in inertial systems." Theoretical Mechanics — by Eugene J. Saletan and Alan H. Cromer (New York: John Wiley & Sons, Inc., 1971), p. 51

* (For advanced students) In addition to the eastward deflection discussed in Footnotes 15-21, there is also a slight southward deflection of a body falling from a tower (in the northern hemisphere). Supposedly this component has even been experimentally detected [Classical Dynamics of Particles & Systems, Third Edition — by Jerry B. Marion and Stephen T. Thornton (Fort Worth: Harcourt Brace Jovanovich College Publishers, 1988), p. 347]. In any case, the southerly component comes about from second-order expansions in the Coriolis derivation.

1 Ernst Mach (1838-1916) disagreed with this. He felt that inertia was dependent on the total amount of matter the universe contains. His view is called "Mach's mechanics." Alfred North Whitehead was skeptical. He said, “I refuse to believe that a little star in its twinkling turned round the Foucault pendulum at the Paris Exposition....”

Dr. Marion has beautifully described the whole fictitious-force situation in these words, “[The centrifugal and Coriolis forces] have been introduced in an artificial manner as a result of our arbitrary requirement that we be able to write an equation which resembles Newton’s equation and which at the same time is valid in a noninertial reference frame....But the ‘requirement’ [that the net force on the body vanish] is an artificial one; it arises solely from an attempt to extend the form of Newton’s equation to a noninertial system, and this can be done only by introducing a fictitious ‘correction force.’ The same comments apply for the Coriolis force; this ‘force’ arises when an attempt is made to describe motion relative to the rotating body” (Classical Dynamics of Particles and Systems, Second Edition — by Jerry B. Marion, Academic Press, Inc., 1970, p. 345).

2 So, to summarize: the fictitious forces are (1) Newton’s F=ma force (simple acceleration), (2) the centrifugal force, and (3) the Coriolis force.

Gustave-Gaspard de Coriolis, May 21, 1792 - Sept. 19, 1843. Reportedly Coriolis was led to his theory of accelerations on rotating frames from his study of water wheels. In any case the famous work giving the correct formula for the Coriolis force was published in 1835: Memoire sur les equations du mouvement relatif des systemes de corps. Coriolis is also known in physics for coining the term kinetic energy (1829).

3 The angular velocity of the earth's rotation is about 7.29 x 10^-5 radians per second.

4 The greater the motion of the body with respect to earth, the greater the Coriolis force. The Coriolis force is the only fictitious force which is dependent on the motion of the body with respect to the non-inertial frame. (Classical Mechanics, Volume 1 — by Edward A. Desloge, John Wiley & Sons, 1982, p. 138 ).

5 The direction "south" is described by any arc of a meridian of longitude with sense of direction from the north pole to the south pole. Which pole is south? That pole from above which the rotation of the earth appears clockwise. What is clockwise? Don't ask. Suffice it to say that if technological man had arisen in the southern hemisphere, clocks would go counterclockwise.

6 This is not as ridiculous as it sounds; in 1918 Paris was bombarded by shells from a cannon 76 miles away (The Guinness Book of Records 1993 — ed. by Peter Matthews et al, Bantam Books, 1993, p. 505), and larger guns could be built if necessary.

7 With respect to the inertial frame of the "far stars." And see Footnote 13 below.

8 Classical Mechanics, Volume 1 — by Edward A. Desloge, John Wiley & Sons, 1982, p. 138

9 Van Nostrand's Scientific Encyclopedia, Fifth Edition — ed. by Douglas M. Considine, Van Nostrand Reinhold Company, 1976, article "Coriolis Effect," p. 684

10 Analytical Mechanics, Second Edition — by Grant Fowles, Holt, Rinehart and Winston, Inc., 1970, pp. 137 and 349).

11 pp. 138-139

12 My account is adapted from Van Nostrand's Scientific Encyclopedia, Fifth Edition — ed. by Douglas M. Considine, Van Nostrand Reinhold Company, 1976, article "Foucault Pendulum," pp. 1092-1093. The rate of deviation is equal to 15 degrees per sidereal hour times the sine of the latitude. The sidereal period of rotation of the earth is 86,163.4 sec. Hence a Foucault pendulum at the north pole would go round — not in 24 hours — but in 23 hrs 56 min 3.4 sec. Here at Ga. Tech (latitude 33° 46’ 36”) it takes about 43 hours, 10 min. for a Foucault pendulum to go round (8.33936° per sidereal hour).

13 James Bradley, who succeeded Halley as Astronomer Royal, had proved the earth's yearly revolution about the sun in 1728 (not using the Coriolis effect; aberration of starlight was used). Galileo's observations of the phases of the planet Venus (1610) had already very strongly supported the view that the earth revolves. As an aside, it’s worth noting that the Cosmic Background Explorer satellite (COBE) proved the earth’s motion around the sun once and for all, by detecting the earth’s annual variation in the frame of the cosmic microwave background (CMB) — “the ultimate vindication of Copernicus” in the words of Bennett, Turner, and White (“The Cosmic Rosetta Stone,” Physics Today, Nov. 1997, p. 32). The rest frame defined by the CMB is about as close as you can come to the “absolute space” of Newton, better even than the frame of the “far stars.”

14 Classical Dynamics of Particles & Systems, Third Edition — by Jerry B. Marion and Stephen T. Thornton (Fort Worth: Harcourt Brace Jovanovich College Publishers, 1988), p. 353 fn.

15 Classical Dynamics of Particles and Systems, Second Edition — by Jerry B. Marion, Academic Press, Inc., 1970, p.356 fn.

16 Classical Dynamics of Particles & Systems, Third Edition — by Jerry B. Marion and Stephen T. Thornton (Fort Worth: Harcourt Brace Jovanovich College Publishers, 1988), p. 347. Why, then, is Newton not credited with the discovery of the Coriolis effect? Because he did not announce his deflection prediction as part of a complete theory of particle motion in accelerated reference frames. I’m sure he could have worked out the complete theory if it had appealed to him.

17 Encyclopedia of Physics, Second Edition — ed. by R. Lerner and G. Trigg, VCH Publishers, Inc., 1991, article "Coriolis Acceleration" by Robert H. March, p. 191

18 Classical Dynamics of Particles & Systems, Third Edition — by Jerry B. Marion and Stephen T. Thornton (Fort Worth: Harcourt Brace Jovanovich College Publishers, 1988), p. 347

19 Classical Dynamics of Particles and Systems, Second Edition — by Jerry B. Marion, Academic Press, Inc., 1970, p.350. Had the experiment been performed at the equator, the deflection would have been about 7/8 of an inch. Had the tower been 1000 meters high (at 25 degrees latitude), the deflection would have been about 60 centimeters [Introductory Mechanics — by Edwin F. Taylor (New York: John Wiley & Sons, Inc., 1963), p. 273].

20 The Guinness Book of Records 1993 — Peter Matthews et al, Bantam Books, 1993, p. 268. Actually, estimates of the height of the Lighthouse (one of the Seven Wonders of the ancient world) vary from 200 to 600 feet. By the way, the Leaning Tower of Pisa, from which Galileo did not drop anything, is a pitiful 179 feet (The 1990 Information Please Almanac — ed. by Otto Johnson et al, Houghton Mifflin Company, 1990, p. 607)

21 Ibid., p. 510

22 Classical Dynamics of Particles & Systems, Third Edition — by Jerry B. Marion and Stephen T. Thornton (Fort Worth: Harcourt Brace Jovanovich College Publishers, 1988), p. 347

23 Classical Mechanics, Volume 1 — by Edward A. Desloge, John Wiley & Sons, 1982, p. 138

24 For example, Classical Dynamics of Particles and Systems, Second Edition — by Jerry B. Marion, Academic Press, Inc., 1970, p. 347

25 Surprisingly, the physicist Edwin Taylor, who should have known better, countenances the "washbasin" effect: Introductory Mechanics — by Edwin F. Taylor (New York: John Wiley & Sons, Inc., 1963), p. 277

26 Concepts in Physics — by Robert K. Adair (New York: Academic Press, 1969), p. 155

27 Dr. David Finkelstein, professor of physics at Ga. Tech, personal communication, April 24, 1996. Dr. Finkelstein has recently been appointed physics tutor to the Dalai Lama.

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