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1 The Strange Tale of Centrifugal Force History of Science Society Annual Meeting Austin, Texas November 2004 William M. Shields Science and Technology Studies Virginia Tech highc.king@verizon.net

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2 What is Centrifugal Force? In classical mechanics, CF is the force experienced by a body in an accelerated reference frame characterized by a central attractive force. Magnitude proportional to the square of the tangential velocity and inversely proportional to the radius of the orbit, direction is radial away from the center of the orbit. F c = mv t 2 /r

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3 Frames of Reference In the inertial frame of reference used by Newton, only centripetal force acts to curve the path of the rotating object, which by Newton’s First Law will move in a straight line unless acted upon by an external force. CF only appears when one considers matters from the accelerated, non-inertial frame of reference.

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4 Inertial Frame

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5 Accelerated Frame

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6 Huygens De Vi Centrifuga, written 1859, published after his death. Viewed circular motion as a balance of forces, attracting towards the center (e.g. gravity) and “fleeing” from the center, or centrifugal. Arrived at correct formula for the magnitude, F c = mv t 2 /r.

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7 Newton In the Principia, Newton abandoned “balance of forces” idea in favor of the principle of inertia (the First Law of Motion) coupled with an inward-directed force such as gravity to explain circular motion. He coined the term “centripetal force” to drive home the distinction between his approach and that of Huygens in dealing with circular motion.

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8 Newton Westfall points out that even Newton could not seem to abandon CF entirely. In illustrating his views on orbital motion, Newton posits a small moon orbiting the earth at the height of “the highest mountains.” In this case, “if it were deprived of all motion with which it proceeds in its orbit, [it would] descend to the earth as a result of the absence of the centrifugal force with which it had remained in its orbit.” (Principia Book 3, Prop. 4, Scholium)

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9 Fictitious…Pseudo…Real Many physics textbooks take a hard line that CF is “fictitious” and insist that even mentioning it in class will confuse everyone. Others look more carefully at the reference frame question and give CF equal treatment with centripetal or central force. Engineering texts present CF as real and use it in exemplars and calculations without apology.

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10 A.P. French, Newtonian Mechanics, (1971) Clear and careful discussion of reference frames, how CF is “fictitious” and how it can be regarded as a legitimate “dynamical principle.” Full section entitled “Centrifugal Force,” another on “Centrifuges.” Introduces in Einstein’s Principle of Equivalence.

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12 Halliday and Resnik, Fundamentals of Physics, (1970) “Centrifugal force” is not mentioned anywhere; there is no Index entry!

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13 Goldstein, Classical Mechanics, (1970) “... the centrifugal force on a particle arising from the earth’s revolution around the sun is appreciable compared to gravity, but it is almost exactly balanced by the gravitational attraction to the sun. It is, of course, just this balance between centrifugal force and gravitational attraction that keeps the earth... in orbit around the sun.” (emphasis added)

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14 Landau and Lifshitz, Mechanics (1976) Employ the Lagrangian approach to calculate the equation of motion in a rotating frame of reference. The resulting equation of motion contains two additional terms not present in an inertial frame. One term represents Coriolis force, the other term “m Ω x (r x Ω) is called the centrifugal force.” [Ω = angular velocity vector.] Calculating the energy of the system, L&L arrive at E = ½ mv 2 + U + ½ m(Ω x r) 2, where the last term “is called the centrifugal potential energy.”

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15 Feynman, Lectures on Physics (1963) CF is a “pseudo force” arising “due to the fact that the observer does not have Newton’s coordinate frame.” “One very important feature of pseudo forces is that they are always proportional to the masses; the same is true of gravity.” Pseudo forces may be connected to gravity by general relativistic ideas.

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16 Twigg, Science for Motor Vehicle Engineers

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17 British Journal Physics Education Search on “centrifugal force” brings up many articles and letters from 1973 to the present. Typical titles: “Centrifugal Force: Fact or Fiction?” (Vol. 24, 1989), Most recent article is by Oxford instructor John Roche, “Introducing Motion in a Circle,” September 2001 issue.

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18 “There is a valid concept of centrifugal force in physics.” “This is the centrifugal force of physics, an entirely fictional force.” “I fully agree that [in teaching physics] the fictional centrifugal force should not even be mentioned.” “However, if the bulk of the class intends to go into engineering surely it would help them to have the engineering concept of centrifugal force explained clearly in their physics class.”

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19 Einstein Theory of general relativity (GR) is based on the Principle of Equivalence: accelerated reference frames and gravitational fields are physically indistinguishable. He gives the example of an observer on a spinning disk sensing “centrifugal force.” Einstein asserts that this observer may regard the disk as stationary and the force as gravitation.

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20 Einstein In Appendix III to The Meaning of Relativity (1920 edition), Einstein calculates the red shift of spectral lines in a gravitational field by returning to the spinning disk model. He carries out the approximation using the “potential of the centrifugal force between the position of the [observer] and the center of the disk.”

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21 Einstein The observer’s clock will then run slower than a clock at the center by an approximate amount: ν = ν o (1 + φ/c 2 ) where φ is the potential of the centrifugal force at the observer’s location, ν o is the rate of a clock at the center of the disk, and ν is the rate of the observer’s clock. Because φ is negative, ν is always < ν o.

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22 Thirring Hans Thirring in two seminal papers (1918 and 1921) showed that a rotating shell of matter induces within it forces analogous to CF and Coriolis force. Thirring thus suggested that CF arises because of the rotation of the mass of fixed stars with respect to the earth, i.e., CF is a component of the local space-time metric. In his 1956 book on relativity, Pauli calls favorable attention to the Thirring’s papers.

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23 Recent GR Work Pfister and Braun, “Induction of Correct Centrifugal Force in a Rotating Mass Shell,” Classical and Quantum Gravitation, 2: 909-918 (1985). They carry out a new calculation of CF and Coriolis forces inside a rotating mass shell. Claim that their solution to Einstein’s field equations coincides in appropriate cases to Thirring’s result and is consistent with “Mach’s idea concerning the relativity of rotation.”

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24 Iyer and Prasanna (I&P), “Centrifugal Force in Kerr Geometry,” Classical and Quantum Gravitation, 10: L13-L16 (1993). They calculate the expression for CF “at the equatorial circumference of a rotating body in the locally non-rotating frame of the Kerr geometry.” In such a metric the sign of centrifugal force reverses!

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25 Questions Is CF “fictitious”? If the answer is yes, what exactly is a fictitious force? How would “force” have to be defined to exclude CF? Can we say that “force” can be defined only where there is an identified physical agent such as a magnet? But if we so define force... what about the Principle of Equivalence and Thirring?

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26 The next time your coffee spills onto your slacks as your car rounds a corner too fast, remember, the stain might be fictitious!

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27 Classical Mechanics Barbour, Julian, The Discovery of Dynamics, Oxford University Press, 2001. Cohen, I. Bernard, The Newtonian Revolution, Cambridge University Press, 1980; The Birth of a New Physics, Norton, 1985. Den Hartog, J.P., Mechanics, McGraw-Hill, 1948, Dover Reprint, 1961. Dugas, René, A History of Mechanics, du Griffon 1955, Dover Reprint 1988. Feynman, R. Leighton, R., Sands, M., Lectures on Physics, Vol. 1, Addison-Wesley, 1963. French, A.P., Newtonian Mechanics, Norton, 1971. Goldstein, Herbert, The Science of Mechanics, Addison-Wesley, 1950, 2nd edition 1980. Halliday David and Resnick, Robert, Fundamentals of Physics, Wiley, 1970. Herivel, J.W. “Newton’s Discovery of the Law of Centrifugal Force,” Isis 51: 546-553 (Dec. 1960). Jammer, Max, Concepts of Mass, Harvard University Press, 1961, Dover Reprint 1997; Concepts of Space, Harvard University Press, 1954, Dover Reprint 1993; Concepts of Force, Harvard University Press, 1957, Dover Reprint 1999. Landau, L.D. and Lifshitz, E.M., Mechanics, Pergamon, 1976. Maxwell, J. Clerk, Matter and Motion, Thoemmes Press 1876, Dover Reprint 1991. Meli, D., “The Relativization of Centrifugal Force,” Isis 81: 23-43 Mar. 1990). Newton, Isaac, The Principia, translated by I. Cohen and A. Whitman, University of California Press, 1999. Tait, Peter and Steele, William, A Treatise on the Dynamics of Particles, MacMillan 1856, Elibron Reprint 2003. Westfall, Richard, Force in Newton’s Physics, Elsevier, 1971.

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28 Teaching Alcaraz, A. and Ramirez, P., “Common on the ‘Role of the Centrifugal Force in Vehicle Roll,’” American Journal of Physics 70: 5-556-557. Roche, J., “Introducing Motion in a Circle,” Physics Education 36: 399-405. Savage, M.D. and Williams, J.S., “Centrifugal Force: fact or Fiction?” Physics Ed. 24: 133-140 (1989). Stinner, A.,“Linking ‘The Book of Nature’ and ‘The Book of Science’: Using Circular Motion as an Exemplar Beyond the Textbook,” Science & Education 10: 323-344 (2001). Taylor, K., “Weight and Centrifugal Force,” Physics Ed. 9: 357-360 (1974). Twigg, Peter, Science for Motor Vehicle Engineers, Butterworth- Heineman, 1996. Ziauddin, S., “Rotational Motion and Centrifugal Force,” Physics Education 8: 77-78 (1973).

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29 General Relativity Eddington, Arthur, Space, Time and Gravitation, Cambridge University Press, 1920, reprint 1987. Einstein, Albert, Relativity, The Special and General Theory, Barnes & Noble, 1920, reprint 2004. Iyer, S. and Prasanna, A.R., “Centrifugal Force in Kerr Geometry,” Classical and Quantum Gravitation 10: L13-L16 (1993). Mach, Ernst, The Science of Mechanics, Open Court, 1893, reprint 1989. Mashoon, B., Hehl, F.W., Theiss, D.S., “On the Gravitational Effects of Rotating Masses: the Thirring-Lense Papers,” General Relativity and Gravitation 16: 711-750 (1984). Pauli, Wolfgang, The Theory of Relativity, Pergamon, 1958, Dover Reprint 1981. Pfister, H and Braun, K.H., “Induction of Correct Centrifugal Force in a Rotating Mass Shell,” Classical and Quantum Gravitation 2: 909- 918 (1985). Weyl, Hermann, Space, Time, Matter, Springer, 1923, Dover Reprint, 1952.

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30 Sidebar: Lense-Thirring Effect Frame Dragging and Gravity Probe B In 1918, Thirring joined with Josef Lense to write another paper on general relativity, “On the Influence of the Proper Rotation of Central Bodies on the Motions of the Planets and Moons According to Einstein’s Theory of Gravitation.” The ideas in this paper became known as “dragging of the inertial frames” or “frame- dragging.” Lense and Thirring conclude that this effect was too small to be measured.

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31 Gravity Probe B

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32 Gravity Probe B uses superconducting quantum interference devices (SQUIDs) to measure tiny changes in the orientations of four perfectly-spherical, quartz gyroscopes as the experiment orbits the Earth. The gyroscopes are housed inside a vacuum chamber and will be maintained at 1.8 Kelvin using liquid helium.

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33 The probe also includes a telescope that will be trained on a distant "guide star" to provide a reference direction for measurements on the gyroscopes. General relativity predicts that the frame-dragging effect will cause the direction of the gyroscopes to change by a tiny 0.041 of an arc second. http://einstein.stanford.edu/

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