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In everyday understanding, the term centrifugal force (from Latin centrum "center" and fugere "to flee") applies to the effects of inertia that arise in connection with rotation and which are experienced as an outward force away from the center of rotation. Centrifugal force is not however restricted to circular motion. In modern science, the name is given is to two distinct, but equally valid concepts relating to rotation.

This article summarizes the several ideas surrounding the concept of centrifugal force.

History of conceptions of centrifugal and centripetal forces

Gottfried Leibniz conceived of centrifugal force as a real outward force which is induced by the circulation of the body upon which the force acts. Leibniz showed that the centrifugal force obeys an inverse cube law. The inverse cube law centrifugal force appears in an equation representing an orbit shaped as a hyperbola, parabola, ellipse or circle, depending on the initial conditions.

There is evidence that Isaac Newton originally conceived of a similar approach to centrifugal force as Leibniz, though he seems to have changed his position at some point – in later years, Newton conceived of centrifugal force as an equal and opposite reaction to centripetal force. According to Newton's third law of "action and reaction", when a centripetal force acts on an object, pushing it into a curved path, the reaction force upon the object supplying the centripetal force is the reactive centrifugal force, i.e. the outward force felt by that object when it is pulling or pushing the other object into a curved path.

It wasn't until the latter half of the 18th century that the modern "fictitious force" understanding of the centrifugal force as an artifact of rotating reference frames took shape. In a 1746 memoir by Daniel Bernoulli, the "idea that the centrifugal force is fictitious emerges unmistakably." Bernoulli, in seeking to describe the motion of an object relative to an arbitrary point, showed that the magnitude of the centrifugal force depended on which arbitrary point was chosen. In other words, the centrifugal force depended on the reference frame of the observer, as opposed to other forces which depended only on the properties of the objects involved in the problem and were independent of the frame. Also in the second half of the 18th century, Joseph Louis Lagrange in his Mécanique Analytique explicitly stated that the centrifugal force depends on the rotation of a system of perpendicular axes.In 1835, Gaspard-Gustave Coriolis analyzed arbitrary motion in rotating systems, specifically in relation to waterwheels. He coined the phrase "compound centrifugal force", because of its similar form and nature, for what later would become known as the Coriolis force.

The common modern conception considers centrifugal force in a rotating reference frame a fictitious force that appears in equations on motion in rotating frames of reference, to explain effects of inertia as seen in such frames. In modern science based on Newtonian mechanics, Leibniz's centrifugal force is a subset of this conception and is a result of his viewing the motion of a planet from the standpoint of a special reference frame co-rotating with the planet.

Reactive vs. fictitious force

The table below compares various facets of the "reactive force" and "fictitious force" concepts of centrifugal force.

The values of the reactive centrifugal force and the fictitious centrifugal force are not in general equal, but can be equal in special cases such as circular motion and a frame of reference co-rotating with the moving object, or for arbitrary smooth paths and a reference frame instantaneously co-rotating about the center of the instantaneous osculating circle.

Reactive centrifugal force

The concept of reactive centrifugal force originated with Isaac Newton in the 17th century. From his third law of motion, Newton concluded that the centripetal force which acts on an object must be balanced by an equal and opposite centrifugal force. In the modern understanding of physics the reactive centrifugal force and the centripetal force do not balance since they do not act on the same body. While the concept of the reactive centrifugal force is not given much attention in modern physics textbooks, it is of interest to engineering texts that deal with internal stresses in rotating solid bodies. For example, in a simple rotating turbine the section of a blade near the shaft exerts an inward (centripetal) force on the outer section of the blade. In accordance with Newton's third law, the outer section also exerts an equal and opposite outward (centrifugal) force on the inner section. This produces an internal stress in the turbine blade.

In some cases, this concept is confused with the rotating reference frame conception. For example, Nelkon & Parker's 1961 edition of Advanced Level Physics, centrifugal force is introduced and explained according to Isaac Newton's action-reaction approach. In the same section, the centrifuge machine is explained using centrifugal force as a real force. However, in the 1971 revision of the same textbook, the centrifugal force section has disappeared and the centrifuge machine is explained using some kind of compound negative centripetal force. This type of confusion still on occasion occurs in modern textbooks.

Fictitious force in a rotating reference frame

From the viewpoint of an observer in a rotating reference frame, centrifugal force is an apparent, or fictitious, or inertial, or non-inertial, or pseudo force that seems to push a body away from the axis of rotation of the frame and is a consequence of the body's mass and the frame's angular rate of rotation. It is zero when the rate of rotation of the reference frame is zero, independent of the motions of objects in the frame.

If objects are moving in a rotating frame, they also experience a Coriolis force, another "fictitious" force; and if the rate of rotation of the frame is changing, objects also experience an Euler force, yet another "fictitious" force. Together, these three fictitious forces allow for the creation of correct equations of motion in complex moving reference frames.

Other topics

The concept of centrifugal force in its more technical aspects introduces several additional topics:

• Reference frames, which compare observations by observers in different states of motion. Among the many possible reference frames the inertial frame of reference are singled out as the frames where physical laws take their simplest form. In this context, physical forces are divided into two groups: real forces that originate in real sources, like electrical force originates in charges, and

• Fictitious forces that do not so originate, but originate instead in the motion of the observer. Naturally, forces that originate in the motion of the observer vary with the motion of the observer, and in particular vanish for some observers, namely those in inertial frames of reference.

Centrifugal force has played a key role in debates over relative versus absolute rotation. These historic arguments are found in the articles:

• Bucket argument: The historic example proposing that explanations of the observed curvature of the surface of water in a rotating bucket are different for different observers, allowing identification of the relative rotation of the observer. In particular, rotating observers must invoke centrifugal force as part of their explanation, while stationary observers do not.

• Rotating spheres: The historic example proposing that the explanation of the the tension in a rope joining two spheres rotating about their center of gravity are different for different observers, allowing identification of the relative rotation of the observer. In particular, rotating observers must invoke centrifugal force as part of their explanation of the tension, while stationary observers do not.

References

1. ^ Roche, John (2001). "Introducing motion in a circle" (PDF). Physics Education. 43 (5): pp. 399–405. ISSN 0031-9120. Retrieved May 7, 2009. {{cite journal}}: |pages= has extra text (help); Unknown parameter |month= ignored (help)
2. Linton, Christopher. From Exodus to Einstein. Cambridge: University Press, 2004, p. 285. ISBN 0521827507
3. Swetz, Frank et al. Learn from the Masters! Mathematical Association of America, 1997, p. 269. ISBN 0883857030
4. Delo E. Mook & Thomas Vargish (1987). Inside relativity. Princeton NJ: Princeton University Press. p. p. 47. ISBN 0691025207. {{cite book}}: |page= has extra text (help)
5. Wilson, Curtis (1994). "Newton's Orbit Problem: A Historian's Response" (PDF). The College Mathematics Journal. 25 (3): pp. 193&ndash200. ISSN 0746-8342. Retrieved May 8, 2009. {{cite journal}}: |pages= has extra text (help); Unknown parameter |month= ignored (help)
6. ^ Meli, Domenico Bertoloni (1990). "The Relativization of Centrifugal Force". Isis. 81 (1): pp. 23&ndash43. ISSN 0021-1753. Retrieved May 8, 2009. {{cite journal}}: |pages= has extra text (help); Unknown parameter |month= ignored (help)
7. Persson, Anders (1998). "How Do We Understand the Coriolis Force?" (PDF). Bulletin of the American Meteorological Society. 79 (7): pp. 1373–1385. ISSN 0003-0007. Retrieved May 9, 2009. {{cite journal}}: |pages= has extra text (help); Unknown parameter |month= ignored (help)
8. Slate, Frederick (1918). The Fundamental Equations of Dynamics and its Main Coordinate Systems Vectorially Treated and Illustrated from Rigid Dynamics. Berkeley, CA: University of California Press. p. 137. ASIN B000ML76V8. Retrieved May 9, 2009.
9. Steinmetz, Charles Proteus (2005). Four Lectures on Relativity and Space. Kessinger Publishing. p. 49. ISBN 1417925302. Retrieved May 9, 2009.
10. Aiton, E.J. (1962). "The celestial mechanics of Leibniz in the light of Newtonian criticism". Annals of Science. 18 (1). Taylor & Francis: pp. 31-41. doi:10.1080/00033796200202682. {{cite journal}}: |pages= has extra text (help); Unknown parameter |day= ignored (help); Unknown parameter |month= ignored (help)
11. R. G. Takwale and P. S. Puranik (1980). Introduction to classical mechanics. Tata McGraw-Hill. p. 248. ISBN 9780070966178.
12. Mark Zachary Jacobson (1980). Fundamentals of atmospheric modeling. Cambridge University Press. p. 80. ISBN 9780521637176.
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• Fictitious forces
• Force
• Mechanics
• Rotation