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In everyday understanding, centrifugal force (from Latin centrum "center" and fugere "to flee") represents the effects of inertia that arise in curved motion and are experienced as an outward force away from the center of curvature of the path or away from a center of rotation. Centrifugal force is not restricted to circular motion, however.

This article summarizes several related but distinct concepts representing different ideas 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 the inverse cube law. The outward inverse cube law centrifugal force appears in a second order differential equation in the radial distance, alongside the inward inverse square law of gravity. The solution to this equation is an orbit shaped as a hyperbola, a parabola, an ellipse, or a circle, depending on the initial conditions.

There is evidence that Isaac Newton originally conceived of a similar approach to centrifugal force as Leibniz. However, he seems to have changed his position at some point. In later years, Newton conceived of centrifugal force as being 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 a centripetal force is the reactive centrifugal force, the outward force felt by that object when it is pulling or pushing another object into a curved path. It wasn't until the latter half of the 18th century that the modern understanding of the centrifugal force as "fictitious force" and the artifact of rotating reference frame took shape as seen in the works of Daniel Bernoulli, who showed that the centrifugal force has to be defined with reference to an arbitrary point, and Joseph Louis Lagrange, who explicitly stated that the centrifugal force depends on the rotation of a system of perpendicular axis, among others.

In an alternate common modern conception, centrifugal force in a rotating reference frame is a fictitious force that appears in equations on motion in rotating frames of reference, to explain effects of inertia as seen in such frames.

Reactive vs. fictitious force

The table below compares various facets of the "reactive force" and "fictitious force" views 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. This approach to centrifugal force appeared in high school textbooks up until the 1960's. Nelkon & Parker's Advanced Level Physics is one example of a textbook which used this approach until the 1960s and then dropped it. In the 1961 edition of this textbook, 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.

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. Linton, Christopher. From Exodus to Einstein. Cambridge: University Press, 2004, p. 285. ISBN 0521827507
2. Swetz, Frank et al. Learn from the Masters! Mathematical Association of America, 1997, p. 269. ISBN 0883857030
3. 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)
4. 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)
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. R. G. Takwale and P. S. Puranik (1980). Introduction to classical mechanics. Tata McGraw-Hill. p. 248. ISBN 9780070966178.
7. Mark Zachary Jacobson (1980). Fundamentals of atmospheric modeling. Cambridge University Press. p. 80. ISBN 9780521637176.
8. Guido Rizzi and Matteo Luca Ruggiero (2004). Relativity in rotating frames. Springer. p. 272. ISBN 9781402018053.
9. Wolfgang Rindler (2006). Relativity. Oxford University Press. p. 7–8. ISBN 9780198567318.
10. Julian B. Barbour and Herbert Pfister (1995). Mach's Principle. Birkhäuser. p. 6–8. ISBN 9780817638238.

• Fictitious forces
• Force
• Mechanics
• Rotation