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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 connection with rotation and which are experienced as an outward force away from the center of rotation. In Newtonian mechanics the term centrifugal force is used to refer to one of two distinct concepts: an inertial force (also called a "fictitious" force) observed in a non-inertial reference frame or a reaction force corresponding to a centripetal force. The term is also used in Lagrangian mechanics to describe certain terms in the generalized force that depend on the choice of generalized coordinates.

Fictitious centrifugal force

Nowadays, centrifugal force is most commonly introduced as a force associated with describing motion in a non-inertial reference frame, and referred to as a fictitious or inertial force (a description that must be understood as a technical usage of these words that means only that the force is not present in a stationary or inertial frame). There are three contexts in which the concept of the fictitious force arises when describing motion using Newtonian mechanics. In the first context, the motion is described relative to a rotating reference frame about a fixed axis at the origin of the coordinate system. For observations made in the rotating frame, all objects appear to be under the influence of a radially outward force that is proportional to the distance from the axis of rotation and to the rate of rotation of the frame. The second context is similar, and describes the motion using an accelerated local reference frame attached to a moving body, for example, the frame of passengers in a car as it rounds a corner. In this case, rotation is again involved, this time about the center of curvature of the path of the moving body. In both these contexts, the centrifugal force is zero when the rate of rotation of the reference frame is zero, independent of the motions of objects in the frame.

The third context is related to the use of generalized coordinates as is done in the Lagrangian formulation of mechanics, discussed below. Here the term "centrifugal force" is an abbreviated substitute for "generalized centrifugal force", which in general has little connection with the Newtonian concept of centrifugal force.

If objects are seen as moving from a rotating frame, this movement results in another fictitious force, the Coriolis force; and if the rate of rotation of the frame is changing, a third fictitious force, the Euler force is experienced. Together, these three fictitious forces allow for the creation of correct equations of motion in a rotating reference frame.

Reactive centrifugal force

A reactive centrifugal force is the reaction force to a centripetal force. A mass undergoing curved motion, such as circular motion, constantly accelerates toward the axis of rotation. This centripetal acceleration is provided by a centripetal force, which is exerted on the mass by some other object. In accordance with Newton's Third Law of Motion, the mass exerts an equal and opposite force on the object. This is the "real" or "reactive" centrifugal force: it is directed away from the center of rotation, and is exerted by the rotating mass on the object that originates the centripetal acceleration.

The concept of the reactive centrifugal force is used often in mechanical engineering sources that deal with internal stresses in rotating solid bodies. Newton's reactive centrifugal force still appears in some sources, and often is referred to as the centrifugal force rather than as the reactive centrifugal force.

Fictitious vs. reactive force

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

Lagrangian formulation of centrifugal force

Lagrangian mechanics formulates mechanics in terms of generalized coordinates {q_k}, which can be as simple as the usual polar coordinates (r, θ) or a much more extensive list of variables. Within this formulation the motion is described in terms of generalized forces, using in place of Newton's laws the Euler-Lagrange equations. Among the generalized forces, those involving the square of the time derivatives {(dq_k/dt)} are called centrifugal forces.

The Lagrangian approach to polar coordinates that treats (r, θ ) as generalized coordinates, ${\displaystyle ({\dot {r}},\ {\dot {\theta }})}$ as generalized velocities and ${\displaystyle ({\ddot {r}},\ {\ddot {\theta }})}$ as generalized accelerations, is outlined in another article, and found in many sources. For the particular case of single-body motion found using the generalized coordinates (r, θ ) in a central force, the Euler-Lagrange equations are the same equations found using Newton's second law in a co-rotating frame. For example, the radial equation is:

${\displaystyle \mu {\ddot {r}}=\mu r{\dot {\theta }}^{2}-{\frac {dU}{dr}}\ ,}$

where U(r) is the central force potential. The left side is a "generalized force" and the first term on the right is the "generalized centrifugal force". However, the left side is not comparable to a Newtonian force, as it does not contain the complete radial acceleration, and likewise, therefore, the terms on the right-hand side are "generalized forces" and cannot be interpreted as Newtonian forces.

The Lagrangian centrifugal force is derived without explicit use of a rotating frame of reference, but in the case of motion in a central potential the result is the same as the fictitious centrifugal force derived in a co-rotating frame. The Lagrangian use of "centrifugal force" in other, more general cases, however, has only a limited connection to the Newtonian definition. Although the two formulations of mechanics must lead to the same equations given the same choice of variables, the connection between them may be obscure, and the same terminology employs different meanings. In particular, "generalized forces" (often referred to without the adjective "generalized") in most cases are not Newtonian forces, and do not transform as vectors. Unlike the fictitious centrifugal force, the Lagrangian centrifugal force may be non-zero even in an inertial frame of reference.

Centrifugal force and absolute rotation

Can absolute rotation be detected? In other words, can one decide whether an observed object is rotating or if it is you, the observer that is rotating? Newton suggested two experiments to resolve this problem. One is the effect of centrifugal force upon the shape of the surface of water rotating in a bucket. The second is the effect of centrifugal force upon the tension in a string joining two spheres rotating about their center of mass. A related third suggestion was that rotation of a sphere (such as a planet) could be detected from its shape (or "figure"), which is formed as a balance between containment by gravitational attraction and dispersal by centrifugal force.

Rotating bucket

Newton suggested the shape of the surface of the water indicates the presence or absence of absolute rotation relative to the fixed stars: rotating water has a curved surface, still water has a flat surface. Because rotating water has a concave surface, if the surface you see is concave, and the water does not seem to you to be rotating, then you are rotating with the water.

Centrifugal force is needed to explain the concavity of the water in a co-rotating frame of reference (one that rotates with the water) because the water appears stationary in this frame, and so should have a flat surface. Thus, observers looking at the stationary water need the centrifugal force to explain why the water surface is concave and not flat. The centrifugal force pushes the water toward the sides of the bucket, where it piles up deeper and deeper, Pile-up is arrested when any further climb costs as much work against gravity as is the energy gained from the greater centrifugal force at larger radius.

If you need a centrifugal force to explain what you see, then you are rotating. Newton's conclusion was that rotation is absolute.

Other thinkers suggest that pure logic implies only relative rotation makes sense. For example, Bishop Berkeley and Ernst Mach (among others) suggested that it is relative rotation with respect to the fixed stars that matters, and rotation of the fixed stars relative to an object has the same effect as rotation of the object with respect to the fixed stars. Newton's arguments do not settle this issue; his arguments may be viewed, however, as establishing centrifugal force as a basis for an operational definition of what we actually mean by absolute rotation.

Rotating spheres

Newton also proposed another experiment to measure one's rate of rotation: using the tension in a cord joining two spheres rotating about their center of mass. Non-zero tension in the string indicates rotation of the spheres, whether or not the observer thinks they are rotating. This experiment is simpler than the bucket experiment in principle, because it need not involve gravity.

Beyond a simple "yes or no" answer to rotation, one may actually calculate one's rotation. To do that, one takes one's measured rate of rotation of the spheres and computes the tension appropriate to this observed rate. This calculated tension then is compared to the measured tension. If the two agree, one is in a stationary (non-rotating) frame. If the two do not agree, to obtain agreement, one must include a centrifugal force in the tension calculation; for example, if the spheres appear to be stationary, but the tension is non-zero, the entire tension is due to centrifugal force. From the necessary centrifugal force, one can determine one's speed of rotation; for example, if the calculated tension is greater than measured, one is rotating in the sense opposite to the spheres, and the larger the discrepancy the faster this rotation.

The centrifugal force is not simply cerebral, but actually is experienced by the rotating observer. That is, forces experienced by the rotating observer are equally real, whether their origin is fundamental or simply in the rotation of the observer.

Figure of the Earth

In a similar fashion, if we did not know the Earth rotates about its axis, we could infer this rotation from the centrifugal force needed to account for the bulging observed at its equator.

In his Principia, Newton proposed the shape of the rotating Earth was that of a homogeneous ellipsoid formed by an equilibrium between the gravitational force holding it together and the centrifugal force pulling it apart. The Earth's surface is an equipotential, that is, no work is done moving upon the Earth's surface, either against gravity or against centrifugal force. Based upon this equilibrium, Newton determined a flattening expressed by the ratio of diameters: 230 to 229. A modern measurement of the Earth's oblateness leads to an equatorial radius of 6378.14 km and a polar radius of 6356.77 km, about 1/10% less oblate than Newton's estimate. A theoretical determination of the precise extent of oblateness in response to a centrifugal force requires an understanding of the make-up of the planet, not only today but during its formation.

History of conceptions of centrifugal and centripetal forces

Christiaan Huygens coined the term "centrifugal force" (vis centrifuga) in his 1673 Horologium Oscillatorium on pendulums, and Newton coined the term "centripetal force" (vis centripita) in his discussions of gravity in his 1684 De Motu Corpurum. Gottfried Leibniz as part of his 'solar vortex theory' conceived of centrifugal force as a real outward force which is induced by the circulation of the body upon which the force acts. The inverse cube law centrifugal force appears in an equation representing planetary orbits, including non-circular ones, as Leibniz described in his 1689 Tentamen de motuum coelestium causis. Leibniz's equation is still used today to solve planetary orbital problems, although the 'solar vortex theory' is no longer used as its basis.

Huygens, who was, along with Leibniz, a neo-Cartesian and critic of Newton, concluded after a long correspondence that Leibniz's writings on celestial mechanics made no sense, and that his invocation of a harmonic vortex was logically redundant, because Leibniz's radial equation of motion follows trivially from Newton's laws. Even the most ardant modern defenders of the cogency of Leibniz's ideas acknowledge that his harmonic vortex as the basis of centrifugal force was dynamically superfluous.

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. When Leibniz produced his equation for planetary orbits in which the centrifugal force appeared as an outward inverse cube law force in the radial direction:

${\displaystyle {\ddot {r}}=-k/r^{2}+l^{2}/r^{3}}$.

Newton himself appears to have previously supported an approach similar to that of Leibniz. Later, Newton in his Principia crucially limited the description of the dynamics of planetary motion to a frame of reference in which the point of attraction is fixed. In this description, Leibniz's centrifugal force was not needed and was replaced by only continually inward forces toward the fixed point. Newton objected to this Liebniz's equation on the grounds that it allowed for the centrifugal force to have a different value from the centripetal force, arguing on the basis of his third law of motion, that the centrifugal force and the centripetal force must constitute an equal and opposite action-reaction pair. In this however, Newton was mistaken, as the reactive centrifugal force which is required by the third law of motion is a completely separate concept from the centrifugal force of Leibniz's equation.

It wasn't until the latter half of the 18th century that the modern "fictitious force" understanding of the centrifugal force as a pseudo-force 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" for a term which bore a similar mathematical expression to that of centrifugal force, albeit that it was multiplied by a factor of two. The force in question was perpendicular to both the velocity of an object relative to a rotating frame of reference and the axis of rotation of the frame. Compound centrifugal force eventually came to be known as the Coriolis Force.

The idea of centrifugal force is closely related to the notion of absolute rotation. In 1707 the Irish bishop George Berkeley took issue with the notion of absolute space, declaring that "motion cannot be understood except in relation to our or some other body". In considering a solitary globe, all forms of motion, uniform and accelerated, are unobservable in an otherwise empty universe. This notion was followed up in modern times by Ernst Mach. For a single body in an empty universe, motion of any kind is inconceivable. Because rotation does not exist, centrifugal force does not exist. Of course, addition of a speck of matter just to establish a reference frame cannot cause the sudden appearance of centrifugal force, so it must be due to rotation relative to the entire mass of the universe. The modern view is that centrifugal force is indeed an indicator of rotation, but relative to those frames of reference that exhibit the simplest laws of physics. Thus, for example, if we wonder how rapidly our galaxy is rotating, we can make a model of the galaxy in which its rotation plays a part. The rate of rotation in this model that makes the observations of (for example) the flatness of the galaxy agree best with physical laws as we know them is the best estimate of the rate of rotation (assuming other observations are in agreement with this assessment, such as isotropy of the background radiation of the universe).

In part I of his 1861 paper On Physical Lines of Force, James Clerk Maxwell used the concept of centrifugal force in order to explain magnetic repulsion. He considered that magnetic lines of force are comprised of molecular vortices aligned along their mutual axes of rotation. When two magnets repel each other, the magnetic lines of force in the space between the like poles spread outwards and away from each other. Maxwell considered that the repulsion is due to centrifugal force acting in the equatorial plane of the molecular vortices.

The modern interpretation is that centrifugal force in a rotating reference frame is a pseudo-force that appears in equations of motion in rotating frames of reference, to explain effects of inertia as seen in such frames. Leibniz's centrifugal force may be understood as an application of this conception, as a result of his viewing the motion of a planet along the radius vector, that is, from the standpoint of a special reference frame rotating with the planet. Leibniz introduced the notions of vis viva (kinetic energy) and action, which eventually found full expression in the Lagrangian formulation of mechanics. In deriving Leibniz's radial equation from the Lagrangian standpoint, a rotating reference frame is not used explicitly, but the result is equivalent to that found using Newtonian vector mechanics in a co-rotating reference frame.

See also

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 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.

The analogy between centrifugal force (sometimes used to create artificial gravity) and gravitational forces led to the equivalence principle of general relativity.

References

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2. Jacobson 1980, p. 80.
3. ^ Bini, D. et al. (1997). "The intrinsic derivative and centrifugal forces in general relativity: I. Theoretical foundations". International Journal of Modern Physics D 6 (1)
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5. Mook & Vargish 1987, p. 47.
6. Signell 2002, "Acceleration and force in circular motion", §5b, p. 7.
7. Mohanty 2004, p. 121.
8. Roche 2001, "Introducing motion in a circle". Retrieved 2009-05-07.
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18. For an introduction see, for example, Cornelius Lanczos (1986). The variational principles of mechanics (Reprint of 1970 University of Toronto ed.). Dover. p. 1. ISBN 0486650677.
19. For a description of generalized coordinates, see Ahmed A. Shabana (2003). "Generalized coordinates and kinematic constraints". Dynamics of Multibody Systems (2 ed.). Cambridge University Press. p. 90 ff. ISBN 0521544114.
20. Christian Ott (2008). Cartesian Impedance Control of Redundant and Flexible-Joint Robots. Springer. p. 23. ISBN 3540692533.
21. Shuzhi S. Ge, Tong Heng Lee, Christopher John Harris (1998). Adaptive Neural Network Control of Robotic Manipulators. World Scientific. p. pp. 47-48. ISBN 981023452X. In the above Euler-Lagrange equations, there are three types of terms. The first involves the second derivative of the generalized co-ordinates. The second is quadratic in ${\displaystyle {\boldsymbol {\dot {q}}}}$ where the coefficients may depend on ${\displaystyle {\boldsymbol {q}}}$. These are further classified into two types. Terms involving a product of the type ${\displaystyle {{\dot {q}}_{i}}^{2}}$ are called centrifugal forces while those involving a product of the type ${\displaystyle {\dot {q}}_{i}{\dot {q}}_{j}}$ for i ≠ j are called Coriolis forces. The third type is functions of ${\displaystyle {\boldsymbol {q}}}$ only and are called gravitational forces. {{cite book}}: |page= has extra text (help)CS1 maint: multiple names: authors list (link)
22. R. K. Mittal, I. J. Nagrath (2003). Robotics and Control. Tata McGraw-Hill. p. 202. ISBN 0070482934.
23. T Yanao & K Takatsuka (2005). "Effects of an intrinsic metric of molecular internal space". In Mikito Toda, Tamiki Komatsuzaki, Stuart A. Rice, Tetsuro Konishi, R. Stephen Berry (ed.). Geometrical Structures Of Phase Space In Multi-dimensional Chaos: Applications to chemical reaction dynamics in complex systems. Wiley. p. 98. ISBN 0471711578. As is evident from the first terms…, which are proportional to the square of ${\displaystyle {\dot {\phi }}}$, a kind of "centrifugal force" arises… We call this force "democratic centrifugal force". Of course, DCF is different from the ordinary centrifugal force, and it arises even in a system of zero angular momentum.{{cite book}}: CS1 maint: multiple names: editors list (link)
24. See, for example, Eq. 8.20 in John R Taylor (2005). op. cit. pp. 299 ff. ISBN 189138922X.
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27. Henry M. Stommel and Dennis W. Moore (1989). An Introduction to the Coriolis Force Columbia University Press. pp 36-38
28. See Edmond T Whittaker. A treatise on the analytical dynamics of particles and rigid bodies (Reprint of 1917 2nd ed.). Cambridge University Press. pp. 40–41. ISBN 0521358833. for an explanation of how the fictitious centrifugal force corresponds to a potential term in the Lagrangian.
29. For example, while the Newtonian picture uses a single frame of reference, the Lagrangian generalized coordinates may refer simultaneously to several different frames, making the connection to the Newtonian picture complex.Shuzhi S. Ge, Tong Heng Lee, Christopher John Harris. op. cit.. p. 136. ISBN 981023452X.{{cite book}}: CS1 maint: multiple names: authors list (link)
30. Max Born and Günther Leibfried. Einstein's Theory of Relativity. Courier Dover Publications. p. 78–79. ISBN 0486607690.
31. BK Ridley (1995). Time, Space, and Things (3 ed.). Cambridge University Press. p. 146. ISBN 0521484863.
32. Rather than justifying a causal link between rotation and centrifugal effects, Newton's arguments may be viewed as defining "absolute rotation" by stating a procedure for its detection and measurement involving centrifugal force. See Robert Disalle (2002). I. Bernard Cohen & George E. Smith (ed.). The Cambridge Companion to Newton. Cambridge University Press. pp. 44–45. ISBN 0521656966.
33. See cartoon: XKCD demonstrates the life and death importance of centrifugal force.
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35. John Clayton Taylor (2001). Hidden unity in nature's laws. Cambridge University Press. p. 26. ISBN 0521659388.
36. Isaac Newton: Principia (July 5, 1687) Book III Proposition XIX Problem III, p. 407 in Andrew Motte translation.
37. See the Principia on line at Andrew Motte Translation. Other sources and some notes are found at Philosophiæ Naturalis Principia Mathematica.
38. Charles D Brown (1998). Spacecraft mission design (2 ed.). American Institute of Aeronautics & Astronomy. p. 58. ISBN 1563472627.
39. This error is the difference in the estimated ratio of diameters. However, a more demanding measure of oblateness is the flattening, defined as f = (a−b)/a where a and b are the semimajor and semiminor axes. Using the cited numbers, the flattening of Newton's prediction differs by 23% from that of modern estimates.
40. Hugh Murray (1837). "Figure and constitution of the Earth deduced from the theory of gravitation". The Encyclopædia of Geography. Vol. vol. 1. Carey, Lea & Blanchard. pp. 124 ff. {{cite book}}: |volume= has extra text (help)
41. Alexander Winchell (1888). World-life; Or, Comparative Geology. SC Griggs & Co. p. 425.
42. Whiteside 2008, pp. 4-5.
43. Gillies 1995, p. 130.
44. Herbert Goldstein 'Classical Mechanics', equation 3-12
45. A. R. Hall, Philosophers at War, 2002, pp 150-151
46. Linton 2004, p. 285.
47. Swetz et al. 1997, p. 268.
48. ^ Linton 2004, pg. 264
49. ^ Swetz et al. 1997, p. 269.
50. Wilson 1994, "Newton's Orbit Problem: A Historian's Response".
51. ^ Meli 1990, "The Relativization of Centrifugal Force".
52. Dugas & Maddox 1988, p. 374.
53. Persson 1998, "How Do We Understand the Coriolis Force?".
54. Slate 1918, p. 137.
55. Edward Robert Harrison (2000). Cosmology (2nd ed.). Cambridge University Press. p. 237. ISBN 052166148X.
56. Ernst Mach (1915). The science of mechanics. The Open Court Publishing Co. p. 33. Try to fix Newton's bucket and rotate the heaven of fixed stars and then prove the absence of centrifugal forces
57. J. F. Kiley, W. E. Carlo (1970). "The epistemology of Albert Einstein". Einstein and Aquinas. Springer. p. 27. ISBN 9024700817.
58. Henning Genz (2001). Nothingness. p. 275. ISBN 0738206105. {{cite book}}: Cite has empty unknown parameter: |unused_data= (help); Text "Da Capo Press" ignored (help)
59. J Garcio-Bellido (2005). "The Paradigm of Inflation". In J. M. T. Thompson (ed.). Advances in Astronomy. Imperial College Press. p. 32, §9. ISBN 1860945775.
60. Steinmetz 2005, p. 49.
61. Aiton 1962, "The celestial mechanics of Leibniz in the light of Newtonian criticism".
62. Bertrand Russell (1992). A Critical Exposition of the Philosophy of Leibniz (Reprint of 1937 2nd ed.). Routledge. p. 96. ISBN 041508296X.
63. Wolfgang Lefèvre (2001). Between Leibniz, Newton, and Kant. Springer. p. 39. ISBN 0792371984.
64. Goldstein 2002, pp.74-77
65. Goldstein 2002, pg. 176
66. Taylor 2005, pp. 358-359
67. Whiting, J.S.S. (1983). "Motion in a central-force field" (PDF). Physics Education. 18 (6): pp. 256–257. ISSN 0031-9120. Retrieved May 7, 2009. {{cite journal}}: |pages= has extra text (help); Unknown parameter |month= ignored (help)
68. Rizzi & Ruggiero 2004, p. 272.
69. Rindler 2006, pp. 7-8.
70. Barbour & Pfister 1995, pp. 6-8.
71. Barbour & Pfister 1995, p. 69.
72. Eriksson 2008, p. 194.

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• Fictitious forces
• Force
• Mechanics
• Rotation