# Centrifugal force

Not to be confused with Centripetal force.

Centrifugal force (from Latin centrum, meaning "center", and fugere, meaning "to flee"[1][2]) is the apparent force that draws a rotating body away from the center of rotation. It is caused by the inertia of the body. 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, and also sometimes to the equal and opposite reaction to a centripetal force in any reference frame (as per Newton's third law of motion).

The concept of centrifugal force is applied in rotating devices such as centrifuges, centrifugal pumps, centrifugal governors, centrifugal clutches, etc., as well as in centrifugal railways, planetary orbits, banked curves, etc. Some aspects of these situations can be analyzed in terms of the fictitious force in the rotating coordinate system, while other aspects additionally require the involvement of the reactive centrifugal force, otherwise called a normal reaction.

## Current meaning

Centrifugal force is an outward force apparent in a rotating reference frame; it does not exist when measurements are made in an inertial frame of reference. This type of force, associated with describing motion in a non-inertial reference frame is 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).[3][4]

In a rotating reference frame, all objects appear to be under the influence of a radially (from the axis of rotation) outward force that is proportional to their mass, the distance from the axis of rotation of the frame, and to the square of the (angular velocity) of the frame.[5]

Motion relative to a rotating frame 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 are necessary for the formulation of correct equations of motion in a rotating reference frame.[6]

## History of conceptions of centrifugal and centripetal forces

The conception of centrifugal force has evolved since the time of Huygens, Newton, Leibniz, and Hooke who expressed early conceptions of it. Its modern conception as a fictitious force arising in a rotating reference frame evolved in the eighteenth and nineteenth centuries.[citation needed]

Centrifugal force has also played a role in debates in classical mechanics about detection of absolute motion. Newton suggested two arguments to answer the question of whether absolute rotation can be detected: the rotating bucket argument, and the rotating spheres argument.[7] According to Newton, in each scenario the centrifugal force would be observed in the object's local frame (the frame where the object is stationary) only if the frame were rotating with respect to absolute space. Nearly two centuries later, Mach's principle was proposed where, instead of absolute rotation, the motion of the distant stars relative to the local inertial frame gives rise through some (hypothetical) physical law to the centrifugal force and other inertia effects. Today's view is based upon the idea of an inertial frame of reference, which privileges observers for which the laws of physics take on their simplest form, and in particular, frames that do not use centrifugal forces in their equations of motion in order to describe motions correctly.

The analogy between centrifugal force (sometimes used to create artificial gravity) and gravitational forces led to the equivalence principle of general relativity.[8][9]

### Reactive centrifugal force

A reactive centrifugal force is a reaction force to a centripetal force. A body undergoing curved motion, such as circular motion, is accelerating toward a center at any particular point in time. This centripetal acceleration is provided by a centripetal force, which is exerted on the body in curved motion by some other body. In accordance with Newton's third law of motion, the body in curved motion exerts an equal and opposite force on the other body. This reactive force is exerted by the body in curved motion on the other body that provides the centripetal force and its direction is from that other body toward the body in curved motion.[10][11] [12][13]

This reaction force is sometimes described as a centrifugal inertial reaction,[14][15] that is, a force that is centrifugally directed, which is a reactive force equal and opposite to the centripetal force that is curving the path of the mass.

The concept of the reactive centrifugal force is sometimes used in mechanics and engineering. It is sometimes referred to as just centrifugal force rather than as reactive centrifugal force.[16][17]

### Use of the term in Lagrangian mechanics

Lagrangian mechanics formulates mechanics in terms of generalized coordinates {qk}, which can be as simple as the usual polar coordinates $(r,\ \theta)$ or a much more extensive list of variables.[18][19] 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 {(dqk   ⁄ dt )2} are sometimes called centrifugal forces.[20][21][22][23]

The Lagrangian approach to polar coordinates that treats $(r,\ \theta)$ as generalized coordinates, $(\dot{r},\ \dot{\theta})$ as generalized velocities and $(\ddot{r},\ \ddot{\theta})$ as generalized accelerations, is outlined in another article, and found in many sources.[24][25][26] For the particular case of single-body motion found using the generalized coordinates $(\dot{r},\ \dot{\theta})$ 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:

$\mu\ddot{r} = \mu r\dot\theta^2 - \frac{\mathrm{d}U}{\mathrm{d}r}$

where $U(r)$ is the central force potential and μ is the mass of the object. 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 acceleration, and likewise, therefore, the terms on the right-hand side are "generalized forces" and cannot be interpreted as Newtonian forces.[27]

The Lagrangian centrifugal force is derived without explicit use of a rotating frame of reference,[28] 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.[29] The Lagrangian use of "centrifugal force" in other, more general cases, however, has only a limited connection to the Newtonian definition.

## References

1. ^ http://www.merriam-webster.com/dictionary/centrifugal
2. ^ http://www.collinsdictionary.com/dictionary/english/centrifugal
3. ^ Takwale, R. G. and P. S. Puranik (1980). Introduction to classical mechanics. Tata McGraw-Hill. ISBN 978-0-07-096617-8, p. 248.
4. ^ Jacobson, Mark Zachary (1980). Fundamentals of atmospheric modeling. Cambridge: University Press. ISBN 978-0-521-63717-6, p. 80.
5. ^ Encyclopaedia Britannica, article on Centrifuge
6. ^ Fetter, Alexander L. and John Dirk Walecka (2003). Theoretical Mechanics of Particles and Continua (Reprint of McGraw-Hill 1980 ed.). Courier Dover Publications. ISBN 0-486-43261-0, pp. 38–39.
7. ^ An English translation is found at Sir Isaac Newton (1934). Philosophiae naturalis principia mathematica (Andrew Motte translation of 1729, revised by Florian Cajori ed.). University of California Press. pp. 10–12.
8. ^ Barbour, Julian B. and Herbert Pfister (1995). Mach's principle: from Newton's bucket to quantum gravity. Birkhäuser. ISBN 0-8176-3823-7, p. 69.
9. ^ Eriksson, Ingrid V. (2008). Science education in the 21st century. Nova Books. ISBN 1-60021-951-9, p. 194.
10. ^ Mook, Delo E. & Thomas Vargish (1987). Inside relativity. Princeton NJ: Princeton University Press. ISBN 0-691-02520-7, p. 47.
11. ^ G. David Scott (1957). "Centrifugal Forces and Newton's Laws of Motion" 25. American Journal of Physics. p. 325.
12. ^ Signell, Peter (2002). "Acceleration and force in circular motion" Physnet. Michigan State University, "Acceleration and force in circular motion", §5b, p. 7.
13. ^ Mohanty, A. K. (2004). Fluid Mechanics. PHI Learning Pvt. Ltd. ISBN 81-203-0894-8, p. 121.
14. ^ Roche, John (September 2001). "Introducing motion in a circle". Physics Education 43 (5), pp. 399-405, "Introducing motion in a circle". Retrieved 2009-05-07.
15. ^ Lloyd William Taylor (1959). Physics, the pioneer science 1. Dover Publications. p. 173.
16. ^ Edward Albert Bowser (1920). An elementary treatise on analytic mechanics: with numerous examples (25th ed.). D. Van Nostrand Company. p. 357.
17. ^ Joseph A. Angelo (2007). Robotics: a reference guide to the new technology. Greenwood Press. p. 267. ISBN 1-57356-337-4.
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 0-486-65067-7.
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 0-521-54411-4.
20. ^ Christian Ott (2008). Cartesian Impedance Control of Redundant and Flexible-Joint Robots. Springer. p. 23. ISBN 3-540-69253-3.
21. ^ Shuzhi S. Ge, Tong Heng Lee, Christopher John Harris (1998). Adaptive Neural Network Control of Robotic Manipulators. World Scientific. pp. 47–48. ISBN 981-02-3452-X. 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 $\boldsymbol{\dot q}$ where the coefficients may depend on $\boldsymbol{q}$. These are further classified into two types. Terms involving a product of the type ${\dot q_i}^2$ are called centrifugal forces while those involving a product of the type $\dot q_i \dot q_j$ for i ≠ j are called Coriolis forces. The third type is functions of $\boldsymbol{q}$ only and are called gravitational forces.
22. ^ R. K. Mittal, I. J. Nagrath (2003). Robotics and Control. Tata McGraw-Hill. p. 202. ISBN 0-07-048293-4.
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. Geometrical Structures Of Phase Space In Multi-dimensional Chaos: Applications to chemical reaction dynamics in complex systems. Wiley. p. 98. ISBN 0-471-71157-8. As is evident from the first terms ..., which are proportional to the square of $\dot\phi$, 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.
24. ^ John R Taylor (2005). Classical Mechanics. Sausalito, California: Univ. Science Books. pp. 299 ff. ISBN 1-891389-22-X.
25. ^ Francis Begnaud Hildebrand (1992). Methods of Applied Mathematics (Reprint of 1965 2nd ed.). Courier Dover Publications. p. 156. ISBN 0-486-67002-3.
26. ^ V. B. Bhatia (1997). Classical Mechanics: With Introduction to Nonlinear Oscillations and Chaos. Alpha Science Int'l Ltd. p. 82. ISBN 81-7319-105-0.
27. ^ Henry M. Stommel and Dennis W. Moore (1989). An Introduction to the Coriolis Force. Columbia University Press. pp. 36–38.
28. ^ Edmond T Whittaker (1988). A treatise on the analytical dynamics of particles and rigid bodies (Reprint of 1917 2nd ed.). Cambridge University Press. pp. 40–41. ISBN 0-521-35883-3.
29. ^ See p. 5 in Donato Bini, Paolo Carini, Robert T Jantzen (1997). "The intrinsic derivative and centrifugal forces in general relativity: I. Theoretical foundations". International Journal of Modern Physics D 6 (1).. The companion paper is Donato Bini, Paolo Carini, Robert T Jantzen (1997). "The intrinsic derivative and centrifugal forces in general relativity: II. Applications to circular orbits in some stationary axisymmetric spacetimes". International Journal of Modern Physics D 6 (1).