Centrifugal force: Difference between revisions

'''Centrifugal force''' is ana [[inertialfictitious force]] in [[Newtonian mechanics]] (also called aan "fictitiousinertial" or "pseudo" force) that appears to act on all objects when viewed in a [[rotating frame of reference]]. It isappears to be directed radially away from the [[Rotation around a fixed axis|axis of rotation]] of the frame. The magnitude of the centrifugal force ''F'' on an object of [[mass]] ''m'' at the distance ''r'' from the axis of rotationa of arotating frame of reference rotating with [[angular velocity]] {{mvar|ω}} is: <math display="block">F = m\omega^2 r</math>

Confusingly, theThe term has sometimes also been used for the ''[[reactive centrifugal force]]'', a real frame-independent Newtonian force that exists as a reaction to a [[centripetal force]] in some scenarios.

From 1659, the [[Neo-Latin]] term ''vi centrifuga'' ("centrifugal force") is attested in [[Christiaan Huygens]]' notes and letters.<ref name=yoeder>{{cite journal | url=http://www.gewina.nl/journals/tractrix/yoder91.pdf | title=Christiaan Huygens' Great Treasure | first=Joella | last=YoederYoder | author-link=Joella Yoder |journal=Tractrix | volume=3 | year=1991 | pages=1–13 | access-date=12 April 2018 | archive-date=13 April 2018 | archive-url=https://web.archive.org/web/20180413044740/http://www.gewina.nl/journals/tractrix/yoder91.pdf | url-status=live }}</ref><ref name="Yoder2013">{{cite book|last=Yoder|first=Joella|url=https://books.google.com/books?id=XGZlIvCOtFsC|title=A Catalogue of the Manuscripts of Christiaan Huygens including a concordance with his Oeuvres Complètes|date=17 May 2013|publisher=BRILL|isbn=9789004235656|access-date=12 April 2018|archive-date=16 March 2020|archive-url=https://web.archive.org/web/20200316011539/https://books.google.com/books?id=XGZlIvCOtFsC|url-status=live}}</ref> Note, that in Latin {{wikt-lang|la|centrum}} means "center" and {{wikt-lang|la|‑fugus}} (from {{wikt-lang|la|fugiō}}) means "fleeing, avoiding". Thus, ''centrifugus'' means "fleeing from the center" in a [[literal translation]].

The same year, [[Isaac Newton]] received Huygens work via [[Henry Oldenburg]] and replied "I pray you return [Mr. Huygens] my humble thanks [...] I am glad we can expect another discourse of the ''vis centrifuga'', which speculation may prove of good use in [[natural philosophy]] and [[astronomy]], as well as [[mechanics]]".{{r|yoeder}}<ref>{{cite book |title=Œuvres complètes de Christiaan Huygens |volume=7 |language=French |date=1897 |location=The Hague |publisher=M. Nijhoff |page=[https://commons.wikimedia.org/w/index.php?title=File:Huygens_-_%C5%92uvres_compl%C3%A8tes,_Tome_7,_1897.djvu&page=353 325] |url=https://commons.wikimedia.org/w/index.php?title=File:Huygens_-_%C5%92uvres_compl%C3%A8tes,_Tome_7,_1897.djvu |access-date=2023-01-14 |archive-date=2023-11-06 |archive-url=https://web.archive.org/web/20231106055244/https://commons.wikimedia.org/w/index.php?title=File:Huygens_-_%C5%92uvres_compl%C3%A8tes,_Tome_7,_1897.djvu |url-status=live }}</ref>

Centrifugal force is an outward force apparent in a [[rotating reference frame]].<ref>{{cite book|author=Richard T. Weidner and Robert L. Sells|title=Mechanics, mechanical waves, kinetic theory, thermodynamics | date=1973 | publisher=Allyn and Bacon|page=123|edition=2}}</ref><ref>{{cite journal |last1=Restuccia |first1=S. |last2=Toroš |first2=M. |last3=Gibson |first3=G. M. |last4=Ulbricht |first4=H. |last5=Faccio |first5=D. |last6=Padgett |first6=M. J. |date=2019 |title=Photon Bunching in a Rotating Reference Frame |url=https://doi.org/10.1103/physrevlett.123.110401 |journal=Physical Review Letters |volume=123 |issue=11 |pages=110401 | doi=10.1103/physrevlett.123.110401|pmid=31573252 |arxiv=1906.03400 |bibcode=2019PhRvL.123k0401R |s2cid=182952610 }}</ref><ref name=Taylor1>{{cite book |title=Classical Mechanics |author=John Robert Taylor |page=Chapter 9, pp. 344 ff |url=https://books.google.com/books?id=P1kCtNr-pJsC&pg=PP1 |isbn=978-1-891389-22-1 |publisher=University Science Books |location=Sausalito CA |year=2004 |no-pp=true |access-date=2020-11-09 |archive-date=2024-10-07 |archive-url=https://web.archive.org/web/20241007141548/https://books.google.com/books?id=P1kCtNr-pJsC&pg=PP1 |url-status=live }}</ref><ref>{{cite journal|last1=Kobayashi|first1=Yukio|title=Remarks on viewing situation in a rotating frame|journal=European Journal of Physics|date=2008|volume=29|issue=3|pages=599–606|doi=10.1088/0143-0807/29/3/019|bibcode=2008EJPh...29..599K|s2cid=120947179 }}</ref> It does not exist when a system is described relative to an [[inertial frame of reference]].

In a reference frame rotating about an axis through its origin, all objects, regardless of their state of motion, appear to be under the influence of a radially (from the axis of rotation) outward force that is proportional to their mass, to the distance from the axis of rotation of the frame, and to the square of the [[angular velocity]] of the frame.<ref>{{cite encyclopedia|url = https://www.britannica.com/EBchecked/topic/102850/centrifuge|encyclopedia = Encyclopædia Britannica|title = Centrifuge|date = April 30, 2015|access-date = June 2, 2022|archive-date = October 7, 2024|archive-url = https://web.archive.org/web/20241007141550/https://www.britannica.com/technology/centrifuge|url-status = live}}</ref><ref>[{{Cite web |url=https://feynmanlectures.caltech.edu/I_12.html#Ch12-S5-p2 |title=The Feynman Lectures on Physics Vol. I Ch. 12: Characteristics of Force] |access-date=2022-05-07 |archive-date=2024-10-07 |archive-url=https://web.archive.org/web/20241007141549/https://www.feynmanlectures.caltech.edu/I_12.html#Ch12-S5-p2 |url-status=live }}</ref> This is the centrifugal force. As humans usually experience centrifugal force from within the rotating reference frame, e.g. on a merry-go-round or vehicle, this is much more well-known than centripetal force.

Motion relative to a rotating frame results in another fictitious force: the [[Coriolis force]]. If the rate of rotation of the frame changes, a third fictitious force (the [[Euler force]]) is required. These fictitious forces are necessary for the formulation of correct equations of motion in a rotating reference frame<ref name=Fetter/><ref name=Marsden>{{cite book | title=Introduction to Mechanics and Symmetry: A Basic Exposition of Classical Mechanical Systems | author1=Jerrold E. Marsden | author2=Tudor S. Ratiu | isbn=978-0-387-98643-2 | year=1999 | publisher=Springer | page=251 | url=https://books.google.com/books?id=I2gH9ZIs-3AC&pg=PA251 | access-date=2020-11-09 | archive-date=2024-10-07 | archive-url=https://web.archive.org/web/20241007141657/https://books.google.com/books?id=I2gH9ZIs-3AC&pg=PA251#v=onepage&q&f=false | url-status=live }}</ref> and allow Newton's laws to be used in their normal form in such a frame (with one exception: the fictitious forces do not obey Newton's third law: they have no equal and opposite counterparts).<ref name=Fetter>{{cite book | title=Theoretical Mechanics of Particles and Continua | author1=Alexander L. Fetter|author-link1=Alexander L. Fetter | author2=John Dirk Walecka | author-link2=John Dirk Walecka | year=2003 | url=https://books.google.com/books?id=olMpStYOlnoC&pg=PA39 | publisher = Courier Dover Publications | isbn=978-0-486-43261-8 | pages=38–39 }}</ref> Newton's third law requires the counterparts to exist within the same frame of reference, hence centrifugal and centripetal force, which do not, are not action and reaction (as is sometimes erroneously contended).

A common experience that gives rise to the idea of a centrifugal force is encountered by passengers riding in a vehicle, such as a car, that is changing direction. If a car is traveling at a constant speed along a straight road, then a passenger inside is not accelerating and, according to [[Newton's laws of motion|Newton's second law of motion]], the net force acting on them is therefore zero (all forces acting on them cancel each other out). If the car enters a curve that bends to the left, the passenger experiences an apparent force that seems to be pulling them towards the right. This is the fictitious centrifugal force. It is needed within the passengers' local frame of reference to explain their sudden tendency to start accelerating to the right relative to the car—a tendency which they must resist by applying a rightward force to the car (for instance, a frictional force against the seat) in order to remain in a fixed position inside. Since they push the seat toward the right, Newton's third law says that the seat pushes them towards the left. The centrifugal force must be included in the passenger's reference frame (in which the passenger remains at rest): it counteracts the leftward force applied to the passenger by the seat, and explains why this otherwise unbalanced force does not cause them to accelerate.<ref name="EB">{{cite web |url=https://www.britannica.com/science/centrifugal-force |title=Centrifugal force |publisher=Encyclopædia Britannica |date=17 August 2016 |access-date=20 April 2017 |archive-date=21 April 2017 |archive-url=https://web.archive.org/web/20170421011514/https://www.britannica.com/science/centrifugal-force |url-status=live }}</ref> However, it would be apparent to a stationary observer watching from an overpass above that the frictional force exerted on the passenger by the seat is not being balanced; it constitutes a net force to the left, causing the passenger to accelerate toward the inside of the curve, as they must in order to keep moving with the car rather than proceeding in a straight line as they otherwise would. Thus the "centrifugal force" they feel is the result of a "centrifugal tendency" caused by inertia.<ref name="Science of Everyday Things">{{cite book |url=https://archive.org/stream/ScienceOfEverydayThingsVol2-RealLifePhysics/ScienceOfEverydayThingsVol.2-Physics365s-o#page/n49/mode/2up/search/Centrifugal+force |title=Centripetal Force |work=Science of Everyday Things, Volume 2: Real-Life Physics |page=47 |editor-first=Neil |editor-last=Schlager |author-first=Judson |author-last=Knight |year=2016 |publisher=Thomson Learning |access-date=19 April 2017}}</ref> Similar effects are encountered in aeroplanes and [[roller coaster]]s where the magnitude of the apparent force is often reported in "[[g-force|G's]]".

The apparent acceleration in the rotating frame is <math> \left[\frac{\mathrm d^2\boldsymbol{r} }{\mathrm dt^2}\right] </math>. An observer unaware of the rotation would expect this to be zero in the absence of outside forces. However, Newton's laws of motion apply only in the inertial frame and describe dynamics in terms of the absolute acceleration <math> \frac{\mathrm d^2\boldsymbol{r} }{\mathrm dt^2} </math>. Therefore, the observer perceives the extra terms as contributions due to fictitious forces. These terms in the apparent acceleration are independent of mass; so it appears that each of these fictitious forces, like gravity, pulls on an object in proportion to its mass. When these forces are added, the equation of motion has the form:<ref>Taylor (2005). p. 342.</ref><ref name=L&L_A> {{cite book |title=Mechanics |author1=LD Landau |author2=LM Lifshitz |page= 128 |url=https://books.google.com/books?id=e-xASAehg1sC&pg=PA40 |edition=Third |year=1976 |isbn=978-0-7506-2896-9 |publisher=Butterworth-Heinemann |location=Oxford |access-date=2020-11-09 |archive-date=2024-10-07 |archive-url=https://web.archive.org/web/20241007141549/https://books.google.com/books?id=e-xASAehg1sC&pg=PA40#v=onepage&q&f=false |url-status=live }}</ref><ref name=Hand_A> {{cite book |title=Analytical Mechanics |author1=Louis N. Hand |author2=Janet D. Finch |page=267 |url=https://books.google.com/books?id=1J2hzvX2Xh8C&q=Hand+inauthor:Finch&pg=PA267 |isbn = 978-0-521-57572-0 |publisher=[[Cambridge University Press]] |year=1998 |access-date=2020-11-09 |archive-date=2024-10-07 |archive-url=https://web.archive.org/web/20241007141658/https://books.google.com/books?id=1J2hzvX2Xh8C&q=Hand+inauthor:Finch&pg=PA267 |url-status=live }}</ref>

<math display="block">\boldsymbol{F} -+ \underbrace{\left(-m\frac{\mathrm{d} \boldsymbol{\omega}}{\mathrm{d}t}\times\boldsymbol{r}\right)}_{\text{Euler}} -+ \underbrace{\left(-2m \boldsymbol{\omega}\times \left[ \frac{\mathrm{d} \boldsymbol{r}}{\mathrm{d}t} \right]\right)}_{\text{Coliolis}} -+ \underbrace{\left(-m\boldsymbol{\omega}\times (\boldsymbol{\omega}\times \boldsymbol{r})\right)}_{\text{centrifugal}} = m\left[ \frac{\mathrm{d}^2 \boldsymbol{r}}{\mathrm{d}t^2} \right] \ .</math>

From the perspective of the rotating frame, the additional force terms are experienced just like the real external forces and contribute to the apparent acceleration.<ref name=Silverman>{{cite book | title = A universe of atoms, an atom in the universe | author = Mark P Silverman | url = https://books.google.com/books?id=-Er5pIsYe_AC&pg=PA249 | page = 249 | isbn = 978-0-387-95437-0 | year = 2002 | publisher = Springer | edition = 2 | access-date = 2020-11-09 | archive-date = 2024-10-07 | archive-url = https://web.archive.org/web/20241007142053/https://books.google.com/books?id=-Er5pIsYe_AC&pg=PA249#v=onepage&q&f=false | url-status = live }}</ref><ref>Taylor (2005). p. 329.</ref> The additional terms on the force side of the equation can be recognized as, reading from left to right, the [[Euler force]] <math>-m \mathrm{d}\boldsymbol{\omega}/\mathrm{d}t \times\boldsymbol{r}</math>, the [[Coriolis force]] <math>-2m \boldsymbol{\omega}\times \left[ \mathrm{d} \boldsymbol{r}/\mathrm{d}t \right]</math>, and the centrifugal force <math>-m\boldsymbol{\omega}\times (\boldsymbol{\omega}\times \boldsymbol{r})</math>, respectively.<ref name=Lanczos_A>{{cite book | url = https://books.google.com/books?id=ZWoYYr8wk2IC&pg=PA103 | title=The Variational Principles of Mechanics | author=Cornelius Lanczos | year=1986 | isbn=978-0-486-65067-8 | publisher=Dover Publications | edition=Reprint of Fourth Edition of 1970 | at = Chapter 4, §5 | no-pp=true }}</ref> Unlike the other two fictitious forces, the centrifugal force always points radially outward from the axis of rotation of the rotating frame, with magnitude <math>m\omega^2r_\perp</math>, where <math>r_\perp</math> is the component of the position vector perpendicular to <math>\boldsymbol{\omega}</math>, and unlike the Coriolis force in particular, it is independent of the motion of the particle in the rotating frame. As expected, for a non-rotating inertial frame of reference <math>(\boldsymbol\omega=0)</math> the centrifugal force and all other fictitious forces disappear.<ref name=Tavel>{{cite book | title=Contemporary Physics and the Limits of Knowledge | page=93 | quote=Noninertial forces, like centrifugal and Coriolis forces, can be eliminated by jumping into a reference frame that moves with constant velocity, the frame that Newton called inertial. | author=Morton Tavel | url=https://books.google.com/books?id=SELS0HbIhjYC&q=Einstein+equivalence+laws+physics+frame&pg=PA95 | isbn=978-0-8135-3077-2 | publisher=[[Rutgers University Press]] | year=2002 | access-date=2020-11-09 2002| archive-date=2024-10-07 | archive-url=https://web.archive.org/web/20241007142054/https://books.google.com/books?id=SELS0HbIhjYC&q=Einstein+equivalence+laws+physics+frame&pg=PA95#v=snippet&q=Einstein%20equivalence%20laws%20physics%20frame&f=false | url-status=live }}</ref> Similarly, as the centrifugal force is proportional to the distance from object to the axis of rotation of the frame, the centrifugal force vanishes for objects that lie upon the axis.

One of these instances occurs in [[Lagrangian mechanics]]. Lagrangian mechanics formulates mechanics in terms of [[generalized coordinates]] {''q_k''}, which can be as simple as the usual polar coordinates <math>(r,\ \theta)</math> or a much more extensive list of variables.<ref name=Lanczos>For an introduction, see for example {{cite book |isbn=978-0-486-65067-8 |title=The variational principles of mechanics |url=https://books.google.com/books?id=ZWoYYr8wk2IC&pg=PR4 |publisher=Dover |edition=Reprint of 1970 University of Toronto |page=1 |author=Cornelius Lanczos |year=1986 |access-date=2020-11-09 |archive-date=2024-10-07 |archive-url=https://web.archive.org/web/20241007142120/https://books.google.com/books?id=ZWoYYr8wk2IC&pg=PR4 |url-status=live }}</ref><ref name=Shabana1>For a description of generalized coordinates, see {{cite book |author= Ahmed A. Shabana |edition=2 |publisher=Cambridge University Press |title=Dynamics of Multibody Systems |chapter-url=https://books.google.com/books?id=zxuG-l7J5rgC |page=90 ''ff'' |chapter=Generalized coordinates and kinematic constraints |year=2003 |isbn=978-0-521-54411-5 |access-date=2020-11-09 |archive-date=2024-10-07 |archive-url=https://web.archive.org/web/20241007142055/https://books.google.com/books?id=zxuG-l7J5rgC |url-status=live }}</ref> Within this formulation the motion is described in terms of ''[[generalized forces]]'', using in place of [[Newton's laws]] the [[Euler–Lagrange equations|Euler&ndash;Lagrange equations]]. Among the generalized forces, those involving the square of the time derivatives {(d''q_k''  ⁄ d''t'' )²} are sometimes called centrifugal forces.<ref name=Ott>{{cite book |title=Cartesian Impedance Control of Redundant and Flexible-Joint Robots |author=Christian Ott |url=https://books.google.com/books?id=wKQvUfwzqjAC&pg=PA23 |page=23 |isbn=978-3-540-69253-9 |year=2008 |publisher=Springer |access-date=2020-11-09 |archive-date=2024-10-07 |archive-url=https://web.archive.org/web/20241007142219/https://books.google.com/books?id=wKQvUfwzqjAC&pg=PA23#v=onepage&q&f=false |url-status=live }}</ref><ref name="Ge">{{cite book |title=Adaptive Neural Network Control of Robotic Manipulators |author1=Shuzhi S. Ge |author2=Tong Heng Lee |author3=Christopher John Harris |isbn=978-981-02-3452-2 |publisher=World Scientific |year=1998 |pages=47–48 |url=https://books.google.com/books?id=cdBENqlY_ucC&q=CHristoffel+centrifugal |quote = In the above [[Euler–Lagrange equations|Euler&ndash;Lagrange equations]], there are three types of terms. The first involves the second derivative of the generalized co-ordinates. The second is quadratic in <math>\boldsymbol{\dot q}</math> where the coefficients may depend on <math>\boldsymbol{q}</math>. These are further classified into two types. Terms involving a product of the type <math>{\dot q_i}^2</math> are called ''centrifugal forces'' while those involving a product of the type <math>\dot q_i \dot q_j</math> for ''i ≠ j'' are called ''Coriolis forces''. The third type is functions of <math>\boldsymbol{q}</math> only and are called ''gravitational forces''.}}</ref><ref name=Nagrath>{{cite book |title=Robotics and Control |url=https://books.google.com/books?id=ZtwMEQzMVlMC&pg=PA202 |page=202 |author1=R. K. Mittal |author2=I. J. Nagrath |isbn=978-0-07-048293-7 |year=2003 |publisher=Tata McGraw-Hill |access-date=2020-11-09 |archive-date=2024-10-07 |archive-url=https://web.archive.org/web/20241007142204/https://books.google.com/books?id=ZtwMEQzMVlMC&pg=PA202 |url-status=live }}</ref><ref name="Toda">{{cite book |title=Geometrical Structures Of Phase Space In Multi-dimensional Chaos: Applications to chemical reaction dynamics in complex systems |author1=T Yanao |author2=K Takatsuka |chapter=Effects of an intrinsic metric of molecular internal space |editor1=Mikito Toda |editor2=Tamiki Komatsuzaki |editor3=Stuart A. Rice |editor4=Tetsuro Konishi |editor5=R. Stephen Berry |quote=As is evident from the first terms ..., which are proportional to the square of <math>\dot\phi</math>, 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. |chapter-url=https://books.google.com/books?id=2M4qIUTITI0C&pg=PA98 |page=98 |isbn=978-0-471-71157-5 |publisher=Wiley |year=2005 |access-date=2020-11-09 |archive-date=2024-10-07 |archive-url=https://web.archive.org/web/20241007142108/https://books.google.com/books?id=2M4qIUTITI0C&pg=PA98 |url-status=live }}</ref> In the case of motion in a central potential the Lagrangian centrifugal force has the same form as the fictitious centrifugal force derived in a co-rotating frame.<ref name=Bini1997>See p. 5 in {{cite journal |title=The intrinsic derivative and centrifugal forces in general relativity: I. Theoretical foundations |author1=Donato Bini |author2=Paolo Carini |author3=Robert T Jantzen |journal= International Journal of Modern Physics D |volume=6 |year=1997 |arxiv=gr-qc/0106014v1 |issue=1 |pages=143–198 |bibcode=1997IJMPD...6..143B |doi=10.1142/S021827189700011X |s2cid=10652293 |url=https://cds.cern.ch/record/503373 |type=Submitted manuscript }}. The companion paper is {{cite journal |title=The intrinsic derivative and centrifugal forces in general relativity: II. Applications to circular orbits in some stationary axisymmetric spacetimes |author1=Donato Bini |author2=Paolo Carini |author3=Robert T Jantzen |journal= International Journal of Modern Physics D |volume=6 |year=1997 |arxiv=gr-qc/0106014v1 |issue=1 |pages=143–198 |bibcode=1997IJMPD...6..143B |doi=10.1142/S021827189700011X |s2cid=10652293 |url=https://cds.cern.ch/record/503373 |type=Submitted manuscript |access-date=2023-06-21 |archive-date=2021-04-29 |archive-url=https://web.archive.org/web/20210429005245/http://cds.cern.ch/record/503373 |url-status=live }}</ref> However, the Lagrangian use of "centrifugal force" in other, more general cases has only a limited connection to the Newtonian definition.

In another instance the term refers to the [[reaction (physics)|reaction]] [[force]] to a centripetal force, or [[reactive centrifugal 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 laws of motion#Newton's third law|Newton's third law of motion]], the body in curved motion exerts an equal and opposite force on the other body. This [[reaction (physics)|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.<ref name=Mook>{{Cite book |last=Mook |first=Delo E. |url= https://books.google.com/books?id=QnJqIyk_dzIC&pg=PA47 |title=Inside relativity |date=1987 |publisher=Princeton University Press | author2 = Thomas Vargish |isbn=0-691-08472-6 | location = Princeton, N.J. | oclc=16089285 | page=47 |access-date=2016-03-11 |archive-date=2024-10-07 |archive-url=https://web.archive.org/web/20241007142711/https://books.google.com/books?id=QnJqIyk_dzIC&pg=PA47#v=onepage&q&f=false |url-status=live 47}}</ref><ref name=Scott>{{cite news | title = Centrifugal Forces and Newton's Laws of Motion | volume = 25 | author = G. David Scott | publisher = American Journal of Physics | year = 1957 | page = 325 | url = http://www.deepdyve.com/lp/american-association-of-physics-teachers/centrifugal-forces-and-newton-s-laws-of-motion-0bO8fgiEUy }}

This reaction force is sometimes described as a ''centrifugal inertial reaction'',<ref name=Roche>{{cite journal |last = Roche |first= John |date= September 2001|url =http://www.iop.org/EJ/article/0031-9120/36/5/305/pe1505.pdf|title =Introducing motion in a circle | journal= Physics Education | volume = 43|number =5|pages = 399–405|doi= 10.1088/0031-9120/36/5/305 |bibcode= 2001PhyEd..36..399R |s2cid= 250827660 }}</ref><ref>{{Cite journal | title = Physics, the pioneer science | journal = American Journal of Physics | volume = 1 | issue = 8 | author = Lloyd William Taylor | year = 1959 | page = 173 | url = https://books.google.com/books?id=fp84AAAAIAAJ&q=%22centrifugal+inertial+reaction%22 | bibcode = 1961AmJPh..29..563T | doi = 10.1119/1.1937847 | url-access = subscription }}</ref> 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<ref name=Bowser>{{cite book | title = An elementary treatise on analytic mechanics: with numerous examples | author = Edward Albert Bowser | publisher = D. Van Nostrand Company | year = 1920 | edition = 25th | page = 357 | url = https://books.google.com/books?id=mE4GAQAAIAAJ&pg=PA357 | access-date = 2020-11-09 | archive-date = 2024-10-07 | archive-url = https://web.archive.org/web/20241007143122/https://books.google.com/books?id=mE4GAQAAIAAJ&pg=PA357#v=onepage&q&f=false | url-status = live }}</ref><ref name=Angelo>{{cite book | title=Robotics: a reference guide to the new technology | url=https://books.google.com/books?id=73kNFV4sDx8C&pg=PA267 | page=267 | author=Joseph A. Angelo | isbn=978-1-57356-337-6 | year=2007 | publisher=Greenwood Press | access-date=2020-11-09 | archive-date=2024-10-07 | archive-url=https://web.archive.org/web/20241007143207/https://books.google.com/books?id=73kNFV4sDx8C&pg=PA267 | url-status=live }}</ref> although this usage is deprecated in elementary mechanics.<ref name = Rogers> {{cite book | title = Physics for the Inquiring Mind | url = https://archive.org/details/physicsforinquir00roge | url-access = registration | author = Eric M Rogers | publisher = Princeton University Press | year = 1960 | page = [https://archive.org/details/physicsforinquir00roge/page/302 302] }}</ref>