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((Short description|Force on objects moving within a reference frame that rotates with respect to an inertial frame.))
((Lead too long|date=October 2023))
((Classical mechanics|cTopic=Core topics))

A '''fictitious force''' is a [[force]] that appears to act on a mass whose motion is described using a [[non-inertial reference frame|non-inertial frame of reference]], such as a linearly accelerating or [[rotating reference frame]].<ref>((cite web |title=What is a "fictitious force"? |url=https://www.scientificamerican.com/article/what-is-a-fictitious-force/ |access-date=2021-12-14 |website=Scientific American |language=en))</ref>
Fictitious forces are invoked to maintain the validity and thus use of [[Newton's laws of motion#Second|Newton's second law of motion]], in frames of reference which are ''not'' inertial.<ref>((cite web |title=Fictitious force - Britannica |url=https://www.britannica.com/science/inertial-force))</ref>

Passengers in a vehicle accelerating in the forward direction may perceive they are acted upon by a force moving them into the direction of the backrest of their seats for instance. An example in a rotating reference frame may be the impression that it is a force which seems to move objects outward toward the rim of a centrifuge or carousel. <!--These apparent forces are examples of fictitious forces.-->

The fictitious force <!--'''F'''--> called a '''pseudo force''' might also be referred to as a [[body force]]. It is due to an object's [[inertia]] when the reference frame does not move inertially any more but begins to accelerate relative to the free object. In terms of the example of the passenger vehicle, a pseudo force seems to be active just before the body touches the backrest of the seat in the car. A person in the car leaning forward first moves a bit backward in relation to the already accelerating car, before touching the backrest. The motion in this short period just seems to be the result of a force on the person; i.e., it is a pseudo force. A pseudo force does not arise from any [[force carrier|physical interaction]] between two objects, such as [[electromagnetism]] or contact forces. It's just a consequence of the acceleration '''a''' of the physical object the [[non-inertial reference frame]] is connected to, i.e. the vehicle in this case. From the viewpoint of the respective accelerating frame, an acceleration of the inert object appears to be present, apparently requiring a "force" for this to have happened.

As stated by Iro:<ref name=Iro>((cite book |author=Harald Iro |title=A Modern Approach to Classical Mechanics |page=180 |url=https://books.google.com/books?id=-L5ckgdxA5YC&q=force+%22non-inertial+%22&pg=PA179 |isbn=981-238-213-5 |year=2002 |publisher=World Scientific))</ref>
((Blockquote|Such an additional force due to nonuniform relative motion of two reference frames is called a ''pseudo-force''.|Harald Iro in ''A Modern Approach to Classical Mechanics'' p. 180))

The pseudo force on an object arises as an imaginary influence when the frame of reference used to describe the object's motion is accelerating compared to a non-accelerating frame. The pseudo force "explains," using Newton's second law mechanics, why an object does not follow Newton's second law and "floats freely" as if weightless. As a frame may accelerate in any arbitrary way, so may pseudo forces also be as arbitrary (but only in direct response to the acceleration of the frame). An example of a pseudo force as defined by Iro is the [[Coriolis force]], maybe better to be called: the Coriolis effect.<ref>Britannica, ((cite web |title=Coriolis force |url=https://www.britannica.com/science/Coriolis-force))</ref><ref>Harvard University lecture demo, ((cite web |title=Coriolis force |url=https://sciencedemonstrations.fas.harvard.edu/presentations/coriolis-force#:~:text=The%20Coriolis%20force%20is%20a,to%20deflect%20its%20motion%20sideways.))</ref><ref>ThoughtCo website, ((cite web |title=Coriolis effect |url=https://www.thoughtco.com/what-is-the-coriolis-effect-1435315))</ref> The [[gravitational field|gravitational force]] would also be a fictitious force (pseudo force), based upon a field model in which particles distort [[spacetime]] due to their mass, such as in the theory of [[general relativity]].

Assuming [[Newton's laws of motion|Newton's second law]] in the form '''F'''&nbsp;=&nbsp;''m'''''a''', fictitious forces are always proportional to the mass ''m''.

The fictitious force that has been called an ''' inertial force'''<ref>((cite web |title=Inertial force - Britannica |url=https://www.britannica.com/science/inertial-force))</ref><ref name=Born>((cite book |title=Einstein's Theory of Relativity |author1=Max Born |author2=Günther Leibfried |url=https://archive.org/details/einsteinstheory000born |url-access=registration |quote=inertial forces. |isbn=0-486-60769-0 |publisher=Courier Dover Publications |location=New York |year=1962 |pages=[https://archive.org/details/einsteinstheory000born/page/76 76]–78))</ref><ref>[https://pwg.gsfc.nasa.gov/stargaze/Sframes2.htm NASA notes:(23) Accelerated Frames of Reference: Inertial Forces]</ref> is also referred to as a [[D'Alembert's principle|'''d'Alembert force''']],<ref name=Lanczos1>((cite book |title=The Variational Principles of Mechanics |url=https://books.google.com/books?id=ZWoYYr8wk2IC&q=%22Euler+force%22&pg=PA103 |page=100 |year=1986 |author=Cornelius Lanczos |isbn=0-486-65067-7 |publisher=Courier Dover Publications |location=New York))</ref><ref>((cite web |last=Seligman |first=Courtney |url=http://cseligman.com/text/physics/fictitious.htm |title=Fictitious Forces |access-date=2007-09-03))</ref> or sometimes as a pseudo force.<ref name="FEYN">[https://feynmanlectures.caltech.edu/I_12.html#Ch12-S5 The Feynman Lectures on Physics Vol. I Ch. 12-5: Pseudo forces]</ref> D'Alembert's principle is just another way of formulating Newton's second law of motion. It defines an inertial force as the negative of the product of mass times acceleration, just for the sake of easier calculations.

(A d'Alembert force is not to be confused with a [[contact force]] arising from the physical interaction between two objects, which is the subject of Newton's third law – 'action is [[Reaction (physics)|reaction]]'.<ref>Physics Forum, ((cite web |url=https://www.physicsforums.com/threads/inertia-and-action-reaction.1000443/ |title=Inertia and Newton's third law|date=3 March 2021 ))</ref><ref>Physics stack exchange, ((cite web |url=https://physics.stackexchange.com/questions/336225/why-is-normal-force-not-the-same-as-newtons-third-law |title=about Newton's third law))</ref> In terms of the example of the passenger vehicle above, a contact force emerges when the body of the passenger touches the backrest of the seat in the car. It is present for as long as the car is accelerated.)

Four fictitious forces have been defined for frames accelerated in commonly occurring ways:
* one caused by any acceleration relative to the origin in a straight line (rectilinear [[acceleration]]);<ref>The term ''d'Alembert force'' often is limited to this case. See Lanczos, for example.</ref>
* two involving rotation: [[centrifugal force]] and [[Coriolis force|Coriolis effect]]
* and a fourth, called the [[Euler force]], caused by a variable rate of rotation, should that occur.

==Background==
((See also|Absolute space and time|Preferred frame))
The role of fictitious forces in Newtonian mechanics is described by [[Marie-Antoinette Tonnelat|Tonnelat]]:<ref name=Tonnelat>((cite book |title=The Principles of Electromagnetic Theory and Relativity |author=Marie-Antoinette Tonnelat |year=2002 |publisher=Springer |isbn=90-277-0107-5 |url=https://books.google.com/books?id=G2KICGm9TbUC&q=%22For+Newton,+the%22 |page=113))</ref>
((Blockquote|For Newton, the appearance of acceleration always indicates the existence of absolute motion – absolute motion of matter where ''real'' forces are concerned; absolute motion of the reference system, where so-called ''fictitious'' forces, such as inertial forces or those of Coriolis, are concerned.|Marie-Antoinette Tonnelat in ''The Principles of Electromagnetic Theory and Relativity'', p.113))

Fictitious forces arise in [[classical mechanics]] and [[special relativity]] in all non-inertial frames.
Inertial frames are [[preferred frame|privileged]] over non-inertial frames because they do not have physics whose causes are outside of the system, while non-inertial frames do. Fictitious forces, or physics whose cause is outside of the system, are no longer necessary in [[general relativity]], since these physics are explained with the [[geodesics in general relativity|geodesics]] of [[spacetime]]: "The field of all possible space-time null geodesics or photon paths unifies the absolute local non-rotation standard throughout space-time.".<ref>((citation|title=Mach's Principle II, p.1, p.9|first1=James G.|last1=Gilson|date=September 1, 2004|arxiv=physics/0409010|bibcode=2004physics...9010G))</ref>

==On Earth==
((See also|Centrifugal force|Coriolis force|Euler force))
The surface of the Earth is a [[rotating reference frame]]. To solve [[classical mechanics]] problems exactly in an Earthbound reference frame, three fictitious forces must be introduced: the [[Coriolis force]], the [[centrifugal force]] (described below) and the [[Euler force]]. The Euler force is typically ignored because the variations in the angular velocity of the rotating surface of the Earth are usually insignificant. Both of the other fictitious forces are weak compared to most typical forces in everyday life, but they can be detected under careful conditions. For example, [[Léon Foucault]] used his [[Foucault pendulum]] to show that a [[Coriolis force]] results from the Earth's rotation.
If the Earth were to rotate twenty times faster (making each day only ~72 minutes long), people could easily get the impression that such fictitious forces were pulling on them, as on a spinning carousel; people in temperate and tropical latitudes would, in fact, need to hold on in order to avoid being launched into orbit by the centrifugal force.

==Detection of non-inertial reference frame==
((See also|Inertial frame of reference))
Observers inside a closed box that is moving with a constant [[velocity]] cannot detect their own motion; however, observers within an accelerating reference frame can detect that they are in a non-inertial reference frame from the fictitious forces that arise. For example, for straight-line acceleration [[Vladimir Arnold]] presents the following theorem:<ref name="Arnol'd">((cite book |title=Mathematical Methods of Classical Mechanics |url=https://books.google.com/books?id=Pd8-s6rOt_cC |author=Vladimir Igorevich Arnold |publisher=Springer |location=Berlin |isbn=0-387-96890-3 |year=1989 |pages=§27 pp. 129 ff))</ref>
((Blockquote|In a coordinate system ''K'' which moves by translation relative to an inertial system ''k'', the motion of a mechanical system takes place as if the coordinate system were inertial, but on every point of mass ''m'' an additional "inertial force" acted: '''F'''&nbsp;((=)) −''m'''''a''', where '''a''' is the acceleration of the system ''K''.))
Other accelerations also give rise to fictitious forces, as described mathematically [[#Mathematical derivation of fictitious forces|below]]. The physical explanation of motions in an inertial frame is the simplest possible, requiring no fictitious forces: fictitious forces are zero, providing a means to distinguish inertial frames from others.<ref name=transformations>As part of the requirement of simplicity, to be an inertial frame, in all other frames that differ only by a uniform rate of translation, the description should be of the same form. However, in the Newtonian system the [[Galilean transformation]] connects these frames and in the special theory of relativity the [[Lorentz transformation]] connects them. The two transformations agree for speeds of translation much less than the [[speed of light]].</ref>

An example of the detection of a non-inertial, rotating reference frame is the precession of a [[Foucault pendulum]]. In the non-inertial frame of the Earth, the fictitious [[Coriolis force]] is necessary to explain observations. In an inertial frame outside the Earth, no such fictitious force is necessary.
<!--
==Example Acceleration in a straight line==-->
<!--[[File:Accelerating car.PNG|thumb|upright=1.2|Figure 1: ''Top panel'': accelerating car of mass ''M'' with passenger of mass ''m''. The force from the axle is (''m'' + ''M'')'''a'''. In the inertial frame, this is the only force on the car and passenger.<br /> ''Center panel'': an exploded view in the inertial frame. The passenger is subject to the accelerating force ''m'''''a'''. The seat (assumed of negligible mass) is compressed between the reaction force –''m'''''a''' and the applied force from the car ''m'''''a'''. The car is subject to the net acceleration force ''M'''''a''' that is the difference between the applied force (''m'' + ''M'')'''a''' from the axle and the reaction from the seat −''m'''''a'''. <br />''Bottom panel'': an exploded view in the non-inertial frame. In the non-inertial frame where the car is not accelerating, the force from the axle is balanced by a fictitious backward force −(''m'' + ''M'')'''a''', a portion −''M'''''a''' applied to the car, and −''m'''''a''' to the passenger. The car is subject to the fictitious force −''M'''''a''' and the force (''m'' + ''M'')'''a''' from the axle. The sum of these forces ''m'''''a''' is applied to the seat, which exerts a reaction −''m'''''a''' upon the car, so zero net force is applied to the car. The seat (assumed massless) transmits the force ''m'''''a''' to the passenger, who is subject also to the fictitious force −''m'''''a''', resulting in zero net force on the passenger. The passenger exerts a reaction force −''m'''''a''' upon the seat, which is therefore compressed. In all frames, the compression of the seat is the same, and the force delivered by the axle is the same.]]-->
<!--Figure 1 (top) shows an [[acceleration|accelerating]] car with a passenger in it being pushed back into the seat already. In an inertial frame of reference attached to the road, there is no physical force moving the rider backward. However, in the rider's non-inertial reference frame attached to the accelerating car, there ''is'' a backward fictitious force. We mention two possible reasons for the force to clarify its (the force's) existence:-->
<!--<ref name=Motz>((cite book |title=The Concepts of Science: From Newton to Einstein |author1=Lloyd Motz |author2=Jefferson Hane Weaver |url=https://books.google.com/books?id=SnufjnGnqD8C&q=history+velocity+%22law+of+inertia%22&pg=PA95 |page=101 |isbn=0-7382-0834-5 |publisher=Basic Books |year=2002))</ref>-->
<!--* Figure 1 (centre panel). To an observer at rest on an [[inertial reference frame]] (like the ground), the car will seem to accelerate. In order for the passenger to stay inside the car, a force must be exerted on the passenger. This force is exerted by the seat, which has started to move forward with the car and is compressed against the passenger until it transmits the full force to keep the passenger moving with the car. Thus, the forces exerted by the seat are unbalanced, so the passenger is accelerating in this frame.
* Figure 1 (bottom panel). From the point of view of the interior of the car, an accelerating reference frame, there is a fictitious force pushing the passenger backward, with a magnitude equal to the [[mass]] of the passenger times the acceleration of the car. This force pushes the passenger back into the seat until the seat compresses and provides an equal and opposite force. Thereafter, the passenger is stationary in this frame, because the fictitious force and the real force of the seat are balanced.

The accelerating frame is discovered to be non-inertial because, in the accelerating frame, everything appears to be subject to zero net force, and nothing moves. Nonetheless, compression of the seat is observed and is explained in the accelerating frame (and in an inertial frame) by the force of acceleration on the seat from the car on one side, and the opposing force of reaction to acceleration by the passenger on the other. Identification of the accelerating frame as non-inertial cannot be based simply on the compression of the seat, which all observers can explain; rather it is based on the ''simplicity'' of the physical explanation for this compression.

The explanation of the seat compression in the accelerating frame requires not only the thrust from the axle of the car but additional (fictitious) forces. In an inertial frame, only the thrust from the axle is necessary. Therefore, the inertial frame has a ''simpler'' physical explanation (not necessarily a simpler mathematical formulation), indicating the accelerating frame is a non-inertial frame of reference. In other words, in the inertial frame, fictitious forces are zero. See [[inertial frame]].

This example illustrates how fictitious forces arise from switching from an inertial to a non-inertial reference frame. Calculations of physical quantities (compression of the seat, required force from the axle) made in any frame give the same answers, but in some cases, calculations are easier to make in a non-inertial frame. (In this simple example, the calculations are equally complex for the two frames described.)
:{|class="toccolours collapsible collapsed" width="60%" style="text-align:left"
!''Animation:'' driving from block to block
|-
|[[File:Carframe.gif|thumb|360px|center|Map and car frame perspectives of physical (red) and fictitious (blue) forces for a car driving from one stop sign to the next]]
In this illustration, the car accelerates after a stop sign until midway up the block, at which point the driver is immediately off the accelerator and onto the brake so as to make the next stop.
|}
-->

==Example concerning Circular motion==
((See also|Centrifugal force|Coriolis force))
[[File:Corioliskraftanimation.gif|frame|In the inertial frame of reference (upper part of the picture), the black ball moves in a straight line. However, the observer (brown dot) who is standing in the rotating/non-inertial frame of reference (lower part of the picture) sees the object as following a curved path due to the Coriolis or centrifugal forces present in this frame.]]
The effect of a fictitious force also occurs when a car takes [[circular motion|the bend]]. Observed from a non-inertial frame of reference attached to the car, the fictitious force called the [[centrifugal force (fictitious)|centrifugal force]] appears. As the car enters a left turn, a suitcase first on the left rear seat slides to the right rear seat and then continues until it comes into contact with the closed door on the right. This motion marks the phase of the fictitious centrifugal force as it is the inertia of the suitcase which plays a role in this piece of movement. It may seem that there must be a force responsible for this movement, but actually, this movement arises because of the inertia of the suitcase, which is (still) a 'free object' within an already accelerating frame of reference.
After the suitcase has come into contact with the closed door of the car, the situation with the emergence of [[contact force]]s becomes current. The centripetal force on the car is now also transferred to the suitcase and the situation of Newton's third law comes into play, with the centripetal force as the action part and with the so-called [[reactive centrifugal force]] as the reaction part. The reactive centrifugal force is also due to the [[inertia]] of the suitcase. Now however the inertia appears in the form of a manifesting resistance to a change in its state of motion.
<ref>Science of everyday things, ((cite web |url=https://archive.org/details/ScienceOfEverydayThingsVol2-RealLifePhysics/page/n51/mode/2up?view=theater |title=centripetal force, pp 48-49))</ref>

Suppose a few miles further the car is moving at constant speed travelling a roundabout, again and again, then the occupants will feel as if they are being pushed to the outside of the vehicle by the (reactive) centrifugal force, away from the centre of the turn.

The situation can be viewed from inertial as well as from non-inertial frames.

* From the viewpoint of an inertial reference frame stationary with respect to the road, the car is accelerating toward the centre of the circle. It is accelerating, because the ''direction'' of the velocity is changing, despite the car having constant speed. This inward acceleration is called [[centripetal acceleration]], it requires a [[centripetal force]] to maintain the circular motion. This force is exerted by the ground upon the wheels, in this case, from the [[friction]] between the wheels and the road.<ref>The force in this example is known as ''[[ground reaction force|ground reaction]],'' and it could exist even without friction, for example, a sledge running down a curve of a bobsled track.</ref> The car is accelerating, due to the unbalanced force, which causes it to move in a circle. (See also [[banked turn]].)
* From the viewpoint of a rotating frame, moving with the car, a fictitious centrifugal force appears to be present pushing the car toward the outside of the road (and pushing the occupants toward the outside of the car). The centrifugal force balances the friction between wheels and the road, making the car stationary in this non-inertial frame.

A classic example of a fictitious force in circular motion is the experiment of [[rotating spheres]] tied by a cord and spinning around their centre of mass. In this case,<!-- as with the linearly accelerating car example, --> the identification of a rotating, non-inertial frame of reference can be based upon the vanishing of fictitious forces. In an inertial frame, fictitious forces are not necessary to explain the tension in the string joining the spheres. In a rotating frame, Coriolis and centrifugal forces must be introduced to predict the observed tension.

In the rotating reference frame perceived on the surface of the Earth, a centrifugal force reduces the apparent force of gravity by about one part in a thousand, depending on latitude. This reduction is zero at the poles, maximum at the [[equator]].
:{|class="toccolours collapsible collapsed" width="60%" style="text-align:left"
!''Animation:'' object released from a carousel
|-
|[[File:Spinframe.gif|thumb|360px|center|Map and spin frame perspectives of physical (red) and fictitious (blue) forces for an object released from a carousel]]
For someone in the map perspective only one force is sufficient to explain the motion: the red arrow: [[centripetal force]]. After release, the number of forces is zero. For someone in the spinning frame the object moves in a complicated way that needs a [[centrifugal force]]: the blue arrow.''Note:'' With some browsers, hitting [Esc] will freeze the motion for more detailed analysis.  However, the page may have to be reloaded to restart.
|}

The fictitious [[Coriolis force]], which is observed in rotational frames, is ordinarily visible only in very large-scale motion like the projectile motion of long-range guns or the circulation of the Earth's atmosphere (see [[Rossby number]]). Neglecting air resistance, an object dropped from a 50-meter-high tower at the equator will fall 7.7 millimetres eastward of the spot below where it is dropped because of the Coriolis force.<ref name=Kleppner>((cite book |author1=Daniel Kleppner |author2=Robert J. Kolenkow |year=1973 |title=An Introduction to Mechanics |publisher=McGraw-Hill |page=[https://archive.org/details/introductiontome00dani/page/363 363] |isbn=0-07-035048-5 |url=https://archive.org/details/introductiontome00dani/page/363))</ref>
<!--In the case of distant objects and a rotating reference frame, what must be taken into account is the resultant force of centrifugal and Coriolis forces. Consider a distant star observed from a rotating spacecraft. In the reference frame co-rotating with the spacecraft, the distant star appears to move along a circular trajectory around the spacecraft. The apparent motion of the star is an apparent centripetal acceleration. Just like in the example above of the car in a circular motion, the centrifugal force has the same magnitude as the fictitious centripetal force but is directed in the opposite, centrifugal direction. In this case, the Coriolis force is twice the magnitude of the centrifugal force, and it points in the centripetal direction. The vector sum of the centrifugal force and the Coriolis force is the total fictitious force, which in this case points in the centripetal direction.-->

==Fictitious forces and work==
Fictitious forces can be considered to do [[mechanical work|work]], provided that they move an object on a [[trajectory]] that changes its [[energy]] from [[potential energy|potential]] to [[kinetic energy|kinetic]]. For example, consider some persons in rotating chairs holding a weight in their outstretched hands. If they pull their hand inward toward their body, from the perspective of the rotating reference frame, they have done work against the centrifugal force. When the weight is let go, it spontaneously flies outward relative to the rotating reference frame, because the centrifugal force does work on the object, converting its potential energy into kinetic. From an inertial viewpoint, of course, the object flies away from them because it is suddenly allowed to move in a straight line. This illustrates that the work done, like the total potential and kinetic energy of an object, can be different in a non-inertial frame than in an inertial one.

==Gravity as a fictitious force==
((Main|General relativity))
The notion of "fictitious force" also arises in Einstein's [[general theory of relativity]].<ref name=Rohrlich>((cite book |title=Fictitional Forces and apparent gravitational fields |author=Fritz Rohrlich |page=40 |url=https://books.google.com/books?id=lJrPrP6L6tQC&q=acceleration+transformation+formula+%22Coriolis+force%22&pg=PA39 |isbn=978-981-270-004-9 |year=2007 |publisher=World Scientific |location=Singapore))</ref><ref name=Stephani>((cite book |title=Introduction to Special and General Relativity - geodesic deviation p. 104-105 |author=Hans Stephani |url=https://books.google.com/books?id=WAW-4nd-OeIC&q=acceleration+transformation+formula+%22Coriolis+force%22&pg=PA104 |page=105 |isbn=0-521-01069-1 |year=2004 |publisher=Cambridge University Press |location=Cambridge UK))</ref> All fictitious forces are proportional to the mass of the object upon which they act, which is also true for [[gravity]].<ref name="Eötvös">

The gravitational mass and the inertial mass are found experimentally to be equal to each other within experimental error.</ref><ref>Motz and Weaver, ((cite book |url=https://books.google.com/books?id=_TQBCAAAQBAJ&q=We+consider+now+a+group&pg=PA81 |title=Example train and gravity, p 101 |isbn=9781489963338 |last1=Motz |first1=Lloyd |last2=Weaver |first2=Jefferson Hane |date=11 November 2013|publisher=Springer ))</ref> This led [[Albert Einstein]] to wonder whether gravity could be modeled as a fictitious force. He noted that a [[freefall]]ing observer in a closed box would not be able to detect the force of gravity; hence, [[Proper acceleration|freefalling reference frames]] are equivalent to inertial reference frames (the [[equivalence principle]]). Developing this insight, Einstein formulated a theory with gravity as a fictitious force, and attributed the apparent acceleration due to gravity to the [[curvature]] of [[spacetime]]. This idea underlies Einstein's theory of [[general relativity]]. See the [[Eötvös experiment]].
:{|class="toccolours collapsible collapsed" width="60%" style="text-align:left"
!''Animation:'' ball that rolls off a cliff
|-
|[[File:Shellframe.gif|thumb|360px|center|Rain and shell frame perspectives of physical (red) and fictitious (blue) forces for an object that rolls off a cliff.]] ''Note:'' The rain frame perspective here, rather than being that of a raindrop, is more like that of a trampoline jumper whose trajectory tops out just as the ball reaches the edge of the cliff.  The shell frame perspective<ref>Edwin F. Taylor and John Archibald Wheeler (2000) ''Exploring black holes'' (Addison Wesley Longman, NY) ((ISBN|0-201-38423-X))</ref> may be familiar to planet dwellers who rely minute by minute on upward physical forces from their environment, to protect them from the geometric acceleration due to curved spacetime.
|}

==Mathematical derivation of fictitious forces ((anchor|coordinates))==
[[File:Moving coordinate system.PNG|thumb|upright=1.2|Figure 2: An object located at '''x'''<sub>A</sub> in inertial frame ''A'' is located at location '''x'''<sub>B</sub> in accelerating frame ''B''. The origin of frame ''B'' is located at '''X'''<sub>AB</sub> in frame ''A''. The orientation of frame ''B'' is determined by the unit vectors along its coordinate directions, '''u'''<sub>j</sub> with ''j'' = 1, 2, 3. Using these axes, the coordinates of the object according to frame ''B'' are '''x'''<sub>B</sub> = ( '''''x'''''<sub>1</sub>, '''''x'''''<sub>2</sub>, '''''x'''''<sub>3</sub>).]]

===General derivation===
Many problems require use of noninertial reference frames, for example, those involving satellites<ref name=Isidon>((cite book |title=Robust Autonomous Guidance: An Internal Model Approach |author1=Alberto Isidori |author2=Lorenzo Marconi |author3=Andrea Serrani |page= 61 |url=https://books.google.com/books?id=Mo0YhjLeTA0C&q=orbit+%22coordinate+system%22&pg=RA2-PA60
|isbn=1-85233-695-1 |publisher=Springer |year=2003))</ref><ref name=Ying>((cite book |title=Advanced Dynamics |author=Shuh-Jing Ying |url=https://archive.org/details/advanceddynamics0000ying |url-access=registration |quote=orbit coordinate system. |page=[https://archive.org/details/advanceddynamics0000ying/page/172 172] |isbn=1-56347-224-4 |publisher=American Institute of Aeronautics, and Astronautics |location=Reston VA |year=1997))</ref> and particle accelerators.<ref name=Bryant>((cite book |title=The Principles of Circular Accelerators and Storage Rings |author1=Philip J. Bryant |author2=Kjell Johnsen |page=xvii |url=https://books.google.com/books?id=6P-UWLmfD4wC&q=orbit+%22coordinate+system%22&pg=PR17 |isbn=0-521-35578-8 |publisher=Cambridge University Press |location=Cambridge UK |year=1993 ))((Dead link|date=April 2024 |bot=InternetArchiveBot |fix-attempted=yes ))</ref> Figure 2 shows a particle with [[mass]] ''m'' and [[position (vector)|position]] [[vector (geometric)|vector]] '''x'''<sub>A</sub>(''t'') in a particular [[inertial frame]] A.  Consider a non-inertial frame B whose origin relative to the inertial one is given by '''X'''<sub>AB</sub>(''t'').  Let the position of the particle in frame B be '''x'''<sub>B</sub>(''t''). What is the force on the particle as expressed in the coordinate system of frame B?<ref name=Fetter>((cite book |author1=Alexander L Fetter |author2=John D Walecka |title=Theoretical Mechanics of Particles and Continua |publisher=Courier Dover Publications |url=https://books.google.com/books?id=olMpStYOlnoC |year=2003 |isbn=0-486-43261-0 |pages=33–39))</ref><ref name=Lim>((cite book |title=Problems and Solutions on Mechanics: Major American Universities Ph.D. Qualifying Questions and Solutions |author1=Yung-kuo Lim |author2=Yuan-qi Qiang |isbn=981-02-1298-4 |page=183 |url=https://books.google.com/books?id=93b3cjVJ2l4C&dq=%22Coriolis+force%22&pg=PA183 |publisher=World Scientific |location=Singapore |year=2001))</ref>

To answer this question, let the coordinate axis in B be represented by unit vectors '''u'''<sub>''j''</sub> with ''j'' any of {&thinsp;1,&thinsp;2,&thinsp;3&thinsp;} for the three coordinate axes. Then
:<math> \mathbf{x}_\mathrm{B} = \sum_{j=1}^3 x_j \mathbf{u}_j \, . </math>

The interpretation of this equation is that '''x'''<sub>B</sub> is the vector displacement of the particle as expressed in terms of the coordinates in frame B at the time ''t''. From frame A the particle is located at:
:<math>\mathbf{x}_\mathrm{A} =\mathbf{X}_\mathrm{AB} + \sum_{j=1}^3 x_j \mathbf{u}_j \, . </math>

As an aside, the unit vectors {&thinsp;'''u'''<sub>''j''</sub>&thinsp;} cannot change magnitude, so derivatives of these vectors express only rotation of the coordinate system B. On the other hand, vector '''X'''<sub>AB</sub> simply locates the origin of frame B relative to frame A, and so cannot include rotation of frame B.

Taking a time derivative, the velocity of the particle is:
:<math> \frac {d \mathbf{x}_\mathrm{A)){dt} =\frac{d \mathbf{X}_\mathrm{AB)){dt} + \sum_{j=1}^3 \frac{dx_j}{dt} \mathbf{u}_j + \sum_{j=1}^3 x_j \frac{d \mathbf{u}_j}{dt} \, . </math>

The second term summation is the velocity of the particle, say '''v'''<sub>B</sub> as measured in frame B. That is:
:<math> \frac {d \mathbf{x}_\mathrm{A)){dt} =\mathbf{v}_\mathrm{AB}+ \mathbf{v}_\mathrm{B} + \sum_{j=1}^3 x_j \frac{d \mathbf{u}_j}{dt}. </math>

The interpretation of this equation is that the velocity of the particle seen by observers in frame A consists of what observers in frame B call the velocity, namely '''v'''<sub>B</sub>, plus two extra terms related to the rate of change of the frame-B coordinate axes. One of these is simply the velocity of the moving origin '''v'''<sub>AB</sub>. The other is a contribution to velocity due to the fact that different locations in the non-inertial frame have different apparent velocities due to the rotation of the frame; a point seen from a rotating frame has a rotational component of velocity that is greater the further the point is from the origin.

To find the acceleration, another time differentiation provides:
:<math> \frac {d^2 \mathbf{x}_\mathrm{A)){dt^2} = \mathbf{a}_\mathrm{AB}+\frac {d\mathbf{v}_\mathrm{B)){dt} + \sum_{j=1}^3 \frac {dx_j}{dt} \frac{d \mathbf{u}_j}{dt} + \sum_{j=1}^3 x_j \frac{d^2 \mathbf{u}_j}{dt^2}. </math>

Using the same formula already used for the time derivative of '''x'''<sub>B</sub>, the velocity derivative on the right is:
:<math>\frac {d\mathbf{v}_\mathrm{B)){dt} =\sum_{j=1}^3 \frac{d v_j}{dt} \mathbf{u}_j+ \sum_{j=1}^3 v_j \frac{d \mathbf{u}_j}{dt} =\mathbf{a}_\mathrm{B} + \sum_{j=1}^3 v_j \frac{d \mathbf{u}_j}{dt}. </math>

((anchor|Eq. 1))Consequently,
((Numbered block||<math display="block"> \frac {d^2 \mathbf{x}_\mathrm{A)){dt^2}=\mathbf{a}_\mathrm{AB}+\mathbf{a}_\mathrm{B} + 2\ \sum_{j=1}^3 v_j \frac{d \mathbf{u}_j}{dt} + \sum_{j=1}^3 x_j \frac{d^2 \mathbf{u}_j}{dt^2}.</math>|((EquationRef|1))))

The interpretation of this equation is as follows: the acceleration of the particle in frame A consists of what observers in frame B call the particle acceleration '''a'''<sub>B</sub>, but in addition, there are three acceleration terms related to the movement of the frame-B coordinate axes: one term related to the acceleration of the origin of frame B, namely '''a'''<sub>AB</sub>, and two terms related to the rotation of frame B. Consequently, observers in B will see the particle motion as possessing "extra" acceleration, which they will attribute to "forces" acting on the particle, but which observers in A say are "fictitious" forces arising simply because observers in B do not recognize the non-inertial nature of frame B.

The factor of two in the Coriolis force arises from two equal contributions: (i) the apparent change of an inertially constant velocity with time because rotation makes the direction of the velocity seem to change (a ''d'''''v'''<sub>B</sub>/d''t'' term) and (ii) an apparent change in the velocity of an object when its position changes, putting it nearer to or further from the axis of rotation (the change in <math display="inline">\sum x_j\, d\mathbf{u}_j/dt</math> due to change in ''x <sub>j</sub>'' ).

To put matters in terms of forces, the accelerations are multiplied by the particle mass:
:<math>\mathbf{F}_\mathrm{A} = \mathbf{F}_\mathrm{B} + m\mathbf{a}_\mathrm{AB}+ 2m \sum_{j=1}^3 v_j \frac{d \mathbf{u}_j}{dt} + m \sum_{j=1}^3 x_j \frac{d^2 \mathbf{u}_j}{dt^2}\ . </math>

The force observed in frame B, '''F'''<sub>B</sub> = ''m'''''a'''<sub>B</sub> is related to the actual force on the particle, '''F'''<sub>A</sub>, by
:<math>\mathbf{F}_\mathrm{B} = \mathbf{F}_\mathrm{A} + \mathbf{F}_\mathrm{fictitious},</math>

where:
:<math> \mathbf{F}_\mathrm{fictitious} = -m\mathbf{a}_\mathrm{AB} - 2m\sum_{j=1}^3 v_j \frac{d \mathbf{u}_j}{dt} - m \sum_{j=1}^3 x_j \frac{d^2 \mathbf{u}_j}{dt^2}\, . </math>

Thus, problems may be solved in frame B by assuming that Newton's second law holds (with respect to quantities in that frame) and treating '''F'''<sub>fictitious</sub> as an additional force.<ref name="Arnol'd"/><ref name=Taylor>((cite book |title=Classical Mechanics |author=John Robert Taylor |location=Sausalito CA |isbn=1-891389-22-X |year=2004 |url=https://books.google.com/books?id=P1kCtNr-pJsC&pg=PP1 |publisher=University Science Books |pages=343–344))</ref><ref>Kleppner pages 62–63</ref>

Below are a number of examples applying this result for fictitious forces. More examples can be found in the article on [[centrifugal force]].

===Rotating coordinate systems===
A common situation in which noninertial reference frames are useful is when the reference frame is rotating. Because such rotational motion is non-inertial, due to the acceleration present in any rotational motion, a fictitious force can always be invoked by using a rotational frame of reference. Despite this complication, the use of fictitious forces often simplifies the calculations involved.

To derive expressions for the fictitious forces, derivatives are needed for the apparent time rate of change of vectors that take into account time-variation of the coordinate axes. If the rotation of frame 'B' is represented by a vector '''Ω''' pointed along the axis of rotation with the orientation given by the [[right-hand rule]], and with magnitude given by
:<math> |\boldsymbol{\Omega} | = \frac {d \theta }{dt} = \omega (t), </math>

then the time derivative of any of the three unit vectors describing frame B is<ref name=Taylor/><ref>See for example, ((cite book |author1=JL Synge |author2=BA Griffith |title=Principles of Mechanics |url=https://archive.org/details/principlesofmech031468mbp |publisher=McGraw-Hill |edition=2nd |year=1949 |pages=[https://archive.org/details/principlesofmech031468mbp/page/n343 348]–349))</ref>
:<math> \frac {d \mathbf{u}_j (t)}{dt} = \boldsymbol{\Omega} \times \mathbf{u}_j (t), </math>

and
:<math>\frac {d^2 \mathbf{u}_j (t)}{dt^2}= \frac{d\boldsymbol{\Omega)){dt} \times \mathbf{u}_j +\boldsymbol{\Omega} \times \frac{d \mathbf{u}_j (t)}{dt} = \frac{d\boldsymbol{\Omega)){dt} \times \mathbf{u}_j+ \boldsymbol{\Omega} \times \left[ \boldsymbol{\Omega} \times \mathbf{u}_j (t) \right], </math>

as is verified using the properties of the [[vector cross product]]. These derivative formulas now are applied to the relationship between acceleration in an inertial frame, and that in a coordinate frame rotating with time-varying angular velocity ω(''t''). From the previous section, where subscript A refers to the inertial frame and B to the rotating frame, setting '''a'''<sub>AB</sub> = 0 to remove any translational acceleration, and focusing on only rotational properties (see [[#Eq. 1|Eq. 1]]):
:<math> \frac {d^2 \mathbf{x}_\mathrm{A)){dt^2}=\mathbf{a}_\mathrm{B} + 2\sum_{j=1}^3 v_j \ \frac{d \mathbf{u}_j}{dt} + \sum_{j=1}^3 x_j \frac{d^2 \mathbf{u}_j}{dt^2},</math>
:<math>\begin{align}
\mathbf{a}_\mathrm{A} &= \mathbf{a}_\mathrm{B} +\ 2\sum_{j=1}^3 v_j \boldsymbol{\Omega} \times \mathbf{u}_j (t) + \sum_{j=1}^3 x_j \frac{d\boldsymbol{\Omega)){dt} \times \mathbf{u}_j \ + \sum_{j=1}^3 x_j \boldsymbol{\Omega} \times \left[ \boldsymbol{\Omega} \times \mathbf{u}_j (t) \right] \\
&=\mathbf{a}_\mathrm{B} + 2 \boldsymbol{\Omega} \times\sum_{j=1}^3 v_j \mathbf{u}_j (t) + \frac{d\boldsymbol{\Omega)){dt} \times \sum_{j=1}^3 x_j \mathbf{u}_j + \boldsymbol{\Omega} \times \left[\boldsymbol{\Omega} \times \sum_{j=1}^3 x_j \mathbf{u}_j (t) \right].
\end{align}</math>
Collecting terms, the result is the so-called ''acceleration transformation formula'':<ref name=Gregory>((cite book |title=Classical Mechanics: An Undergraduate Text |url=https://books.google.com/books?id=uAfUQmQbzOkC&q=%22rigid+body+kinematics%22 |author=R. Douglas Gregory |year=2006 |isbn=0-521-82678-0 |publisher=Cambridge University Press |location=Cambridge UK |pages=Eq. (17.16), p. 475))</ref>
:<math>\mathbf{a}_\mathrm{A}=\mathbf{a}_\mathrm{B} + 2\boldsymbol{\Omega} \times\mathbf{v}_\mathrm{B} + \frac{d\boldsymbol{\Omega)){dt} \times \mathbf{x}_\mathrm{B} + \boldsymbol{\Omega} \times \left(\boldsymbol{\Omega} \times \mathbf{x}_\mathrm{B} \right)\, .</math>

The [[proper acceleration|physical acceleration]] '''a'''<sub>A</sub> due to what observers in the inertial frame A call ''real external forces'' on the object is, therefore, not simply the acceleration '''a'''<sub>B</sub> seen by observers in the rotational frame B, but has several additional geometric acceleration terms associated with the rotation of B. As seen in the rotational frame, the acceleration '''a'''<sub>B</sub> of the particle is given by rearrangement of the above equation as:
:<math>
\mathbf{a}_\mathrm{B} = \mathbf{a}_\mathrm{A} - 2\boldsymbol{\Omega} \times \mathbf{v}_\mathrm{B} - \boldsymbol{\Omega} \times (\boldsymbol\Omega \times  \mathbf{x}_\mathrm{B}) - \frac{d \boldsymbol\Omega}{dt} \times \mathbf{x}_\mathrm{B}.
</math>

The net force upon the object according to observers in the rotating frame is '''F'''<sub>B</sub> = ''m'''''a'''<sub>B</sub>.  If their observations are to result in the correct force on the object when using Newton's laws, they must consider that the additional force '''F'''<sub>fict</sub> is present, so the end result is '''F'''<sub>B</sub> = '''F'''<sub>A</sub> + '''F'''<sub>fict</sub>. Thus, the fictitious force used by observers in B to get the correct behaviour of the object from Newton's laws equals:
:<math>
\mathbf{F}_{\mathrm{fict)) = - 2 m \boldsymbol\Omega \times \mathbf{v}_\mathrm{B} - m \boldsymbol\Omega \times (\boldsymbol\Omega \times \mathbf{x}_\mathrm{B}) - m \frac{d \boldsymbol\Omega}{dt} \times \mathbf{x}_\mathrm{B}.
</math>

Here, the first term is the ''[[Coriolis force]]'',<ref name=Joos>((cite book |title=Theoretical Physics |author1=Georg Joos |author2=Ira M. Freeman |url=https://books.google.com/books?id=vIw5m2XuvpIC&dq=%22Coriolis+force%22&pg=PA233 |page=233 |isbn=0-486-65227-0 |publisher=Courier Dover Publications |location=New York |year=1986))</ref> the second term is the ''[[centrifugal force (fictitious)|centrifugal force]]'',<ref name=Smith>((cite book |title=Theoretical Mechanics |author=[[Percey F. Smith]] & William Raymond Longley |url=https://archive.org/details/theoreticalmech00longgoog |quote=centrifugal force theoretical. |page=[https://archive.org/details/theoreticalmech00longgoog/page/n135 118] |year=1910 |publisher=Gin |location=Boston))</ref> and the third term is the ''[[Euler force]]''.<ref name=Lanczos>((cite book |author=Cornelius Lanczos |title=The Variational Principles of Mechanics |url=https://books.google.com/books?id=ZWoYYr8wk2IC&q=%22Euler+force%22&pg=PA103 |page=103 |year=1986 |publisher=Courier Dover Publications |isbn=0-486-65067-7 |location=New York))</ref><ref name=Marsden>((cite book |author1=Jerold E. Marsden |author2=Tudor.S. Ratiu |year=1999 |edition=2nd |title= Introduction to Mechanics and Symmetry: A Basic Exposition of Classical Mechanical Systems: Texts in applied mathematics, 17 |publisher=Springer-Verlag |location =NY |url=https://books.google.com/books?id=I2gH9ZIs-3AC |isbn=0-387-98643-X |page=251))</ref>

===Orbiting coordinate systems===
((Unreferenced section|date=July 2008))
[[File:Orbiter.PNG|thumb|upright=1.2|Figure 3: An orbiting but fixed orientation coordinate system ''B'', shown at three different times. The unit vectors '''u'''<sub>j</sub>, j = 1, 2, 3 do ''not'' rotate, but maintain a fixed orientation, while the origin of the coordinate system ''B'' moves at constant angular rate ω about the fixed axis '''Ω'''. Axis '''Ω''' passes through the origin of inertial frame ''A'', so the origin of frame ''B'' is a fixed distance ''R'' from the origin of inertial frame ''A''.]]
As a related example, suppose the moving coordinate system ''B'' rotates with a constant angular speed ω in a circle of radius ''R'' about the fixed origin of inertial frame ''A'', but maintains its coordinate axes fixed in orientation, as in Figure 3. The acceleration of an observed body is now (see [[#Eq. 1|Eq. 1]]):
:<math>\begin{align}
\frac {d^2 \mathbf{x}_{A)){dt^2} &= \mathbf{a}_{AB}+\mathbf{a}_{B} + 2\ \sum_{j=1}^3 v_j \ \frac{d \mathbf{u}_j}{dt} + \sum_{j=1}^3 x_j \ \frac{d^2 \mathbf{u}_j}{dt^2} \\
&=\mathbf{a}_{AB}\ +\mathbf{a}_B\ ,
\end{align}</math>
where the summations are zero inasmuch as the unit vectors have no time dependence. The origin of the system ''B'' is located according to frame ''A'' at:
:<math>\mathbf{X}_{AB} = R \left( \cos ( \omega t) , \ \sin (\omega t) \right) \ ,</math>
leading to a velocity of the origin of frame ''B'' as:
:<math>\mathbf{v}_{AB} = \frac{d}{dt} \mathbf{X}_{AB} = \mathbf{\Omega \times X}_{AB} \ , </math>

leading to an acceleration of the origin of ''B'' given by:
:<math>\mathbf{a}_{AB} = \frac{d^2}{dt^2} \mathbf{X}_{AB} = \mathbf{ \Omega \ \times } \left( \mathbf{ \Omega \times X}_{AB}\right) = - \omega^2 \mathbf{X}_{AB} \, .</math>
Because the first term, which is
<math display="block">\mathbf{ \Omega \ \times } \left( \mathbf{ \Omega \times X}_{AB}\right)\, , </math>
is of the same form as the normal centrifugal force expression:
<math display="block">\boldsymbol{\Omega} \times \left( \boldsymbol{\Omega} \times \mathbf{x}_B \right)\, ,</math>
it is a natural extension of standard terminology (although there is no standard terminology for this case) to call this term a "centrifugal force". Whatever terminology is adopted, the observers in frame ''B'' must introduce a fictitious force, this time due to the acceleration from the orbital motion of their entire coordinate frame, that is radially outward away from the centre of rotation of the origin of their coordinate system:
:<math>\mathbf{F}_{\mathrm{fict)) = m \omega^2 \mathbf{X}_{AB} \, , </math>
and of magnitude:
:<math>|\mathbf{F}_{\mathrm{fict))| = m \omega^2 R \, . </math>

This "centrifugal force" has differences from the case of a rotating frame. In the rotating frame the centrifugal force is related to the distance of the object from the origin of frame ''B'', while in the case of an orbiting frame, the centrifugal force is independent of the distance of the object from the origin of frame ''B'', but instead depends upon the distance of the origin of frame ''B'' from ''its'' centre of rotation, resulting in the ''same'' centrifugal fictitious force for ''all'' objects observed in frame ''B''.

===Orbiting and rotating===
((See also|Clohessy–Wiltshire equations))
[[File:Center-facing orbiting coordinate system.PNG|thumb|upright=1.2|Figure 4: An orbiting coordinate system ''B'' similar to Figure 3, but in which unit vectors '''u'''<sub>j</sub>, j = 1, 2, 3 rotate to face the rotational axis, while the origin of the coordinate system ''B'' moves at constant angular rate ω about the fixed axis '''Ω'''.]]
As a combination example, Figure 4 shows a coordinate system ''B'' that orbits inertial frame ''A'' as in Figure 3, but the coordinate axes in frame ''B'' turn so unit vector '''u'''<sub>1</sub> always points toward the centre of rotation. This example might apply to a test tube in a centrifuge, where vector '''u'''<sub>1</sub> points along the axis of the tube toward its opening at its top. It also resembles the Earth–Moon system, where the Moon always presents the same face to the Earth.<ref name=Newcomb>However, the Earth–Moon system rotates about its [[barycentric coordinates (astronomy)|barycenter]], not the Earth's center; see ((cite book |author=Simon Newcomb |title=Popular Astronomy |page=307 |url=https://books.google.com/books?id=VS7aS8QS91oC&q=centrifugal+revolution+and+rotation+date:1970-2009&pg=PA307 |isbn=978-1-4067-4574-0 |year=2007 |publisher=Read Books))</ref> In this example, unit vector '''u'''<sub>3</sub> retains a fixed orientation, while vectors '''u'''<sub>1</sub>, '''u'''<sub>2</sub> rotate at the same rate as the origin of coordinates. That is,
:<math>\mathbf{u}_1 = (-\cos \omega t ,\ -\sin \omega t )\ ;\ </math>&ensp;<math>\mathbf{u}_2 = (\sin \omega t ,\ -\cos \omega t ) \, . </math>
:<math>\frac{d}{dt}\mathbf{u}_1 = \mathbf{\Omega \times u_1}= \omega\mathbf{u}_2\ ;</math>&ensp;<math> \ \frac{d}{dt}\mathbf{u}_2 = \mathbf{\Omega \times u_2} = -\omega\mathbf{u}_1\ \ .</math>
Hence, the acceleration of a moving object is expressed as (see [[#Eq. 1|Eq. 1]]):
:<math>\begin{align}
\frac {d^2 \mathbf{x}_{A)){dt^2}&=\mathbf{a}_{AB}+\mathbf{a}_B + 2\ \sum_{j=1}^3 v_j \ \frac{d \mathbf{u}_j}{dt} + \ \sum_{j=1}^3 x_j \ \frac{d^2 \mathbf{u}_j}{dt^2}\\
&=\mathbf{ \Omega \ \times } \left( \mathbf{ \Omega \times X}_{AB}\right) +\mathbf{a}_B + 2\ \sum_{j=1}^3 v_j\ \mathbf{\Omega \times u_j} \ +\ \sum_{j=1}^3 x_j\ \boldsymbol{\Omega} \times \left( \boldsymbol{\Omega} \times \mathbf{u}_j \right)\\
&=\mathbf{ \Omega \ \times } \left( \mathbf{ \Omega \times X}_{AB}\right) + \mathbf{a}_B + 2\ \boldsymbol{\Omega} \times\mathbf{v}_B\ \ +\ \boldsymbol{\Omega} \times \left( \boldsymbol{\Omega} \times \mathbf{x}_B \right)\\
&=\mathbf{ \Omega \ \times } \left( \mathbf{ \Omega \times} (\mathbf{ X}_{AB}+\mathbf{x}_B) \right) + \mathbf{a}_B + 2\ \boldsymbol{\Omega} \times\mathbf{v}_B\ \, ,
\end{align}</math>
where the angular acceleration term is zero for the constant rate of rotation.
Because the first term, which is
<math display="block">\mathbf{ \Omega \ \times } \left( \mathbf{ \Omega \times} (\mathbf{ X}_{AB}+\mathbf{x}_B) \right)\, , </math>
is of the same form as the normal centrifugal force expression:
<math display="block">\boldsymbol{\Omega} \times \left( \boldsymbol{\Omega} \times \mathbf{x}_B \right)\, ,</math>
it is a natural extension of standard terminology (although there is no standard terminology for this case) to call this term the "centrifugal force". Applying this terminology to the example of a tube in a centrifuge, if the tube is far enough from the center of rotation, |'''X'''<sub>AB</sub>| = ''R'' ≫ |'''x'''<sub>B</sub>|, all the matter in the test tube sees the same acceleration (the same centrifugal force). Thus, in this case, the fictitious force is primarily a uniform centrifugal force along the axis of the tube, away from the centre of rotation, with a value |'''F'''<sub>fict</sub>| = ω<sup>2</sup> ''R'', where ''R'' is the distance of the matter in the tube from the centre of the centrifuge. It is the standard specification of a centrifuge to use the "effective" radius of the centrifuge to estimate its ability to provide centrifugal force. Thus, the first estimate of centrifugal force in a centrifuge can be based upon the distance of the tubes from the centre of rotation, and corrections applied if needed.<ref name=Singh>((cite book |title=Constitutive and Centrifuge Modelling: Two Extremes |author1=Bea K Lalmahomed |author2=Sarah Springman |author3=Bhawani Singh |isbn=90-5809-361-1 |year=2002 |publisher=Taylor and Francis |page=82 |url=https://books.google.com/books?id=MJkz_IBZZS0C&q=centrifuge))</ref><ref name=Nen>((cite book |title=Consolidation of Soils: Testing and Evaluation: a Symposium |author=Raymond Nen |isbn=0-8031-0446-4 |year=1986 |publisher=ASTM International |page=590 |url=https://books.google.com/books?id=a-BKqGTXA6kC&q=radius+centrifuge+effective&pg=PA590))</ref>

Also, the test tube confines motion to the direction down the length of the tube, so '''v'''<sub>B</sub> is opposite to '''u'''<sub>1</sub> and the Coriolis force is opposite to '''u'''<sub>2</sub>, that is, against the wall of the tube. If the tube is spun for a long enough time, the velocity '''v'''<sub>B</sub> drops to zero as the matter comes to an equilibrium distribution. For more details, see the articles on [[sedimentation]] and the [[Lamm equation]].

A related problem is that of centrifugal forces for the Earth–Moon–Sun system, where three rotations appear: the daily rotation of the Earth about its axis, the lunar-month rotation of the Earth–Moon system about its centre of mass, and the annual revolution of the Earth–Moon system about the Sun. These three motions influence the [[tide]]s.<ref name=Appleton>((cite book |title=The Popular Science Monthly |year=1877 |author=D Appleton |page=276 |url=https://books.google.com/books?id=YO0KAAAAYAAJ&q=rotation+revolution+%22centrifugal+force%22&pg=PA276))</ref>

===Crossing a carousel===
((See also|Coriolis force#Tossed ball on a rotating carousel))
[[File:Carousel walk.PNG|thumb|upright=1.5|Figure 5: Crossing a rotating carousel walking at a constant speed from the centre of the carousel to its edge, a spiral is traced out in the inertial frame, while a simple straight radial path is seen in the frame of the carousel.]]
Figure 5 shows another example comparing the observations of an inertial observer with those of an observer on a rotating [[carousel]].<ref name= Giancoli>For a similar example, see ((cite book |title=A Handbook for Wireless/ RF, EMC, and High-Speed Electronics, Part of the EDN Series for Design Engineers |author=Ron Schmitt |year=2002 |publisher=Newnes |isbn=0-7506-7403-2 |url=https://books.google.com/books?id=fUBPN8T9bwUC&q=spheres+rotating++Coriolis&pg=PA61 |pages=60–61)), and ((cite book |title=Physics for Scientists And Engineers With Modern Physics |author=Douglas C. Giancoli |page=301 |isbn=978-0-13-149508-1 |year=2007 |publisher=Pearson Prentice-Hall |url=https://books.google.com/books?id=xz-UEdtRmzkC&q=spheres+rotating++Coriolis&pg=PA301))</ref> The carousel rotates at a constant angular velocity represented by the vector '''Ω''' with magnitude ''ω'', pointing upward according to the [[right-hand rule]]. A rider on the carousel walks radially across it at a constant speed, in what appears to the walker to be the straight line path inclined at 45° in Figure 5. To the stationary observer, however, the walker travels a spiral path. The points identified on both paths in Figure 5 correspond to the same times spaced at equal time intervals. We ask how two observers, one on the carousel and one in an inertial frame, formulate what they see using Newton's laws.

====Inertial observer====
The observer at rest describes the path followed by the walker as a spiral. Adopting the coordinate system shown in Figure 5, the trajectory is described by '''r'''(''t''):
:<math>\mathbf{r}(t) =R(t)\mathbf{u}_R = \begin{bmatrix} x(t) \\ y(t) \end{bmatrix} = \begin{bmatrix} R(t)\cos (\omega t + \pi/4) \\ R(t)\sin (\omega t + \pi/4) \end{bmatrix}, </math>
where the added π/4 sets the path angle at 45° to start with (just an arbitrary choice of direction), '''u'''<sub>''R''</sub> is a unit vector in the radial direction pointing from the centre of the carousel to the walker at the time ''t''.  The radial distance ''R''(''t'') increases steadily with time according to:
:<math>R(t) = s t,</math>
with ''s'' the speed of walking. According to simple kinematics, the velocity is then the first derivative of the trajectory:
:<math>\begin{align}
\mathbf{v}(t) &= \frac{dR}{dt} \begin{bmatrix} \cos (\omega t + \pi/4) \\ \sin (\omega t + \pi/4) \end{bmatrix} + \omega R(t) \begin{bmatrix} -\sin(\omega t + \pi/4) \\ \cos (\omega t + \pi/4) \end{bmatrix} \\
&= \frac{dR}{dt} \mathbf{u}_R + \omega R(t) \mathbf{u}_{\theta},
\end{align}</math>
with '''u'''<sub>θ</sub> a unit vector perpendicular to '''u'''<sub>R</sub> at time ''t'' (as can be verified by noticing that the vector [[dot product]] with the radial vector is zero) and pointing in the direction of travel.
The acceleration is the first derivative of the velocity:
:<math>\begin{align}
\mathbf{a}(t) &= \frac{d^2 R}{dt^2} \begin{bmatrix} \cos (\omega t + \pi/4) \\ \sin (\omega t + \pi/4) \end{bmatrix} + 2 \frac {dR}{dt} \omega \begin{bmatrix} -\sin(\omega t + \pi/4) \\ \cos (\omega t + \pi/4) \end{bmatrix} - \omega^2 R(t) \begin{bmatrix} \cos (\omega t + \pi/4) \\ \sin (\omega t + \pi/4) \end{bmatrix} \\
&=2s\omega \begin{bmatrix} -\sin(\omega t + \pi/4) \\ \cos (\omega t + \pi/4) \end{bmatrix} -\omega^2 R(t) \begin{bmatrix} \cos (\omega t + \pi/4) \\ \sin (\omega t + \pi/4) \end{bmatrix} \\
&=2s\ \omega \ \mathbf{u}_{\theta}-\omega^2 R(t)\ \mathbf{u}_R \, .
\end{align}</math>
The last term in the acceleration is radially inward of magnitude ω<sup>2</sup> ''R'', which is therefore the instantaneous [[centripetal force|centripetal acceleration]] of [[circular motion]].<ref>((Anchor|Note1))'''Note''': There is a subtlety here: the distance ''R'' is the instantaneous distance from the rotational axis ''of the carousel''. However, it is not the [[osculating circle|radius of curvature]] ''of the walker's trajectory'' as seen by the inertial observer, and the unit vector '''u'''<sub>R</sub> is not perpendicular to the path. Thus, the designation "centripetal acceleration" is an approximate use of this term. See, for example, ((cite book |title=[[Orbital Mechanics for Engineering Students]] |author=Howard D. Curtis |isbn=0-7506-6169-0 |publisher=Butterworth-Heinemann |year=2005 |page=5)) and ((cite book |title=Accelerator physics |author=S. Y. Lee |page= 37 |url=https://books.google.com/books?id=VTc8Sdld5S8C&q=orbit+%22coordinate+system%22&pg=PA37 |isbn=981-256-182-X |publisher=World Scientific |location=Hackensack NJ |edition=2nd |year=2004))</ref> The first term is perpendicular to the radial direction, and pointing in the direction of travel. Its magnitude is 2''sω'', and it represents the acceleration of the walker as the edge of the carousel is neared, and the arc of the circle travelled in a fixed time increases, as can be seen by the increased spacing between points for equal time steps on the spiral in Figure 5 as the outer edge of the carousel is approached.

Applying Newton's laws, multiplying the acceleration by the mass of the walker, the inertial observer concludes that the walker is subject to two forces: the inward radially directed centripetal force and another force perpendicular to the radial direction that is proportional to the speed of the walker.

====Rotating observer====
The rotating observer sees the walker travel a straight line from the centre of the carousel to the periphery, as shown in Figure 5. Moreover, the rotating observer sees that the walker moves at a constant speed in the same direction, so applying Newton's law of inertia, there is ''zero'' force upon the walker. These conclusions do not agree with the inertial observer. To obtain agreement, the rotating observer has to introduce fictitious forces that appear to exist in the rotating world, even though there is no apparent reason for them, no apparent gravitational mass, electric charge or what have you, that could account for these fictitious forces.

To agree with the inertial observer, the forces applied to the walker must be exactly those found above. They can be related to the general formulas already derived, namely:
:<math>
\mathbf{F}_{\mathrm{fict)) =
- 2 m \boldsymbol\Omega \times \mathbf{v}_\mathrm{B} - m \boldsymbol\Omega \times (\boldsymbol\Omega \times \mathbf{x}_\mathrm{B} ) - m \frac{d \boldsymbol\Omega}{dt} \times \mathbf{x}_\mathrm{B}.
</math>
In this example, the velocity seen in the rotating frame is:
:<math>\mathbf{v}_\mathrm{B} = s \mathbf{u}_R, </math>
with '''u'''<sub>R</sub> a unit vector in the radial direction. The position of the walker as seen on the carousel is:
:<math>\mathbf{x}_\mathrm{B} = R(t)\mathbf{u}_R, </math>
and the time derivative of '''Ω''' is zero for uniform angular rotation. Noticing that
:<math>\boldsymbol\Omega \times \mathbf{u}_R =\omega \mathbf{u}_{\theta} </math>
and
:<math>\boldsymbol\Omega \times \mathbf{u}_{\theta} =-\omega \mathbf{u}_R \, ,</math>
we find:
:<math>\mathbf{F}_{\mathrm{fict)) = - 2 m \omega s \mathbf{u}_{\theta} + m \omega^2 R(t) \mathbf{u}_R.</math>
To obtain a [[straight-line motion]] in the rotating world, a force exactly opposite in sign to the fictitious force must be applied to reduce the net force on the walker to zero, so Newton's law of inertia will predict a straight line motion, in agreement with what the rotating observer sees. The fictitious forces that must be combated are the [[Coriolis force]] (first term) and the [[centrifugal force]] (second term). (These terms are approximate.<ref>A circle about the axis of rotation is not the [[osculating circle]] of the walker's trajectory, so "centrifugal" and "Coriolis" are approximate uses for these terms. [[#Note1|See note]].</ref>) By applying forces to counter these two fictitious forces, the rotating observer ends up applying exactly the same forces upon the walker that the inertial observer predicted were needed.

Because they differ only by the constant walking velocity, the walker and the rotational observer see the same accelerations. From the walker's perspective, the fictitious force is experienced as real, and combating this force is necessary to stay on a straight line radial path holding a constant speed. It's like battling a crosswind while being thrown to the edge of the carousel.
<ref name="coordinate">In this connection, it may be noted that a change in the coordinate system, for example, from Cartesian to polar, if implemented without any change in relative motion, does not cause the appearance of rotational fictitious forces, despite the fact that the form of the laws of motion varies from one type of curvilinear coordinate system to another, depending from the (purely spatial) delta-curvature: <math>\ddot x^j + \Gamma^j{}_{lk} \dot x^l\dot x^k = F^j</math>, where <math>F^j</math> are the contravariant components of the force per unit mass, and <math>\Gamma^j{}_{lk}</math> are the [[Christoffel symbols]] of the second kind, see, for instance: David, Kay, ''Tensor Calculus'' (1988) McGraw-Hill Book Company ((ISBN|0-07-033484-6)), Section 11.4; or: Adler, R., Bazin, M., & Schiffer, M. ''Introduction to General Relativity'' (New York, 1965). This could be the first hint of the crisis of the non-relativistic physics: in "non-inertial" frames using non-Euclidean and not flat metrics, fictitious forces transform into force exchanged with "objects" that do not follow the geodesic trajectory (simply with a relative speed respect it). In any case this generalized "Newton's second law" must wait for the [[general relativity]] to obtain curvature in spacetime according to [[stress–energy tensor]] by [[Einstein field equations]] and a spacetime form that uses the [[four-force]] density tensor that is derived from the covariant divergence of the energy-momentum tensor.</ref>

===Observation===
Notice that this [[kinematics|kinematical]] discussion does not delve into the mechanism by which the required forces are generated. That is the subject of [[kinetics (physics)|kinetics]]. In the case of the carousel, the kinetic discussion would involve perhaps a study of the walker's shoes and the friction they need to generate against the floor of the carousel, or perhaps the dynamics of skateboarding if the walker switched to travel by skateboard. Whatever the means of travel across the carousel, the forces calculated above must be realized. A very rough analogy is heating your house: you must have a certain temperature to be comfortable, but whether you heat by burning gas or by burning coal is another problem. Kinematics sets the thermostat, kinetics fires the furnace.

==See also==
((Div col|colwidth=22em))
* [[Analytical mechanics]]
* [[Applied mechanics]]
* [[Centrifugal force (fictitious)|Centrifugal force]]
* [[Centripetal force]]
* [[Circular motion]]
* [[Classical mechanics]]
* [[Coriolis force]]
* ((section link|Curvilinear coordinates|Fictitious forces in general curvilinear coordinates))
* [[d'Alembert's principle]] of inertial forces
* [[Dynamics (physics)]]
* [[Free motion equation]]
* [[Frenet–Serret formulas]]
* [[General relativity]]
* [[Generalized coordinates]]
* [[Generalized force]]
* [[Gravity]]
* [[Inertial reference frame]]
* [[Kinematics]]
* [[Kinetics (physics)]]
* [[Newton's laws of motion]]
* [[Non-inertial reference frame]]
* [[Orthogonal coordinates]]
* [[Rotating reference frame]]
* [[Statics]]
* [[Uniform circular motion]]
((Div col end))
((Portal|Physics))

==Notes==
((Reflist))

==Further reading==
* ((cite book |author=[[Lev D. Landau]] and E. M. Lifshitz |isbn=0-7506-2896-0 |edition=3rd |year=1976 |title=Mechanics |volume=1 |series=[[Course of Theoretical Physics]] |url=https://books.google.com/books?id=LmAV8q_OOOgC |publisher=Butterworth-Heinenan |pages=128–130))
* ((cite book |author=Keith Symon |year=1971 |title=Mechanics |publisher=Addison-Wesley |edition=3rd |isbn=0-201-07392-7))
* ((cite book |author=Jerry B. Marion |year=1970 |title=Classical Dynamics of Particles and Systems |publisher=Academic Press |isbn=0-12-472252-0 |url=https://archive.org/details/classicaldynamic00mari))
* ((cite book |author=Marcel J. Sidi |title=Spacecraft Dynamics and Control: A Practical Engineering Approach |isbn=0-521-78780-7 |year=1997 |publisher=Cambridge University Press |url=https://books.google.com/books?id=xQpZJMtDehQC&q=orbit+%22coordinate+system%22&pg=RA1-PA101 |no-pp=true |pages=Chapter 4.8))
<!-- The text referencing these must have been deleted. Kept as a comment in case somebody knows where they are relevant:
# ((note|kandk2)) Kleppner, pages 355-360.
# ((note|fetter)) Alexander Fetter and John Walecka, ''Theoretical Mechanics of particles and continua'', McGraw-Hill, pages 33–39.
# ((note|landau)) [[Lev D. Landau]] and E. M. Lifshitz, (1976) ''Mechanics'', Butterworth-Heinenan, pages 128–130.
# Jerry B. Marion, (1970), ''Classical Dynamics of Particles and Systems'', Academic Press.
# Keith Symon, (1971), ''Mechanics'', Addison-Wesley
-->

==External links==
* [https://web.archive.org/web/20050501210856/http://www.hcc.hawaii.edu/~rickb/SciColumns/FictForce.04Feb96.html Q and A from Richard C. Brill, Honolulu Community College]
* [https://pwg.gsfc.nasa.gov/stargaze/Sframes2.htm NASA's David Stern: Lesson Plans for Teachers #23 on ''Inertial Forces'']
* [https://scienceworld.wolfram.com/physics/CoriolisForce.html Coriolis Force]
* [https://web.archive.org/web/20060511131009/http://mensch.org/physlets/merry.html Motion over a flat surface] Java physlet by Brian Fiedler illustrating fictitious forces. The physlet shows both the perspective as seen from a rotating and from a non-rotating point of view.
* [https://web.archive.org/web/20060321094211/http://www.mensch.org/physlets/inosc.html Motion over a parabolic surface] Java physlet by Brian Fiedler illustrating fictitious forces. The physlet shows both the perspective as seen from a rotating and as seen from a non-rotating point of view.

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