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Afictitious force, also known as aninertial force orpseudo-force, is aforce that appears to act on an object when itsmotion is described or experienced from anon-inertial frame of reference. Unlike real forces, which result from physical interactions between objects, fictitious forces occur due to theacceleration of the observer’sframe of reference rather than any actual force acting on a body. These forces are necessary for describing motion correctly within an accelerating frame, ensuring thatNewton's second law of motion remains applicable.[1][2]
Common examples of fictitious forces include thecentrifugal force, which appears to push objects outward in a rotating system; theCoriolis force, which affects objects moving relative to the rotating frame, such as a wind parcel on Earth; and theEuler force, which arises when a rotating system changes itsangular velocity (i.e., due toangular acceleration).
While these forces are not real in the sense of being caused by physical interactions, they are essential for accurately analyzing motion within accelerating reference frames, particularly in disciplines such as classical mechanics, meteorology, and astrophysics.Fictitious forces play a crucial role in understanding everyday phenomena, such as weather patterns influenced by the Coriolis effect and the perceived weightlessness experienced by astronauts in free-fall orbits. They are also fundamental in engineering applications, including navigation systems and rotating machinery.
According togeneral relativity theory we perceivegravitational force whenspacetime isbending near heavy objects, so even this might be called afictitious force.
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.
The fictitious force called apseudo force might also be referred to as abody force. It is due to an object'sinertia 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 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 anyphysical interaction between two objects, such aselectromagnetism or contact forces. It is only a consequence of the acceleration of the physical object thenon-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:[3]
Such an additional force due to nonuniform relative motion of two reference frames is called apseudo-force.
— Harald Iro inA 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 theCoriolis force, maybe better to be called: the Coriolis effect.[4][5][6] Thegravitational force would also be a fictitious force (pseudo force) in a field model in which particles distortspacetime due to their mass, such as in the theory ofgeneral relativity.
AssumingNewton's second law in the formF = ma, fictitious forces are always proportional to the massm.
The fictitious force that has been called an inertial force[7][8][9] is also referred to as ad'Alembert force,[10][11] or sometimes as a pseudo force.[12] 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 acontact force arising from the physical interaction between two objects, which is the subject of Newton's third law – 'action isreaction'.[13][14] 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:
The role of fictitious forces in Newtonian mechanics is described byTonnelat:[16]
For Newton, the appearance of acceleration always indicates the existence of absolute motion – absolute motion of matter wherereal forces are concerned; absolute motion of the reference system, where so-calledfictitious forces, such as inertial forces or those of Coriolis, are concerned.
— Marie-Antoinette Tonnelat inThe Principles of Electromagnetic Theory and Relativity, p.113
Fictitious forces arise inclassical mechanics andspecial relativity in all non-inertial frames.Inertial frames areprivileged 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 ingeneral relativity, since these physics are explained with thegeodesics ofspacetime: "The field of all possible space-time null geodesics or photon paths unifies the absolute local non-rotation standard throughout space-time.".[17]
The surface of the Earth is arotating reference frame. To solveclassical mechanics problems exactly in an Earthbound reference frame, three fictitious forces must be introduced: theCoriolis force, thecentrifugal force (described below) and theEuler 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 hisFoucault pendulum to show that the 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.
When moving along the equator in a ship heading in an easterly direction, objects appear to be slightly lighter than on the way back. This phenomenon has been observed and is called theEötvös effect.
Observers inside a closed box that is moving with a constantvelocity 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 accelerationVladimir Arnold presents the following theorem:[18]
In a coordinate systemK which moves by translation relative to an inertial systemk, the motion of a mechanical system takes place as if the coordinate system were inertial, but on every point of massm an additional "inertial force" acted:F = −ma, wherea is the acceleration of the systemK.
Other accelerations also give rise to fictitious forces, as described mathematicallybelow. 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.[19]
An example of the detection of a non-inertial, rotating reference frame is the precession of aFoucault pendulum. In the non-inertial frame of the Earth, the fictitiousCoriolis force is necessary to explain observations. In an inertial frame outside the Earth, no such fictitious force is necessary.
The effect of a fictitious force also occurs when a car takesthe bend. Observed from a non-inertial frame of reference attached to the car, the fictitious force called thecentrifugal 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 ofcontact forces 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-calledreactive centrifugal force as the reaction part. The reactive centrifugal force is also due to theinertia of the suitcase. Now however the inertia appears in the form of a manifesting resistance to a change in its state of motion.[20]
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.
A classic example of a fictitious force in circular motion is the experiment ofrotating spheres tied by a cord and spinning around their centre of mass. In this case, 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 theequator.
| Animation: object released from a carousel |
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 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 acentrifugal 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 fictitiousCoriolis 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 (seeRossby 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.[22]
Fictitious forces can be considered to dowork, provided that they move an object on atrajectory that changes itsenergy frompotential tokinetic. 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.
The notion of "fictitious force" also arises in Einstein'sgeneral theory of relativity.[23][24] All fictitious forces are proportional to the mass of the object upon which they act, which is also true forgravity.[25][26] This ledAlbert Einstein to wonder whether gravity could be modeled as a fictitious force. He noted that afreefalling observer in a closed box would not be able to detect the force of gravity; hence,freefalling reference frames are equivalent to inertial reference frames (theequivalence principle). Developing this insight, Einstein formulated a theory with gravity as a fictitious force, and attributed the apparent acceleration due to gravity to thecurvature ofspacetime. This idea underlies Einstein's theory ofgeneral relativity. See theEötvös experiment.
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Many problems require use of noninertial reference frames, for example, those involving satellites[28][29] and particle accelerators.[30] Figure 2 shows a particle withmassm andpositionvectorxA(t) in a particularinertial frame A. Consider a non-inertial frame B whose origin relative to the inertial one is given byXAB(t). Let the position of the particle in frame B bexB(t). What is the force on the particle as expressed in the coordinate system of frame B?[31][32]
To answer this question, let the coordinate axis in B be represented by unit vectorsuj withj any of { 1, 2, 3 } for the three coordinate axes. Then
The interpretation of this equation is thatxB is the vector displacement of the particle as expressed in terms of the coordinates in frame B at the timet. From frame A the particle is located at:
As an aside, the unit vectors {uj } cannot change magnitude, so derivatives of these vectors express only rotation of the coordinate system B. On the other hand, vectorXAB 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:
The second term summation is the velocity of the particle, sayvB as measured in frame B. That is:
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, namelyvB, 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 originvAB. 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:
Using the same formula already used for the time derivative ofxB, the velocity derivative on the right is:
Consequently,
| 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 accelerationaB, 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, namelyaAB, 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 (advB/dt 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 due to change inxj ).
To put matters in terms of forces, the accelerations are multiplied by the particle mass:
The force observed in frame B,FB =maB is related to the actual force on the particle,FA, bywhere:
Thus, problems may be solved in frame B by assuming that Newton's second law holds (with respect to quantities in that frame) and treatingFfictitious as an additional force.[18][33][34]
Below are a number of examples applying this result for fictitious forces. More examples can be found in the article oncentrifugal force.
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 vectorΩ pointed along the axis of rotation with the orientation given by theright-hand rule, and with magnitude given by
then the time derivative of any of the three unit vectors describing frame B is[33][35]andas is verified using the properties of thevector 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, settingaAB = 0 to remove any translational acceleration, and focusing on only rotational properties (seeEq. 1):Collecting terms, the result is the so-calledacceleration transformation formula:[36]
Thephysical accelerationaA due to what observers in the inertial frame A callreal external forces on the object is, therefore, not simply the accelerationaB 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 accelerationaB of the particle is given by rearrangement of the above equation as:
The net force upon the object according to observers in the rotating frame isFB =maB. If their observations are to result in the correct force on the object when using Newton's laws, they must consider that the additional forceFfict is present, so the end result isFB =FA +Ffict. Thus, the fictitious force used by observers in B to get the correct behaviour of the object from Newton's laws equals:
Here, the first term is theCoriolis force,[37] the second term is thecentrifugal force,[38] and the third term is theEuler force.[39][40]
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As a related example, suppose the moving coordinate systemB rotates with a constant angular speed ω in a circle of radiusR about the fixed origin of inertial frameA, but maintains its coordinate axes fixed in orientation, as in Figure 3. The acceleration of an observed body is now (seeEq. 1):where the summations are zero inasmuch as the unit vectors have no time dependence. The origin of the systemB is located according to frameA at:leading to a velocity of the origin of frameB as:leading to an acceleration of the origin ofB given by:Because the first term, which isis of the same form as the normal centrifugal force expression: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 frameB 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:and of magnitude:
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 frameB, while in the case of an orbiting frame, the centrifugal force is independent of the distance of the object from the origin of frameB, but instead depends upon the distance of the origin of frameB fromits centre of rotation, resulting in thesame centrifugal fictitious force forall objects observed in frameB.
As a combination example, Figure 4 shows a coordinate systemB that orbits inertial frameA as in Figure 3, but the coordinate axes in frameB turn so unit vectoru1 always points toward the centre of rotation. This example might apply to a test tube in a centrifuge, where vectoru1 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.[41] In this example, unit vectoru3 retains a fixed orientation, while vectorsu1,u2 rotate at the same rate as the origin of coordinates. That is,Hence, the acceleration of a moving object is expressed as (seeEq. 1):where the angular acceleration term is zero for the constant rate of rotation.Because the first term, which isis of the same form as the normal centrifugal force expression: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, |XAB| =R ≫ |xB|, 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 |Ffict| = ω2R, whereR 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.[42][43]
Also, the test tube confines motion to the direction down the length of the tube, sovB is opposite tou1 and the Coriolis force is opposite tou2, that is, against the wall of the tube. If the tube is spun for a long enough time, the velocityvB drops to zero as the matter comes to an equilibrium distribution. For more details, see the articles onsedimentation and theLamm 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 thetides.[44]
Figure 5 shows another example comparing the observations of an inertial observer with those of an observer on a rotatingcarousel.[45] The carousel rotates at a constant angular velocity represented by the vectorΩ with magnitudeω, pointing upward according to theright-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.
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 byr(t):where the added π/4 sets the path angle at 45° to start with (just an arbitrary choice of direction),uR is a unit vector in the radial direction pointing from the centre of the carousel to the walker at the timet. The radial distanceR(t) increases steadily with time according to:withs the speed of walking. According to simple kinematics, the velocity is then the first derivative of the trajectory:withuθ a unit vector perpendicular touR at timet (as can be verified by noticing that the vectordot product with the radial vector is zero) and pointing in the direction of travel.The acceleration is the first derivative of the velocity:The last term in the acceleration is radially inward of magnitude ω2R, which is therefore the instantaneouscentripetal acceleration ofcircular motion.[46] The first term is perpendicular to the radial direction, and pointing in the direction of travel. Its magnitude is 2sω, 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.
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 iszero 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:In this example, the velocity seen in the rotating frame is:withuR a unit vector in the radial direction. The position of the walker as seen on the carousel is:and the time derivative ofΩ is zero for uniform angular rotation. Noticing thatandwe find:To obtain astraight-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 theCoriolis force (first term) and thecentrifugal force (second term). (These terms are approximate.[47]) 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 is like battling a crosswind while being thrown to the edge of the carousel.[48]
Notice that thiskinematical discussion does not delve into the mechanism by which the required forces are generated. That is the subject ofkinetics. 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.
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