A fictitious force, also known as an inertial force or pseudo-force, is a force that appears to act on an object when its motion is described or experienced from a non-inertial frame of reference. Unlike real forces, which result from physical interactions between objects, fictitious forces occur due to the acceleration of the observer's frame of reference rather than any actual force acting on a body. These forces are necessary for describing motion correctly within an accelerating frame, ensuring that Newton's second law of motion remains applicable. Common examples of fictitious forces include the centrifugal force, which appears to push objects outward in a rotating system; the Coriolis force, which affects objects moving relative to the rotating frame, such as a wind parcel on Earth; and the Euler force, which arises when a rotating system changes its angular velocity (i.e., due to angular 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 to general relativity theory we perceive gravitational force when spacetime is bending near heavy objects, so even this might be called a fictitious force.
Measurable examples of fictitious forces 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 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 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 physical interaction between two objects, such as electromagnetism or contact forces. It is only a consequence of the acceleration 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:
Such an additional force due to nonuniform relative motion of two reference frames is called a pseudo-force. 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. The gravitational force would also be a fictitious force (pseudo force) in a field model in which particles distort spacetime due to their mass, such as in the theory of general relativity. Assuming Newton's second law in the form F = ma, fictitious forces are always proportional to the mass m. The fictitious force that has been called an inertial force is also referred to as a d'Alembert force, or sometimes as a pseudo force. 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'. 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); two involving rotation: centrifugal force and Coriolis effect and a fourth, called the Euler force, caused by a variable rate of rotation, should that occur.
Background
The role of fictitious forces in Newtonian mechanics is described by Tonnelat:
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. Fictitious forces arise in classical mechanics and special relativity in all non-inertial frames. Inertial frames are 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 of spacetime: "The field of all possible space-time null geodesics or photon paths unifies the absolute local non-rotation standard throughout space-time.".
On Earth 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 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 the Eötvös effect.
Detection of non-inertial reference frame
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:
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 = −ma, where a is the acceleration of the system K. Other accelerations also give rise to fictitious forces, as described mathematically 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. 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 concerning circular motion
The effect of a fictitious force also occurs when a car takes the bend. Observed from a non-inertial frame of reference attached to the car, the fictitious force called the 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 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-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. 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. 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, 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.
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.
Fictitious forces and work Fictitious forces can be considered to do work, provided that they move an object on a trajectory that changes its energy from potential to 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
The notion of "fictitious force" also arises in Einstein's general theory of relativity. All fictitious forces are proportional to the mass of the object upon which they act, which is also true for gravity. This led Albert Einstein to wonder whether gravity could be modeled as a fictitious force. He noted that a freefalling 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 (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.
Mathematical derivation of fictitious forces
General derivation Many problems require use of noninertial reference frames, for example, those involving satellites and particle accelerators. Figure 2 shows a particle with mass m and position vector xA(t) in a particular inertial frame A. Consider a non-inertial frame B whose origin relative to the inertial one is given by XAB(t). Let the position of the particle in frame B be xB(t). What is the force on the particle as expressed in the coordinate system of frame B? To answer this question, let the coordinate axis in B be represented by unit vectors uj with j any of { 1, 2, 3 } for the three coordinate axes. Then
x B = ∑ j = 1 3 x j u j . {\displaystyle \mathbf {x} _{\mathrm {B} }=\sum _{j=1}^{3}x_{j}\mathbf {u} _{j}\,.}
The interpretation of this equation is that xB 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:
x A = X A B + ∑ j = 1 3 x j u j . {\displaystyle \mathbf {x} _{\mathrm {A} }=\mathbf {X} _{\mathrm {AB} }+\sum _{j=1}^{3}x_{j}\mathbf {u} _{j}\,.}
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, vector XAB 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:
d x A d t = d X A B d t + ∑ j = 1 3 d x j d t u j + ∑ j = 1 3 x j d u j d t . {\displaystyle {\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}}\,.}
The second term summation is the velocity of the particle, say vB as measured in frame B. That is:
d x A d t = v A B + v B + ∑ j = 1 3 x j d u j d t . {\displaystyle {\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}}.}
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 vB, 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 vAB. 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:
d 2 x A d t 2 = a A B + d v B d t + ∑ j = 1 3 d x j d t d u j d t + ∑ j = 1 3 x j d 2 u j d t 2 . {\displaystyle {\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}}}.}
Using the same formula already used for the time derivative of xB, the velocity derivative on the right is:
d v B d t = ∑ j = 1 3 d v j d t u j + ∑ j = 1 3 v j d u j d t = a B + ∑ j = 1 3 v j d u j d t . {\displaystyle {\frac {d\mathbf {v} _{\mathrm {B} }}{dt}}=\sum _{j=1}^{3}{\frac {dv_{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}}.}
Consequently,
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 aB, 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 aAB, 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 dvB/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 ∑ x j d u j / d t {\textstyle \sum x_{j}\,d\mathbf {u} _{j}/dt} due to change in x j ). To put matters in terms of forces, the accelerations are multiplied by the particle mass:
F A = F B + m a A B + 2 m ∑ j = 1 3 v j d u j d t + m ∑ j = 1 3 x j d 2 u j d t 2 . {\displaystyle \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}}}\ .}
The force observed in frame B, FB = maB is related to the actual force on the particle, FA, by
F B = F A + F f i c t i t i o u s , {\displaystyle \mathbf {F} _{\mathrm {B} }=\mathbf {F} _{\mathrm {A} }+\mathbf {F} _{\mathrm {fictitious} },}
where:
F f i c t i t i o u s = − m a A B − 2 m ∑ j = 1 3 v j d u j d t − m ∑ j = 1 3 x j d 2 u j d t 2 . {\displaystyle \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}}}\,.}
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 Ffictitious as an additional force. 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
| Ω | = d θ d t = ω ( t ) , {\displaystyle |{\boldsymbol {\Omega }}|={\frac {d\theta }{dt}}=\omega (t),}
then the time derivative of any of the three unit vectors describing frame B is
d u j ( t ) d t = Ω × u j ( t ) , {\displaystyle {\frac {d\mathbf {u} _{j}(t)}{dt}}={\boldsymbol {\Omega }}\times \mathbf {u} _{j}(t),}
and
d 2 u j ( t ) d t 2 = d Ω d t × u j + Ω × d u j ( t ) d t = d Ω d t × u j + Ω × [ Ω × u j ( t ) ] , {\displaystyle {\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],}
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 aAB = 0 to remove any translational acceleration, and focusing on only rotational properties (see Eq. 1):
d 2 x A d t 2 = a B + 2 ∑ j = 1 3 v j d u j d t + ∑ j = 1 3 x j d 2 u j d t 2 , {\displaystyle {\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}}},}
a A = a B + 2 ∑ j = 1 3 v j Ω × u j ( t ) + ∑ j = 1