Fictitious

Fictitious force

A fictitious force, also called a pseudo force^[1], d'Alembert force^[2]^[3] or inertial force^[4]^[5], is an apparent force that acts on all masses in a non-inertial frame of reference, such as a rotating reference frame. The force F does not arise from any physical interaction but rather from the acceleration a of the non-inertial reference frame itself. Due to Newton's second law in the form F = m a, fictitious forces always are proportional to the mass m acted upon.

Non-inertial reference frames may be a convenience in calculation, or an intuitively appealing explanation of everyday life, but whenever using a non-inertial reference frame, its acceleration results in fictitious forces. Four fictitious forces are defined in accelerated frames: one caused by any acceleration of the origin in a straight line,^[6] two caused by any rotation (centrifugal force and Coriolis force) and a fourth, called the Euler force, caused by a variable rate of rotation, should that occur.

1 Fictitious forces on Earth
2 Detection of non-inertial reference frame
3 Examples of fictitious forces
4 Gravity as a fictitious force
5 Mathematical derivation of fictitious forces
- 5.1 General derivation
- 5.2 Rotating coordinate systems
6 References
7 See also
8 External links

Fictitious forces on Earth

See also: centrifugal force, reactive centrifugal force, and Coriolis force

The surface of the Earth is a rotating reference frame. To solve classical mechanics problems exactly in an Earth-bound reference frame, two fictitious forces must be introduced, the Coriolis force and the centrifugal force (described below). Both of these fictitious forces are weak compared to most typical forces in everyday life, but they can be detected under careful conditions. For example, Léon Foucault was able to show the Coriolis force that results from the Earth's rotation using the Foucault pendulum. If the Earth were to rotate a thousandfold faster (making each day only ~86 seconds long), people could easily get the impression that such fictitious forces are pulling on them, as on a spinning carousel.

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.^[7] They can even map out the magnitude and direction of the acceleration at every point with a plumb bob and a protractor. Another example of the detection of a non-inertial reference frame is the way a Foucault pendulum precesses.

Examples of fictitious forces

Acceleration in a straight line

When a car accelerates hard, the common human response is to feel "pushed back into the seat." 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 ways of analyzing the problem:

From the viewpoint of an inertial reference frame with constant velocity matching the initial motion of the car, the car is accelerating. In order for the passenger to stay inside the car, a force must be exerted on him. This force is exerted by the seat, which has started to move forward with the car and compressed against the passenger until it transmits the full force to keep the passenger inside. Thus, the passenger is accelerating in this frame, due to the unbalanced force of the seat.
From the point of view of the interior of the car, an accelerating reference frame, there is a fictitious force pushing the passenger backwards, with 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.

Animation: driving from block to block
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.

This serves as an illustration of the manner in which fictitious forces arise from switching to a non-inertial reference frame. Calculations of physical quantities 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.)

Circular motion

See also: Circular motion, Uniform circular motion, and Time derivative

A similar effect occurs in circular motion, circular for the standpoint of an inertial frame of reference attached to the road, with the fictitious force called the centrifugal force, fictitious when seen from a non-inertial frame of reference. If a car is moving at constant speed around a circular section of road, the occupants will feel pushed outside, away from the center of the turn. Again the situation can be viewed from inertial or non-inertial frames:

From the viewpoint of an inertial reference frame stationary with respect to the road, the car is accelerating toward the center of the circle. This is called centripetal acceleration and requires a centripetal force to maintain the motion. This force is exerted by the ground on the wheels, in this case thanks to the friction between the wheels and the road.^[8] The car is accelerating, due to the unbalanced force, which causes it to move in a circle.
From the viewpoint of a rotating frame, moving with the car, there is a fictitious centrifugal force that tends to push the car toward the outside of the road (and the occupants toward the outside of the car). The centrifugal force balances out the friction between wheels and road, making the car stationary in this non-inertial frame.

Animation: object released from a carousel

Map and spin frame perspectives of physical (red) and fictitious (blue) forces for an object released from a carousel.

From the map frame perspective, what's dangerous on losing centripetal acceleration may be your speed. From the spin frame perspective, the danger instead may lie with the geometric acceleration which gives rise to that fictitious force.Note: With some browsers, you can hit [Esc] to freeze the motion for more detailed analysis. However you may have to reload the page to restart.

To consider another example, taking as our reference frame the surface of the rotating Earth, centrifugal force reduces the apparent force of gravity by about one part in a thousand, depending on latitude. This is zero at the poles, maximum at the equator.

Another fictitious force that arises in rotating frames is the Coriolis force, which 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. Neglecting air resistance, an object dropped from a 50-meter-high tower at the equator will fall 7.7 millimeters eastward of the spot below where it is dropped because of the Coriolis force.^[9]

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 force. 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 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 centripetal direction. The vector sum of the centrifugal force and the Coriolis force is the total fictitious force, which in this case points in centripetal direction.

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 a person in a rotating chair holding a weight in his outstretched arm. If he pulls his arm inward, from the perspective of his rotating reference frame he has done work against centrifugal force. If he now lets go of the weight, from his perspective it spontaneously flies outward, because centrifugal force has done work on the object, converting its potential energy into kinetic. From an inertial viewpoint, of course, the object flies away from him 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 an inertial one.

Gravity as a fictitious force

Main article: General 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 was a fictitious force as well. 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 an inertial reference frame (the equivalence principle). Following up on this insight, Einstein was able to formulate a theory with gravity as a fictitious force; attributing the apparent acceleration to the curvature of spacetime. This is the essential physics of Einstein's theory of general relativity.

Animation: ball that rolls off a cliff

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^[10] 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

General derivation

Consider a particle with mass m and position vector x_A(t) in a particular inertial frame A. Consider a non-inertial frame B whose position relative to the inertial one is given by X_AB(t). Let the position of the particle in frame B be x_B(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 u_j with j any of { 1, 2, 3 } for the three coordinate axes. Then

$\mathbf{x}_{B} = \sum_{j=1}^3 x_j\ \mathbf{u}_j \ .$

The interpretation of this equation is that x_B is the vector displacement of the particle as expressed in terms of the coordinates in frame B at time t. From frame A the particle is located at:

$\mathbf{x}_A =\mathbf{X}_{AB} + \sum_{j=1}^3 x_j\ \mathbf{u}_j \ .$

As an aside, the unit vectors { u_j } cannot change magnitude, so derivatives of these vectors express only rotation of the coordinate system B. On the other hand, vector X_AB 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:

$\frac {d \mathbf{x}_{A}}{dt} =\frac{d \mathbf{X}_{AB}}{dt}+ \sum_{j=1}^3 \frac{dx_j}{dt} \ \mathbf{u}_j + \sum_{j=1}^3 x_j \ \frac{d \mathbf{u}_j}{dt} \ .$

The first sum is the velocity of the particle, say v_B as measured in frame B. That is:

$\frac {d \mathbf{x}_{A}}{dt} =\mathbf{v}_{AB}+ \mathbf{v}_B + \sum_{j=1}^3 x_j \ \frac{d \mathbf{u}_j}{dt} \ .$

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 v_B, plus extra terms related to the rate of change of the frame-B coordinate axes.

To find the acceleration, another time differentiation provides:

$\frac {d^2 \mathbf{x}_{A}}{dt^2} = \mathbf{a}_{AB}+\frac {d\mathbf{v}_B}{dt} + \sum_{j=1}^3 \frac {dx_j}{dt} \ \frac{d \mathbf{u}_j}{dt} + \sum_{j=1}^3 x_j \ \frac{d^2 \mathbf{u}_j}{dt^2}\ .$

The velocity derivative on the right, using the same formula already used for the time derivative of x_B is:

$\frac {d\mathbf{v}_B}{dt} =\sum_{j=1}^3 \frac{d v_j}{dt} \ \mathbf{u}_j+ \sum_{j=1}^3 v_j \ \frac{d \mathbf{u}_j}{dt} =\mathbf{a}_B + \sum_{j=1}^3 v_j \ \frac{d \mathbf{u}_j}{dt} \ .$

Consequently,

$\frac {d^2 \mathbf{x}_{A}}{dt^2}=\mathbf{a}_{AB}+\mathbf{a}_B + 2\ \sum_{j=1}^3 v_j \ \frac{d \mathbf{u}_j}{dt} + \sum_{j=1}^3 x_j \ \frac{d^2 \mathbf{u}_j}{dt^2}\ .$

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_B, 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_AB, and two terms related to 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. To put matters in terms of forces, the accelerations are multiplied by the particle mass:

$\mathbf{F}_A = \mathbf{F}_B + m\ \mathbf{a}_{AB}+ 2m\ \sum_{j=1}^3 v_j \ \frac{d \mathbf{u}_j}{dt} + m\ \sum_{j=1}^3 x_j \ \frac{d^2 \mathbf{u}_j}{dt^2}\ .$

The force observed in frame B, F_B = m a_B is related to the actual force on the particle, F_A, by:

$\mathbf{F}_B = \mathbf{F}_A + \mathbf{F}_{\mbox{fictitious}} \ .$

where:

$\mathbf{F}_{\mbox{fictitious}} =-m\ \mathbf{a}_{AB} -2m\ \sum_{j=1}^3 v_j \ \frac{d \mathbf{u}_j}{dt} - m\ \sum_{j=1}^3 x_j \ \frac{d^2 \mathbf{u}_j}{dt^2}\ .$

Thus, we can solve problems in frame B by assuming that Newton's second law holds (with respect to quantities in that frame) and treating F_fictitious as an additional force.^[11]^[12]^[13]

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 is represented by a vector Ω pointed along the axis of rotation with orientation given by the right-hand rule, and with magnitude given by

$|\boldsymbol{\Omega} | = \frac {d \theta }{dt} = \omega (t) \ ,$

then the time derivative of any unit vector is:^[11]^[14]

$\frac {d \mathbf{u}_j (t)}{dt} = \boldsymbol{\Omega} \times \mathbf{u}_j (t) \ ,$

and

$\frac {d^2 \mathbf{u}_j (t)}{dt^2}= \frac{d\boldsymbol{\Omega}}{dt} \times \mathbf{u}_j +\boldsymbol{\Omega} \times \frac{d \mathbf{u}_j (t)}{dt} = \frac{d\boldsymbol{\Omega}}{dt} \times \mathbf{u}_j+ \boldsymbol{\Omega} \times \left( \boldsymbol{\Omega} \times \mathbf{u}_j (t) \right)\ ,$

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 a_AB = 0 to remove any translational acceleration, and focusing on only rotational properties:

$\frac {d^2 \mathbf{x}_{A}}{dt^2}=\mathbf{a}_B + 2\ \sum_{j=1}^3 v_j \ \frac{d \mathbf{u}_j}{dt} + \sum_{j=1}^3 x_j \ \frac{d^2 \mathbf{u}_j}{dt^2}\ ,$

$\mathbf{a}_A=\mathbf{a}_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}_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)\$

$=\mathbf{a}_B + 2\ \boldsymbol{\Omega} \times\mathbf{v}_B\ + \frac{d\boldsymbol{\Omega}}{dt} \times \mathbf{x}_B \ + \boldsymbol{\Omega} \times \left( \boldsymbol{\Omega} \times \mathbf{x}_B \right)\ .$

The physical acceleration a_A due to what observers in the inertial frame A call real external forces on the object is. therefore, not simply the acceleration a_B seen by observers in the rotational frame B , but has in addition several additional geometric acceleration terms associated with the rotation of B. As seen in the rotational frame, the acceleration a_B of the particle is given by rearrangement of the above equation as:

$\mathbf{a}_{B} = \mathbf{a}_A - 2 \boldsymbol\Omega \times \mathbf{v}_{B} - \boldsymbol\Omega \times (\boldsymbol\Omega \times \mathbf{x}_B ) - \frac{d \boldsymbol\Omega}{dt} \times \mathbf{x}_B \ .$

The net force upon the object according to observers in the rotating frame is F_B = m a_B. 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_fict is present, so the end result is F_B = F_A + F_fict. Thus, the fictitious force used by observers in B to get the correct behavior of the object from Newton's laws equals:

$\mathbf{F}_{\mathrm{fict}} = - 2 m \boldsymbol\Omega \times \mathbf{v}_{B} - m \boldsymbol\Omega \times (\boldsymbol\Omega \times \mathbf{x}_B ) - m \frac{d \boldsymbol\Omega }{dt} \times \mathbf{x}_B \ .$

Here, the first term is the Coriolis force, the second term is the centrifugal force, and the third term is the Euler force.^[15]^[16] When the rate of rotation doesn't change, as is typically the case for a planet, the Euler force is zero.

References

^ Richard Phillips Feynman, Leighton R B & Sands M L (2006). The Feynman Lectures on Physics. San Francisco: Pearson/Addison-Wesley, Vol. I, section 12-5. ISBN 0805390499.
^ Cornelius Lanczos (1986). The Variational Principles of Mechanics. New York: Courier Dover Publications, p. 100. ISBN 0486650677.
^ Seligman, Courtney. Fictitious Forces. Retrieved on 2007-09-03.
^ Max Born & Günther Leibfried (1962). Einstein's Theory of Relativity. New York: Courier Dover Publications, pp.76-78. ISBN 0486607690.
^ (23) Accelerated Frames of Reference: Inertial Forces
^ The term d'Alembert force often is limited to this case. See Lanczos, for example.
^ Vladimir Igorevich Arnolʹd (1989). Mathematical Methods of Classical Mechanics. Berlin: Springer, §27 pp. 129 ff.. ISBN 0387968903.
^ The force in this example is known as ground reaction, and it could exist even without friction, e.g., a sled running down a curve of a bobsled track.
^ Daniel Kleppner and Robert J. Kolenkow (1973). An Introduction to Mechanics. McGraw-Hill, page 363. ISBN 0070350485.
^ Edwin F. Taylor and John Archibald Wheeler (2000) Exploring black holes (Addison Wesley Longman, NY) ISBN 0-201-38423-X
^ ^a ^b John Robert Taylor (2004). Classical Mechanics. Sausalito CA: University Science Books, pp. 343-344. ISBN 189138922X.
^ Vladimir Igorevich Arnolʹd (1989). Mathematical Methods of Classical Mechanics. Berlin: Springer, §27 pp. 129 ff.. ISBN 0387968903.
^ Kleppner, pages 62-63
^ See for example, JL Synge & BA Griffith (1949). Principles of Mechanics, 2nd Edition, McGraw-Hill, pp. 348-349.
^ Cornelius Lanczos (1986). The Variational Principles of Mechanics. New York: Courier Dover Publications, p. 103. ISBN 0486650677.
^ Jerold E. Marsden & Tudor.S. Ratiu (1999). Introduction to Mechanics and Symmetry: A Basic Exposition of Classical Mechanical Systems: Texts in applied mathematics, 17, 2nd Edition, NY: Springer-Verlag, page 251. ISBN 038798643X.

External links

Q and A from Richard C. Brill, Honolulu Community College
NASA
Coriolis Force
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.
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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