A fictitious force is an apparent force that acts on all masses in a non-inertial frame of reference, e.g., 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 F = ma, fictitious forces are always proportional to the mass m being acted upon. In physics, force is an influence that may cause a body to accelerate. ... In theoretical physics, a common coordinate system or frame of reference, that refers to a non-inertial state of motion can be referred to as a noninertial frame (as opposed to an inertial frame). ... A rotating frame of reference is a coordinate system that describes how physics appears when measured against a hypothetical network of rigid rulers extending from a rotating body. ... In theoretical physics, a common coordinate system or frame of reference, that refers to a non-inertial state of motion can be referred to as a noninertial frame (as opposed to an inertial frame). ... Newtons First and Second laws, in Latin, from the original 1687 edition of the Principia Mathematica. ...

Role as calculational tool

It is sometimes convenient to solve physical problems in a non-inertial reference frame. In such cases, it is necessary to introduce fictitious forces to account for the acceleration of the reference frame. For example, 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), of which the Coriolis force is dominant on Earth. 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. In theoretical physics, a common coordinate system or frame of reference, that refers to a non-inertial state of motion can be referred to as a noninertial frame (as opposed to an inertial frame). ... A frame of reference in physics is a set of axes which enable an observer to measure the aspect, position and motion of all points in a system relative to the reference frame. ... Adjectives: Terrestrial, Terran, Telluric, Tellurian, Earthly Atmosphere Surface pressure: 101. ... A rotating frame of reference is a coordinate system that describes how physics appears when measured against a hypothetical network of rigid rulers extending from a rotating body. ... Classical mechanics is a branch of physics which studies the deterministic motion of objects. ... In physics, the Coriolis effect is an inertial force first described by Gaspard-Gustave Coriolis, a French scientist, in 1835. ... Centrifugal force (from Latin centrum center and fugere to flee) is a term which may refer to two different forces which are related to rotation. ... In physics, the Coriolis effect is an inertial force first described by Gaspard-Gustave Coriolis, a French scientist, in 1835. ... ... In physics, the Coriolis effect is an inertial force first described by Gaspard-Gustave Coriolis, a French scientist, in 1835. ... Foucaults Pendulum in the Panthéon, Paris A Foucault pendulum, or Foucaults pendulum, named after the French physicist Léon Foucault, was conceived as an experiment to demonstrate the rotation of the Earth; its action is a result of the Coriolis effect. ... A carousel in a summer festival in London, with traditional animal mounts, barley twist poles and fairy lights. ...

Detection of non-inertial reference frame

Philosophers^{[attribution needed]} sometimes conjecture that a person living inside a closed box that is rotating (or otherwise accelerating) could not detect their own acceleration (rotation is a form of acceleration). That is not true, a person could not detect their velocity, but they could detect their acceleration (their change in velocity). Careful observers within the box can detect that they are in a non-inertial reference frame from the fictitious forces that arise from the acceleration of the box. They can even map out the magnitude and direction of the acceleration at every point within the box. The method by which this can be done is by detecting the movement of the box in a different rate/direction than their own for at least a short while. In theoretical physics, a common coordinate system or frame of reference, that refers to a non-inertial state of motion can be referred to as a noninertial frame (as opposed to an inertial frame). ...

This is similar to the lurch you would feel if you are driving and the driver suddenly speeds up: for a short while during and after the acceleration, the car is moving faster than you. Once your body catches up through the force of the seat pushing forward against your thigh or back, you stop feeling the lurch or shift (which is, in actuality, the force acting upon you). Another example of the detection of a non-inertial reference frame is the way a Foucault pendulum in a science museum will precess in exactly the same manner, regardless of whether the museum has walls or not. Foucaults Pendulum in the Panthéon, Paris A Foucault pendulum, or Foucaults pendulum, named after the French physicist Léon Foucault, was conceived as an experiment to demonstrate the rotation of the Earth; its action is a result of the Coriolis effect. ...

In the box example, a person standing in the center of a very large box (for demonstration purposes, let's assume the box is large enough that they have some space around them and they're not touching a wall) has a velocity V, which is the same velocity of the box. If the box accelerated backwards rapidly, their feet would accelerate backwards with the box, faster than their body would accelerate, and they would trip forward. Although it might seem to those untrained in physics that a force had pushed him forward, that is an example of a fictitious force that relies on the movement of the box as a frame of reference (a frame of reference which is non-inertial). Someone with physics knowledge would probably be able to tell that the entire box had accelerated backwards, and their feet with it. A rule of thumb is that if you get lurched or turned in one direction, much like the man's feet were lurched backwards, it means the frame of reference you are in is accelerating in that direction.

Observers living inside a closed box that is moving uniformly (i.e., without acceleration) cannot detect their own motion. That is the essential physics of Newton's first two laws of motion. Newtons First and Second laws, in Latin, from the original 1687 edition of the Principia Mathematica. ...

Newtonian 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: Acceleration is the time rate of change of velocity, and at any point on a velocity-time graph, it is given by the slope of the tangent to that point In physics or physical science, acceleration (symbol: a) is defined as the rate of change (or derivative with respect to... In theoretical physics, a common coordinate system or frame of reference, that refers to a non-inertial state of motion can be referred to as a noninertial frame (as opposed to an inertial frame). ...

1. 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.
2. 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.

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 easy in either of the two frames described.) In physics, an inertial frame of reference, or inertial frame for short (also descibed as absolute frame of reference), is a frame of reference in which the observers move without the influence of any accelerating or decelerating force. ... In science, a magnitude is the numerical size of something: see orders of magnitude. ... Unsolved problems in physics: What causes anything to have mass? Mass is a property of a physical object that quantifies the amount of matter and energy it is equivalent to. ...

Circular motion

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: Centrifugal force (from Latin centrum center and fugere to flee) is a term which may refer to two different forces which are related to rotation. ...

1. 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 maintained by the friction of the wheels on the road. The car is accelerating, due to the unbalanced force, which causes it to move in a circle.
2. 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 is balanced by the acceleration of the tires inward, making the car stationary in this non-inertial frame.

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. A centripetal force is a force pulling an object toward the center of a circular path as the object goes around the circle. ... The centripetal force is the external force required to make the body move in a circular path with uniform speed and directed towards the center. ... Friction is the force that opposes the relative motion or tendency of such motion of two surfaces in contact. ... Centrifugal force (from Latin centrum center and fugere to flee) is a term which may refer to two different forces which are related to rotation. ...

Another fictitious force that arises in the case of circular motion 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 m high tower at the equator will fall 7.7 mm eastward of the spot below where it was dropped because of the Coriolis force.^[1] In physics, the Coriolis effect is an inertial force first described by Gaspard-Gustave Coriolis, a French scientist, in 1835. ... Layers of Atmosphere (NOAA) Air redirects here. ... The metre, or meter (symbol: m) is the SI base unit of length. ... World map showing the equator in red The Equator is an imaginary circle drawn around a planet (or other astronomical object) at a distance halfway between the poles. ... A millimetre (American spelling: millimeter), symbol mm is an SI unit of length that is equal to one thousandth of a metre. ...

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. This article is about the astronomical object. ... A spacecraft is a vessel, craft or device designed to operate beyond the surface of the Earth in outer space. ...

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. Mechanical work is a force applied through a distance, defined mathematically as the line integral of a scalar product of force and displacement vectors. ... Mathematically the term trajectory refers to the ordered set of states which are assumed by a dynamical system over time (see e. ... In the physical sciences, potential energy is energy which is captured within a physical system by virtue of the relative positions or configurations of objects, and which has the potential to be released when the system is allowed to attain a configuration with a lower energy state. ... Kinetic energy is the energy by virtue of the motion of an object. ...

Gravity as a fictitious force

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, free falling reference frames are equivalent to an inertial reference frame (the equivalence principle). Following up on this insight, Einstein was able to show (after ~9 years of work) that gravity is indeed a fictitious force; the apparent acceleration is actually inertial motion in curved spacetime. This is the essential physics of Einstein's theory of general relativity. Gravity is a force of attraction that acts between bodies that have mass. ... Albert Einstein ( ) (March 14, 1879 – April 18, 1955) was a German-born theoretical physicist who is widely considered one of the greatest physicists of all time. ... Freefall or free fall in the strict sense is the condition of acceleration which is due only to gravity. ... Freefall or free fall in the strict sense is the condition of acceleration which is due only to gravity. ... In physics, an inertial frame of reference, or inertial frame for short (also descibed as absolute frame of reference), is a frame of reference in which the observers move without the influence of any accelerating or decelerating force. ... In relativity, the equivalence principle is applied to several related concepts dealing with gravitation and the uniformity of physical measurements in different frames of reference. ... Curvature refers to a number of loosely related concepts in different areas of geometry. ... In physics, spacetime is a mathematical model that combines three-dimensional space and one-dimensional time into a single construct called the space-time continuum, in which time plays the role of the 4th dimension. ... General relativity (GR) is the geometrical theory of gravitation published by Albert Einstein in 1915/16. ...

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(t). Since B is non-inertial, we must have that d²X/dt² (the acceleration of frame B with respect to frame A) is non-zero. Let the position of the particle in frame B be x_b(t). Then we have Unsolved problems in physics: What causes anything to have mass? Mass is a property of a physical object that quantifies the amount of matter and energy it is equivalent to. ... Look up position in Wiktionary, the free dictionary. ... This article is about vectors that have a particular relation to the spatial coordinates. ... In physics, an inertial frame of reference, or inertial frame for short (also descibed as absolute frame of reference), is a frame of reference in which the observers move without the influence of any accelerating or decelerating force. ... Acceleration is the time rate of change of velocity, and at any point on a velocity-time graph, it is given by the slope of the tangent to that point In physics or physical science, acceleration (symbol: a) is defined as the rate of change (or derivative with respect to...

Taking two time derivatives, this gives In mathematics, a derivative is the rate of change of a quantity. ...

Now consider the forces in the problem. By Newton's Second Law, F = ma. The true force is of course the one in frame A (the inertial one), so In physics, force is an influence that may cause a body to accelerate. ... Newtons laws of motion are the three scientific laws which Isaac Newton discovered concerning the behaviour of moving bodies. ...

However, suppose we are working to solve a problem in frame B. It may be useful to consider the apparent force in this frame, which is given by

Now we define

giving finally:

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.^[2]

Rotating coordinate systems

A common situation in which noninertial reference frames are useful is when the reference frame is rotating. Since 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.

The relationship between acceleration in an inertial frame, and that in a coordinate frame rotating with angular velocity can be expressed as

where we have used the relationship for the time derivative of a vector in rotating coordinates

, for any vector

Since , the acceleration becomes

or, equivalently,

The acceleration in the rotating frame equals

Since the force in the rotating frame is and, by definition, , the fictitious force equals

Here, the first term is the Coriolis force, the second term is the centrifugal force, and the third term is the Euler force.^[3] When the rate of rotation doesn't change, as is typically the case for a planet, the Euler force is zero. In physics, the Coriolis effect is an inertial force first described by Gaspard-Gustave Coriolis, a French scientist, in 1835. ... Centrifugal force (from Latin centrum center and fugere to flee) is a term which may refer to two different forces which are related to rotation. ...

References

1. ↑  Daniel Kleppner and Robert J. Kolenkow, (1973) An Introduction to Mechanics, McGraw-Hill, page 363.
2. ↑  Kleppner, pages 62-63.
3. ↑  Jerold E. Marsden and Tudor.S. Ratiu, (1994), Introduction to Mechanics and Symmetry: A Basic Exposition of Classical Mechanical Systems, Springer-Verlag, ISBN 0-387-97275-7, page 233.
4. ↑  Kleppner, pages 355-360.
5. ↑  Alexander Fetter and John Walecka, Theoretical Mechanics of particles and continua, McGraw-Hill, pages 33-39.
6. ↑  Lev D. Landau and E. M. Lifshitz, (1976) Mechanics, Butterworth-Heinenan, pages 128-130.
7. Jerry B. Marion, (1970), Classical Dynamics of Particles and Systems, Academic Press.
8. Keith Symon, (1971), Mechanics, Addison-Wesley

Lev Davidovich Landau (Ле́в Дави́дович Ланда́у) (January 22, 1908 – April 1, 1968) was a prominent Soviet physicist and winner of the Nobel Prize for Physics whose broad field of work included...

See also

• Newton's laws of motion
• inertial reference frame
• non-inertial reference frame
• rotating reference frame
• Coriolis force
• Centrifugal force
• Gravity
• General relativity
• d'Alembert's principle of inertial forces
• Centripetal force

Newtons First and Second laws, in Latin, from the original 1687 edition of the Principia Mathematica. ... In physics, an inertial frame of reference, or inertial frame for short (also descibed as absolute frame of reference), is a frame of reference in which the observers move without the influence of any accelerating or decelerating force. ... In theoretical physics, a common coordinate system or frame of reference, that refers to a non-inertial state of motion can be referred to as a noninertial frame (as opposed to an inertial frame). ... A rotating frame of reference is a coordinate system that describes how physics appears when measured against a hypothetical network of rigid rulers extending from a rotating body. ... In physics, the Coriolis effect is an inertial force first described by Gaspard-Gustave Coriolis, a French scientist, in 1835. ... Centrifugal force (from Latin centrum center and fugere to flee) is a term which may refer to two different forces which are related to rotation. ... Gravity is a force of attraction that acts between bodies that have mass. ... General relativity (GR) is the geometrical theory of gravitation published by Albert Einstein in 1915/16. ... DAlembert DAlemberts-Lagrange principle is a statement of the fundamental classical laws of motion. ... The centripetal force is the external force required to make the body move in a circular path with uniform speed and directed towards the center. ...

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.