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Centrifugal force
Classical mechanics 

<math>\vec{F} = m\vec{a}</math> 
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In Newtonian mechanics, the term centrifugal force is used to refer to an inertial force (also called a "fictitious" force) that appears to act on all objects when viewed in a rotating reference frame, drawing them away form the axis. The concept of centrifugal force is applied in rotating devices such as centrifuges, centrifugal pumps, centrifugal governors, centrifugal clutches, etc., as well as in centrifugal railways, planetary orbits, banked curves, etc. These situations can be analyzed in terms of the fictitious force in the rotating coordinate system. The name has historically sometimes also been used to refer to the reaction force to the centripetal force. ContentsCurrent meaningMain article: Centrifugal force (rotating reference frame)
Centrifugal force is an outward force apparent in a rotating reference frame; it does not exist when measurements are made in an inertial frame of reference. This type of force, associated with describing motion in a noninertial reference frame is referred to as a fictitious or inertial force (a description that must be understood as a technical usage of these words that means only that the force is not present in a stationary or inertial frame).^{[1]}^{[2]} In a rotating reference frame, all objects appear to be under the influence of a radially (from the axis of rotation) outward force that is proportional to their mass, the distance from the axis of rotation of the frame, and to the square of the (angular velocity) of the frame.^{[3]} ^{[4]} Motion relative to a rotating frame results in another fictitious force, the Coriolis force; and if the rate of rotation of the frame is changing, a third fictitious force, the Euler force is experienced. Together, these three fictitious forces are necessary for the formulation of correct equations of motion in a rotating reference frame.^{[5]} History of conceptions of centrifugal and centripetal forcesMain article: History of centrifugal and centripetal forces
The conception of centrifugal force has evolved since the time of Huygens, Newton, Leibniz, and Hooke who expressed early conceptions of it. Its modern conception as a fictitious force arising in a rotating reference frame evolved in the eighteenth and nineteenth centuries.^{[citation needed]} Centrifugal force has also played a role in debates in classical mechanics about detection of absolute motion. Newton suggested two arguments to answer the question of whether absolute rotation can be detected: the rotating bucket argument, and the rotating spheres argument.^{[6]} According to Newton, in each scenario the centrifugal force would be observed in the object's local frame (the frame where the object is stationary) only if the frame were rotating with respect to absolute space. Nearly two centuries later, Mach's principle was proposed where, instead of absolute rotation, the motion of the distant stars relative to the local inertial frame gives rise through some (hypothetical) physical law to the centrifugal force and other inertia effects. Today's view is based upon the idea of an inertial frame of reference, which privileges observers for which the laws of physics take on their simplest form, and in particular, frames that do not use centrifugal forces in their equations of motion in order to describe motions correctly. The analogy between centrifugal force (sometimes used to create artificial gravity) and gravitational forces led to the equivalence principle of general relativity.^{[7]}^{[8]} Reactive centrifugal forceMain article: Reactive centrifugal force
A reactive centrifugal force is a reaction force to a centripetal force. A body undergoing curved motion, such as circular motion, is accelerating toward a center at any particular point in time. This centripetal acceleration is provided by a centripetal force, which is exerted on the body in curved motion by some other body. In accordance with Newton's third law of motion, the body in curved motion exerts an equal and opposite force on the other body. This reactive force is exerted by the body in curved motion on the other body that provides the centripetal force and its direction is from that other body toward the body in curved motion.^{[9]}^{[10]} ^{[11]}^{[12]} This reaction force is sometimes described as a centrifugal inertial reaction,^{[13]}^{[14]} that is, a force that is centrifugally directed, which is a reactive force equal and opposite to the centripetal force that is curving the path of the mass. The concept of the reactive centrifugal force is sometimes used in mechanics and engineering. It is sometimes referred to as just centrifugal force rather than as reactive centrifugal force.^{[15]}^{[16]} Use of the term in Lagrangian mechanicsSee also: Lagrangian and Mechanics of planar particle motion
Lagrangian mechanics formulates mechanics in terms of generalized coordinates {q_{k}}, which can be as simple as the usual polar coordinates <math>(r,\ \theta)</math> or a much more extensive list of variables.^{[17]}^{[18]} Within this formulation the motion is described in terms of generalized forces, using in place of Newton's laws the Euler–Lagrange equations. Among the generalized forces, those involving the square of the time derivatives {(dq_{k} ⁄ dt )^{2}} are sometimes called centrifugal forces.^{[19]}^{[20]}^{[21]}^{[22]} The Lagrangian approach to polar coordinates that treats <math>(r,\ \theta)</math> as generalized coordinates, <math>(\dot{r},\ \dot{\theta})</math> as generalized velocities and <math>(\ddot{r},\ \ddot{\theta})</math> as generalized accelerations, is outlined in another article, and found in many sources.^{[23]}^{[24]}^{[25]} For the particular case of singlebody motion found using the generalized coordinates <math>(\dot{r},\ \dot{\theta})</math> in a central force, the Euler–Lagrange equations are the same equations found using Newton's second law in a corotating frame. For example, the radial equation is:
where <math>U(r)</math> is the central force potential and μ is the mass of the object. The left side is a "generalized force" and the first term on the right is the "generalized centrifugal force". However, the left side is not comparable to a Newtonian force, as it does not contain the complete acceleration, and likewise, therefore, the terms on the righthand side are "generalized forces" and cannot be interpreted as Newtonian forces.^{[26]} The Lagrangian centrifugal force is derived without explicit use of a rotating frame of reference,^{[27]} but in the case of motion in a central potential the result is the same as the fictitious centrifugal force derived in a corotating frame.^{[28]} The Lagrangian use of "centrifugal force" in other, more general cases, however, has only a limited connection to the Newtonian definition. References

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