When the trajectory of an object travels on a closed path about a point -- either circular or elliptical -- it does so because there is a force pulling the object in the direction of that point. That force is defined as the CENTRIPETAL force. It has not been more simply, or directly stated than by one Isaac Newton in his famous "Principia" (definition 5): "A centripetal force is that by which bodies are drawn or impelled, or any way tend, towards a point as to a center."
This force can be demonstrated by twirling a ball on a string, and either actually or conceptually cutting the string. The ball's trajectory is then a straight line tangential to the closed trajectory at the instant the string is cut. This is also illustrated by what happens to the ball in the "hammer throw" of track and field. The athlete spins the heavy ball around several times then lets it fly. It takes off in a straight line (not quite, because the hammer is actually not spun parallel to the ground, but that is not relevant).
That is really all that is necessary. The term CENRTIFUGAL force appears to have come about because of a mistaken perception that there is a force that operates in the opposite direction as the CENTRIPETAL force. But that is a misconception. The "pull" that is felt by the ball on a string or by the hammer thrower is the force that has to be applied toward the center, to keep the ball from flying off tangentially, not radially.
Unfortunately, the terms are often used interchangably, or incorrectly. Newton's term, which I think should take the prize is CENTRIPETAL.
centripetal force and centrifugal force
centripetal force and centrifugal force, action-reaction force pair associated with circular motion. According to Newton's first law of motion, a moving body travels along a straight path with constant speed (i.e., has constant velocity) unless it is acted on by an outside force. For circular motion to occur there must be a constant force acting on a body, pushing it toward the center of the circular path. This force is the centripetal (“center-seeking”) force. For a planet orbiting the sun, the force is gravitational; for an object twirled on a string, the force is mechanical; for an electron orbiting an atom, it is electrical. The magnitude F of the centripetal force is equal to the mass m of the body times its velocity squared v 2 divided by the radius r of its path: F=mv2/r. According to Newton's third law of motion, for every action there is an equal and opposite reaction. The centripetal force, the action, is balanced by a reaction force, the centrifugal (“center-fleeing”) force. The two forces are equal in magnitude and opposite in direction. The centrifugal force does not act on the body in motion; the only force acting on the body in motion is the centripetal force. The centrifugal force acts on the source of the centripetal force to displace it radially from the center of the path. Thus, in twirling a mass on a string, the centripetal force transmitted by the string pulls in on the mass to keep it in its circular path, while the centrifugal force transmitted by the string pulls outward on its point of attachment at the center of the path. The centrifugal force is often mistakenly thought to cause a body to fly out of its circular path when it is released; rather, it is the removal of the centripetal force that allows the body to travel in a straight line as required by Newton's first law. If there were in fact a force acting to force the body out of its circular path, its path when released would not be the straight tangential course that is always observed.
The Columbia Electronic Encyclopedia, 6th ed. Copyright © 2007, Columbia University Press. All rights reserved.
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Not to be confused with Centripetal force.
Centrifugal force (from Latin centrum, meaning "center", and fugere, meaning "to flee") is the apparent outward force that draws a rotating body away from the center of rotation and is caused by the inertia of the body. In Newtonian mechanics, the term centrifugal force is used to refer to one of two distinct concepts: an inertial force (also called a "fictitious" force) observed in a non-inertial reference frame, and a reaction force corresponding to a centripetal force.
The term is also sometimes used in Lagrangian mechanics to describe certain terms in the generalized force that depend on the choice of generalized coordinates.
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 devices and situations can be analyzed either in terms of the fictitious force in the rotating coordinate system of the motion relative to a center, or in terms of the centripetal and reactive centrifugal forces seen from a non-rotating frame of reference; these different forces are equal in magnitude, but centrifugal and reactive centrifugal forces are opposite in direction to the centripetal force.
History of conceptions of centrifugal and centripetal forces
Main 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. The modern conception as a fictitious force or pseudo force due to a rotating reference frame as described above evolved in the eighteenth and nineteenth centuries.
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. 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.
Action and Reaction are general terms for the interaction of bodies in dynamic situations. Thus principally they refer to situations in which both force and velocity are involved, that is acceleration and velocity. Newton's conceepts of force and quantitiy of motion as Measures were ahead of his time, but not quite what we are taught today, even as Newtonian Principles.
We have to distinguish between the conception and the measure of the same name. The concept of the measure as a conjunction of measures is different to the force as a concept of liveliness. Thus the liveliness in a dynamic situation Newton called in translation an sction or a reaction, and observed that they were a pair, that there was generally no action without an equal reaction in the opposite direction.
But Newton;s disciples were eventually convinced to modify this principle subtly because of circular motion.
Why this occurred is because no one including Newton accepted "action at a distance". There was no model that explained it convincingly and every model which otherwise worked involved some tensile rope or resistive wall.
The centripetal(pressure) and centrigugal(attractive) forces acted the same way according to Cotes, when clearly and experimentally they did not. circular motion existed in dynamic situations where a centre seeking force( a curvature force) was balanced by a centre fleeing force(an evolute force). In Newton's day such forces were not thought to exist in space. Only collisions and that mysterious sticky behaviour of particles in crystals, ropes etc which were lumped under the notion of tensile forces.
Consequently a lot of obfuscation started to cover over Newton's insight. Clearly magnetism, and electrostatics were not considered mechanically until Faraday.
Since classical antiquity, it has been known that some materials such as amber attract lightweight particles after rubbing. The Greek word for amber, ήλεκτρον electron, was the source of the word 'electricity'. Electrostatic phenomena arise from the forces that electric charges exert on each other. Such forces are described by Coulomb's law. Even though electrostatically induced forces seem to be rather weak, the electrostatic force between e.g. an electron and a proton, that together make up a hydrogen atom, is about 40 orders of magnitude stronger than the gravitational force acting between them.......
Coulomb's torsion balance
As reported by the ancient Greek philosopher Thales of Miletus around 600 BC, charge (or electricity) could be accumulated by rubbing fur on various substances, such as amber. The Greeks noted that the charged amber buttons could attract light objects such as hair. They also noted that if they rubbed the amber for long enough, they could even get an electric spark to jump. This property derives from the triboelectric effect.
In 1600, the English scientist William Gilbert returned to the subject in De Magnete, and coined the New Latin word electricus from ηλεκτρον (elektron), the Greek word for "amber", which soon gave rise to the English words "electric" and "electricity." He was followed in 1660 by Otto von Guericke, who invented what was probably the first electrostatic generator. Other European pioneers were Robert Boyle, who in 1675 stated that electric attraction and repulsion can act across a vacuum; Stephen Gray, who in 1729 classified materials as conductors and insulators; and C. F. du Fay, who proposed in 1733 that electricity comes in two varieties that cancel each other, and expressed this in terms of a two-fluid theory. When glass was rubbed with silk, du Fay said that the glass was charged with vitreous electricity, and, when amber was rubbed with fur, the amber was said to be charged with resinous electricity. In 1839, Michael Faraday showed that the apparent division between static electricity, current electricity, and bioelectricity was incorrect, and all were a consequence of the behavior of a single kind of electricity appearing in opposite polarities. It is arbitrary which polarity is called positive and which is called negative. Positive charge can be defined as the charge left on a glass rod after being rubbed with silk.
One of the foremost experts on electricity in the 18th century was Benjamin Franklin, who argued in favour of a one-fluid theory of electricity. Franklin imagined electricity as being a type of invisible fluid present in all matter; for example, he believed that it was the glass in a Leyden jar that held the accumulated charge. He posited that rubbing insulating surfaces together caused this fluid to change location, and that a flow of this fluid constitutes an electric current. He also posited that when matter contained too little of the fluid it was "negatively" charged, and when it had an excess it was "positively" charged. For a reason that was not recorded, he identified the term "positive" with vitreous electricity and "negative" with resinous electricity. William Watson arrived at the same explanation at about the same time.
Boyle's air pump
Boyle's great merit as a scientific investigator is that he carried out the principles which Francis Bacon espoused in the Novum Organum. Yet he would not avow himself a follower of Bacon, or indeed of any other teacher. On several occasions he mentions that in order to keep his judgment as unprepossessed as might be with any of the modern theories of philosophy, until he was "provided of experiments" to help him judge of them, he refrained from any study of the Atomical and the Cartesian systems, and even of the Novum Organum itself, though he admits to "transiently consulting" them about a few particulars. Nothing was more alien to his mental temperament than the spinning of hypotheses. He regarded the acquisition of knowledge as an end in itself, and in consequence he gained a wider outlook on the aims of scientific inquiry than had been enjoyed by his predecessors for many centuries. This, however, did not mean that he paid no attention to the practical application of science nor that he despised knowledge which tended to use.
Boyle was an alchemist; and believing the transmutation of metals to be a possibility, he carried out experiments in the hope of achieving it; and he was instrumental in obtaining the repeal, in 1689, of the statute of Henry IV against multiplying gold and silver. With all the important work he accomplished in physics – the enunciation of Boyle's law, the discovery of the part taken by air in the propagation of sound, and investigations on the expansive force of freezing water, on specific gravities and refractive powers, on crystals, on electricity, on colour, on hydrostatics, etc. – chemistry was his peculiar and favourite study. His first book on the subject was The Sceptical Chymist, published in 1661, in which he criticised the "experiments whereby vulgar Spagyrists are wont to endeavour to evince their Salt, Sulphur and Mercury to be the true Principles of Things." For him chemistry was the science of the composition of substances, not merely an adjunct to the arts of the alchemist or the physician. He endorsed the view of elements as the undecomposable constituents of material bodies; and made the distinction between mixtures and compounds. He made considerable progress in the technique of detecting their ingredients, a process which he designated by the term "analysis". He further supposed that the elements were ultimately composed of particles of various sorts and sizes, into which, however, they were not to be resolved in any known way. He studied the chemistry of combustion and of respiration, and conducted experiments in physiology, where, however, he was hampered by the "tenderness of his nature" which kept him from anatomical dissections, especially vivisections, though he knew them to be "most instructing".......
1674 – two volumes of tracts on the Saltiness of the Sea, Suspicions about the Hidden Realities of the Air, Cold, Celestial Magnets, Animadversions on Hobbes's Problemata de Vacuo
1676 – Experiments and Notes about the Mechanical Origin or Production of Particular Qualities, including some notes on electricity and magnetism
Thus we find that Boyles speculations were not thought relevant in Newton's time to the problem of orbits, and not until Coulomb who immediately recognised the Newtonian gravitational form. No one however was willing to accept, nor even remebered Boyles observation about action in Vacuo. It was considered as some kind of fluid for a while until Faraday reolaced he idea of a fluid model with the conception of a field.
The difference is profound, for the field acts like a fluid, like space, and yet there is no fluid observable. There is no gas abservable or containable etc etc, and it worked in vacuo as Boyle said, In vacuo meant all the gas was extracted.
Now we are not so naive as in Boyle's time, neither was Boyle naive to think all gas could be extracted from a space. But there was and is a quandary as to what is transmitting the force. The posit of a small particle called an electron by Lorentz, and its subsequent discovery after exhaustive experimentation and arguments provided a possible gas particle for the so called vacuum. In the meantime, the "Field" terminology of Faraday gained popularity
In modern textbooks there is a great polisemy as regards the meaning of electric
fi'eld and magnetic/ield, 1'1], ,  and . Field appears defined as a region
of space, as a vectorial function, as something which propagates in space, as
something which stores or contains energy and momentum, as a substance that
mediates interactions between gross bodies etc.
Here we anaJyze how the field concept was presented by Faraday and Max-
well, as these two authors are normally considered the modern initiators of this
concept. Although we restrict our analysis to these famous scientists, we agree
with lIeilbron when he mentioned that "the electricians of 1780 lacked the
word but not the concept, which they called 'sphere of influence', sphaera activitatis,or Wirkungkreis", .
In the rapid pace of Theoretical, technological and experimental development much of the old theoretical or hypothetical or metaphysical models were considered obsolete. In particular the universal acceptance of a medium called an aether was simply by passed to put an end to egregious controversy. Action at a distance was subtly changed to field effect, and by so doing the conceptions of both antagonistic schools of thought had a common ground of agreement on which to proceed.
This led to the fractured description of "Sphaerae Activitas " we have today.
along the way the third law of Newton was severely modified in the case of circular motion to make centrifugal force fictitious, without reviewing the new discoveries in Electromagnetism as rrelevant to the issue..
A force that pushes away is a centrifugal force in circular motion. A force that pulls toward a centre is a centripetal force. Both such forces exist in the electrostatic model. The fluid/field description obscures this obvious experimental fact. Thus for a dipole we have the kind of force field needed to explain circular motion. Newton's third law is applicable in this way to the dynamic situation, and the weak torque action can be modeleled by centrifugal and centripetal models of tensile ropes and circular boundary "pressure" surfaces.