Centripetal and Centrifugal Forces

The Latin word “petitus” means inclining towards, and the Latin word “fugo” means to drive away. Hence the European scientists of the 17th century (who customarily composed their scholarly treatise in Latin) used the terms centripetal and centrifugal to refer to effects directed towards or away from (respectively) some central point. For example, the effect of the Sun’s gravity was said to be centripetal because it compels an orbiting planet toward the center of the orbit. On the other hand, the inertia of an orbiting object is said to have a centrifugal effect, because continued motion in a straight line would tend to carry the object away from the center. In a circular orbit these two effects are equal, so the object maintains a constant distance from the center.

The meanings of the words “centripetal” (inward) and “centrifugal” (outward) are fairly clear and free of ambiguity, provided both the “central” point and the location of the effect are adequately specified. However, these words are often conjoined with the word “force”, the meaning of which has been the subject of philosophical debate since ancient times. As a result, the meanings of the terms “centripetal force” and (especially) “centrifugal force” have sometimes been obscured.

According to Newton, a material object moves with constant velocity unless acted upon by a force, in which case the object undergoes an acceleration proportional to the applied force. (This is understood to apply only if the velocities and accelerations are defined in terms of a Cartesian inertial coordinate system.) Hence we attribute any change in the velocity of an object to the action of a force on that object. In addition, Newton asserted that “to any action there is always an opposite and equal reaction”. For example, the Sun acts on a planet by applying a centripetal force that continually accelerates the planet, holding it in a (roughly) circular orbit; and likewise the planet acts on the Sun by applying a centripetal force that continually accelerates the Sun, holding it in a (roughly) circular orbit. Newton was the first to recognize that the center of the Sun is not actually the center of the orbits of the planets, because the Sun itself orbits the true center of mass of the solar system. Of course, the radius of the Sun’s orbit is extremely small compared with the radii of the planetary orbits, but nevertheless the Sun does “orbit the Earth” in this sense, just as the Earth orbits the Sun. (Eppur si muove!)

Notice that we have not mentioned any “centrifugal force” in our description of orbiting bodies. This is consistent with the fact that the only forces involved are the mutual forces of gravity that the Sun and planet exert on each other, and these forces compel each body toward the other, and therefore toward the center of their orbits. We could, however, find some ambiguity in the direction of the forces based on the ambiguity in the location of those forces. For example, the planet is “pulling” the Sun inward (i.e., toward their common center of mass) by a force acting in the direction from the Sun toward the planet. If we regard this force as existing at the Sun’s location, then the indicated direction is indeed “inward”, but if we regard this force as existing at the planet, then the indicated direction of the force is actually “outward”. On that basis, we might claim that the force of the planet on the Sun should be called centrifugal rather than centripetal, and the same argument could be made for the force of the Sun on the planet. This ambiguity is especially acute in the context of the Newton-Cotes concept of gravity as an instantaneous “force at a distance”, which makes the location of forces indeterminate. However, it is generally agreed (in the Newtonian context) to identify a force with the associated action, i.e., the deviation of an object from an inertial path. Thus the force exerted by the planet on the Sun is regarded as being located at the Sun, and therefore the force is properly called centripetal (as is the force of the Sun on the planet).

There are, however, examples of genuine centrifugal forces. For example, two electrons repel each other, so the forces could be termed centrifugal, meaning the forces tend to drive the objects away from the “center”. Of course, in such cases, the concept of a “center” is less clear than in the case of closed orbital motion, but it still seems legitimate to regard the source of the repulsion as the “center”. Interestingly, Newton included a brief mention of this kind of “centrifugal force” in the Principia. First he explained that the motion of an object subject to a central force, whose magnitude varies inversely as the square of the distance from the “central” point, is a conic section with the central point at one focus. Then, in Book I, Section 3, Proposition 12, after proving this for one branch of a hyperbolic path, he notes that “if this centripetal force is turned into a centrifugal force, a body will move in the opposite branch of the hyperbola”, as illustrated in the figure below.

Another situation in which centrifugal forces arise is in cases exemplified by a marble rolling around the stationary circular housing of a roulette “wheel”. This is similar to the case of a planet moving in a circular orbit, except that the reaction force exerted by the marble on the housing is applied at the point of contact, and hence is directed outwardly from that point, as shown in the figure below.

Thus the marble exerts a genuine centrifugal (i.e., outward) force on the housing, balancing the centripetal (i.e., inward) force exerted by the housing on the marble, so that the marble maintains a constant distance from the center of the wheel. Indeed Newton used the term “centrifugal force” to describe precisely this kind of force in Principia. (See the Scholium following Proposition 4 of Section 2, Book 1.) However, it must be remembered that Newton was writing in Latin, using the word centrifugal simply as a literally descriptive adjective signifying the direction away from the center.

Confusion about the meaning of the term “centrifugal force” in modern English usage comes about because that Latin word has been adopted to refer to something entirely different than the literal outward force described above. Just to re-iterate, in the preceding example the housing exerts a centripetal force on the marble, which causes the marble to undergo centripetal acceleration, continually diverting it inward from its inertial path, and compelling it to follow a circular path. This is the only force (in the Newtonian sense of the word) being applied to the marble. Admittedly the marble is, in turn, exerting a centrifugal force on the housing, but there is no centrifugal (i.e., outward) force on the marble. The confusion arises if we try to view the situation in terms of a system of coordinates rotating (about the center of the roulette wheel) in such a way that the marble is stationary. The housing is still exerting an inward force on the marble and yet, in terms of this rotating coordinate system, the marble is not accelerating. Needless to say, this is not a violation of Newton’s second law, because Newton’s laws apply only to motions described in terms of inertial coordinate systems, whereas our rotating coordinate system is clearly not an inertial coordinate system.

We might just choose to leave it at that, but in some circumstances there is a desire to make use of Newton’s laws (formally) while working in terms of a non-inertial coordinate system. This can actually be done by introducing certain fictitious forces. For example, in the rotating system of coordinates we must posit a centrifugal force on every (stationary) particle, dependent on the rotational speed of the coordinate system, and varying in proportion to the distance from the center of rotation. This fictitious force exactly balances the inward force on the marble, so the absence of acceleration (in terms of these rotating coordinates) is made formally consistent with Newton’s second law. In other words, we explain why the marble is not accelerating by saying that the net radial force on the marble is zero. Of course, in the inertial sense, the marble actually is accelerating inward, but we are accounting for one fiction by means of another. We are pretending, first, that the marble is not accelerating, and second, that the marble is subject to an outward (centrifugal) force – which explains why it is “not accelerating”.

In general we can consider a Cartesian coordinate system whose origin is at R(t) relative to the origin of some inertial coordinate system, and whose absolute angular velocity is w(t). It isn’t difficult to show that Newton’s second law for a particle of mass m in terms of this coordinate system is

where r is the position vector of the particle relative to this coordinate system. If the origin has no translational acceleration, and if the rotation vector is constant, two of the terms drop out and this equation can be written as

The first term on the left side is the net force applied to the subject particle, and the right hand side is just the familiar “ma” from Newton’s second law. If we define fictitious “inertial forces” equal to the other two terms on the left side of this equation, then we can say, in a formal sense, that Newton’s second law applies in terms of these coordinates. The second term is sometimes called “centrifugal force”, and the third term is sometimes called the “Coriolis force”. Both of these are actually accelerations from the right side of the original equation, but we have brought them over to the left side and called them “forces”, purely in order to maintain the “simple” form of Newton’s second law for this non-inertial coordinate system.

Newton himself made use of the term “centrifugal force” in this fictitious sense. The Scholium following Proposition 4 at the beginning of Book 3 explains the reasoning by which Newton realized that the moon is held in its orbit around the earth by the force of gravity, i.e., the same force that pulls terrestrial objects (like apples) to the ground. He imagines several moons orbiting the earth at different radii, and notes that Kepler’s law for orbiting bodies implies that the inward acceleration is proportional to the inverse square of the orbital radius. If we then imagine the lowest of these moons being at the radius of the mountain tops on earth, we find that its downward acceleration (in accord with Kepler’s law) has the very same value as the downward acceleration of an apple at the top of the mountain. We must therefore conclude that the inward force on orbiting bodies like the moon must be nothing other than the very same force of gravity that pulls apples to the ground. Newton wrote

This centripetal force [implied by Kepler’s law] would cause this little moon, if it were deprived of all the motion with which it had remained in its orbit, to descend to the earth – as a result of the absence of the centrifugal force with which it had remained in its orbit – and to do so with the same velocity with which heavy bodies fall on the tops of those mountains…

Here we see that Newton has tacitly asserted the existence of a centrifugal (outward) force given to the “little moon” by its orbital motion, and he conceives of this centrifugal force as balancing the centripetal force, thereby maintaining the moon at its normal distance. This illustrates how psychologically natural it is for us to “abstract away” the acceleration of an object in circular motion and to conceive of a fictitious (in Newtonian terms) outward force on the object to balance the real inward force. (The same tendency can be seen underlying Galileo’s difficulty in freeing himself from the idea that purely circular motion represented a kind of force-free motion – the idea that prevented him from clearly articulating the rectilinear law of inertia.)

Thus Newton uses the term “centrifugal force” in the Principia to describe three very distinct concepts. First, he uses it to refer to a hypothetical repulsive force (such as the force between two electrons), which would result in a hyperbolic path, accelerating away from the source of the “central” repulsive force. Second, he uses the term to refer to the outward force exerted by a revolving object on some framework (such as the force exerted by a roulette marble on the housing). Third, he uses the term to refer to the “fictitious” outward force on a revolving object when viewed from a revolving frame of reference.

Of course, if the second derivative of R is not zero, the term in the original equation involving that quantity can also be brought over to the left side and treated as a fictitious force. Interestingly, a fictitious force of that kind is found to behave exactly like a “real” homogeneous gravitational field. This fact, due to the proportionality of inertial and gravitational mass established by Galileo and Newton, served as the inspiration for the Equivalence Principle, which led Einstein to the general theory of relativity. According to that theory there is no local physical difference between a “real gravitational force” and a “fictitious inertial force”, because free motion in a gravitational field is understood to be purely inertial motion (locally). In other words, the gravitational field and the inertial field are one and the same, characterized by the ten metric tensor coefficients at each point of spacetime. For more on this topic, see Vis Inertiae.

The question of whether inertial forces are “real” or merely “fictitious” has sometimes been passionately debated – as is usual for matters of definition. One hears it stated confidently that fictitious forces may be distinguished from real forces by the (alleged) fact that the latter are mutually exerted between objects whereas the former – being supposedly just an artifact of a choice of an accelerating coordinate system, are not. However, strictly speaking, the assertion that inertia is intrinsic to each body, rather than being a result of interactions with other objects in the universe, is only a conjecture. Some scientists, notably Ernst Mach, have maintained that inertia actually does arise from interactions with other objects, albeit interactions of a kind different from those with which we are most familiar. Indeed Einstein's general theory of relativity provides some (limited) support for this view, since the inertial behavior of each object is affected by the presence of other objects. Whether it is possible to account for all inertia in this way is an open question, and depends on subtle issues of boundary conditions and the topology of the universe. (A prominent advocate of this view was the late American physicist John Wheeler.) In the context of the "standard model" of quantum field theory, there are presently intense efforts underway to detect the so-called Higgs particle, which is a hypothetical particle responsible for the inertial masses of all other particles. So far the Higgs has never been detected. In the absence of any definitive theory of the origin of inertia, it is customary to disregard these issues, especially in elementary discussions, and simply accept uncritically the Newtonian view that there is such a thing as absolute acceleration (and we know it when we see it), independent of the mean state of motion of all the matter in the universe. Only on this naïve basis can we assert that inertial forces are “fictitious”, i.e., that they do not arise from interactions. General relativity clearly undermines this distinction between real and fictitious forces, because it teaches us that the metric field responsible for the “real” force of gravity is identical with the metric field responsible for the “fictitious” force of inertia.

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