History of centrifugal and centripetal forces

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In physics, the history of centrifugal and centripetal forces illustrates a long and complex evolution of thought about the nature of forces, relativity, and the nature of physical laws.

Contents

Huygens, Leibniz, Newton, and Hooke

Early scientific ideas about centrifugal force were based upon intuitive perception, and circular motion was considered somehow more "natural" than straight-line motion. According to Domenico Bertoloni-Meli:

For Huygens and Newton centrifugal force was the result of a curvilinear motion of a body; hence it was located in nature, in the object of investigation. According to a more recent formulation of classical mechanics, centrifugal force depends on the choice of how phenomena can be conveniently represented. Hence it is not located in nature, but is the result of a choice by the observer. In the first case a mathematical formulation mirrors centrifugal force; in the second it creates it. [1]

Christiaan Huygens coined the term "centrifugal force" in his 1659 De Vi Centrifuga [2] and wrote of it in his 1673 Horologium Oscillatorium on pendulums. In 1676–77, Isaac Newton combined Kepler's laws of planetary motion with Huygens' ideas and found

the proposition that by a centrifugal force reciprocally as the square of the distance a planet must revolve in an ellipsis about the center of the force placed in the lower umbilicus of the ellipsis, and with a radius drawn to that center, describe areas proportional to the times. [3]

Newton coined the term "centripetal force" (vis centripeta) in his discussions of gravity in his De motu corporum in gyrum , a 1684 manuscript which he sent to Edmond Halley. [4]

Gottfried Leibniz as part of his "solar vortex theory" conceived of centrifugal force as a real outward force which is induced by the circulation of the body upon which the force acts. An inverse cube law centrifugal force appears in an equation representing planetary orbits, including non-circular ones, as Leibniz described in his 1689 Tentamen de motuum coelestium causis. [5] Leibniz's equation is still used today to solve planetary orbital problems, although his solar vortex theory is no longer used as its basis. [6]

Leibniz produced an equation for planetary orbits in which the centrifugal force appeared as an outward inverse cube law force in the radial direction: [7]

.

Newton himself appears to have previously supported an approach similar to that of Leibniz. [8] Later, Newton in his Principia crucially limited the description of the dynamics of planetary motion to a frame of reference in which the point of attraction is fixed. In this description, Leibniz's centrifugal force was not needed and was replaced by only continually inward forces toward the fixed point. [7] Newton objected to Leibniz's equation on the grounds that it allowed for the centrifugal force to have a different value from the centripetal force, arguing on the basis of his third law of motion, that the centrifugal force and the centripetal force must constitute an equal and opposite action-reaction pair. In this however, Newton was mistaken, as the reactive centrifugal force which is required by the third law of motion is a completely separate concept from the centrifugal force of Leibniz's equation. [8] [9]

Huygens, who was, along with Leibniz, a neo-Cartesian and critic of Newton, concluded after a long correspondence that Leibniz's writings on celestial mechanics made no sense, and that his invocation of a harmonic vortex was logically redundant, because Leibniz's radial equation of motion follows trivially from Newton's laws. Even the most ardent modern defenders of the cogency of Leibniz's ideas acknowledge that his harmonic vortex as the basis of centrifugal force was dynamically superfluous. [10]

It has been suggested that the idea of circular motion as caused by a single force was introduced to Newton by Robert Hooke. [9]

Newton described the role of centrifugal force upon the height of the oceans near the equator in the Principia :

Since the centrifugal force of the parts of the earth, arising from the earth's diurnal motion, which is to the force of gravity as 1 to 289, raises the waters under the equator to a height exceeding that under the poles by 85472 Paris feet, as above, in Prop. XIX., the force of the sun, which we have now shewed to be to the force of gravity as 1 to 12868200, and therefore is to that centrifugal force as 289 to 12868200, or as 1 to 44527, will be able to raise the waters in the places directly under and directly opposed to the sun to a height exceeding that in the places which are 90 degrees removed from the sun only by one Paris foot and 113 V inches; for this measure is to the measure of 85472 feet as 1 to 44527.

— Newton: Principia Corollary to Book II, Proposition XXXVI. Problem XVII

The effect of centrifugal force in countering gravity, as in this behavior of the tides, has led centrifugal force sometimes to be called "false gravity" or "imitation gravity" or "quasi-gravity". [11]

Eighteenth century

It wasn't until the latter half of the 18th century that the modern "fictitious force" understanding of the centrifugal force as a pseudo-force artifact of rotating reference frames took shape. [12] In a 1746 memoir by Daniel Bernoulli, "the idea that the centrifugal force is fictitious emerges unmistakably." [13] Bernoulli, in seeking to describe the motion of an object relative to an arbitrary point, showed that the magnitude of the centrifugal force depended on which arbitrary point was chosen to measure circular motion about. Later in the 18th century Joseph Louis Lagrange in his Mécanique Analytique explicitly stated that the centrifugal force depends on the rotation of a system of perpendicular axes. [13] In 1835, Gaspard-Gustave Coriolis analyzed arbitrary motion in rotating systems, specifically in relation to waterwheels. He coined the phrase "compound centrifugal force" for a term which bore a similar mathematical expression to that of centrifugal force, albeit that it was multiplied by a factor of two. [14] The force in question was perpendicular to both the velocity of an object relative to a rotating frame of reference and the axis of rotation of the frame. Compound centrifugal force eventually came to be known as the Coriolis Force. [15] [16]

Absolute versus relative rotation

The idea of centrifugal force is closely related to the notion of absolute rotation. In 1707 the Irish bishop George Berkeley took issue with the notion of absolute space, declaring that "motion cannot be understood except in relation to our or some other body". In considering a solitary globe, all forms of motion, uniform and accelerated, are unobservable in an otherwise empty universe. [17] This notion was followed up in modern times by Ernst Mach. For a single body in an empty universe, motion of any kind is inconceivable. Because rotation does not exist, centrifugal force does not exist. Of course, addition of a speck of matter just to establish a reference frame cannot cause the sudden appearance of centrifugal force, so it must be due to rotation relative to the entire mass of the universe. [18] The modern view is that centrifugal force is indeed an indicator of rotation, but relative to those frames of reference that exhibit the simplest laws of physics. [19] Thus, for example, if we wonder how rapidly our galaxy is rotating, we can make a model of the galaxy in which its rotation plays a part. The rate of rotation in this model that makes the observations of (for example) the flatness of the galaxy agree best with physical laws as we know them is the best estimate of the rate of rotation [20] (assuming other observations are in agreement with this assessment, such as isotropy of the background radiation of the universe). [21]

Role in developing the idea of inertial frames and relativity

In the rotating bucket experiment, Newton observed the shape of the surface of water in a bucket as the bucket was spun on a rope. At first the water is flat, then, as it acquires the same rotation as the bucket, it becomes parabolic. Newton took this change as evidence that one could detect rotation relative to "absolute space" experimentally, in this instance by looking at the shape of the surface of the water.

Later scientists pointed out (as did Newton) that the laws of mechanics were the same for all observers that differed only by uniform translation; that is, all observers that differed in motion only by a constant velocity. Hence, "absolute space" was not preferred, but only one of a set of frames related by Galilean transformations. [22]

By the end of the nineteenth century, some physicists had concluded that the concept of absolute space is not really needed...they used the law of inertia to define the entire class of inertial frames. Purged of the concept of absolute space, Newton's laws do single out the class of inertial frames of reference, but assert their complete equality for the description of all mechanical phenomena.

— Laurie M. Brown, Abraham Pais, A. B. Pippard: Twentieth Century Physics, pp. 256-257

Ultimately this notion of the transformation properties of physical laws between frames played a more and more central role. [23] It was noted that accelerating frames exhibited "fictitious forces" like the centrifugal force. These forces did not behave under transformation like other forces, providing a means of distinguishing them. This peculiarity of these forces led to the names inertial forces, pseudo-forces or fictitious forces. In particular, fictitious forces did not appear at all in some frames: those frames differing from that of the fixed stars by only a constant velocity. In short, a frame tied to the "fixed stars" is merely a member of the class of "inertial frames", and absolute space is an unnecessary and logically untenable concept. The preferred, or "inertial frames", were identifiable by the absence of fictitious forces. [24] [25] [26]

The effect of his being in the noninertial frame is to require the observer to introduce a fictitious force into his calculations….

— Sidney Borowitz and Lawrence A Bornstein in A Contemporary View of Elementary Physics, p. 138

The equations of motion in a non-inertial system differ from the equations in an inertial system by additional terms called inertial forces. This allows us to detect experimentally the non-inertial nature of a system.

— V. I. Arnol'd: Mathematical Methods of Classical Mechanics Second Edition, p. 129

The idea of an inertial frame was extended further in the special theory of relativity. This theory posited that all physical laws should appear of the same form in inertial frames, not just the laws of mechanics. In particular, Maxwell's equations should apply in all frames. Because Maxwell's equations implied the same speed of light in the vacuum of free space for all inertial frames, inertial frames now were found to be related not by Galilean transformations, but by Poincaré transformations, of which a subset is the Lorentz transformations. That posit led to many ramifications, including Lorentz contractions and relativity of simultaneity. Einstein succeeded, through many clever thought experiments, in showing that these apparently odd ramifications in fact had very natural explanation upon looking at just how measurements and clocks actually were used. That is, these ideas flowed from operational definitions of measurement coupled with the experimental confirmation of the constancy of the speed of light.

Later the general theory of relativity further generalized the idea of frame independence of the laws of physics, and abolished the special position of inertial frames, at the cost of introducing curved space-time. Following an analogy with centrifugal force (sometimes called "artificial gravity" or "false gravity"), gravity itself became a fictitious force, [27] as enunciated in the equivalence principle. [28]

The principle of equivalence: There is no experiment observers can perform to distinguish whether an acceleration arises because of a gravitational force or because their reference frame is accelerating

— Douglas C. Giancoli Physics for Scientists and Engineers with Modern Physics, p. 155

In short, centrifugal force played a key early role in establishing the set of inertial frames of reference and the significance of fictitious forces, even aiding in the development of general relativity.

The modern conception

The modern interpretation is that centrifugal force in a rotating reference frame is a pseudo-force that appears in equations of motion in rotating frames of reference, to explain effects of inertia as seen in such frames. [29]

Leibniz's centrifugal force may be understood as an application of this conception, as a result of his viewing the motion of a planet along the radius vector, that is, from the standpoint of a special reference frame rotating with the planet. [7] [8] [30] Leibniz introduced the notions of vis viva (kinetic energy) [31] and action, [32] which eventually found full expression in the Lagrangian formulation of mechanics. In deriving Leibniz's radial equation from the Lagrangian standpoint, a rotating reference frame is not used explicitly, but the result is equivalent to that found using Newtonian vector mechanics in a co-rotating reference frame. [33] [34] [35]

Related Research Articles

Isaac Newton's rotating bucket argument was designed to demonstrate that true rotational motion cannot be defined as the relative rotation of the body with respect to the immediately surrounding bodies. It is one of five arguments from the "properties, causes, and effects" of "true motion and rest" that support his contention that, in general, true motion and rest cannot be defined as special instances of motion or rest relative to other bodies, but instead can be defined only by reference to absolute space. Alternatively, these experiments provide an operational definition of what is meant by "absolute rotation", and do not pretend to address the question of "rotation relative to what?" General relativity dispenses with absolute space and with physics whose cause is external to the system, with the concept of geodesics of spacetime.

<span class="mw-page-title-main">Coriolis force</span> Force on objects moving within a reference frame that rotates with respect to an inertial frame

In physics, the Coriolis force is an inertial force that acts on objects in motion within a frame of reference that rotates with respect to an inertial frame. In a reference frame with clockwise rotation, the force acts to the left of the motion of the object. In one with anticlockwise rotation, the force acts to the right. Deflection of an object due to the Coriolis force is called the Coriolis effect. Though recognized previously by others, the mathematical expression for the Coriolis force appeared in an 1835 paper by French scientist Gaspard-Gustave de Coriolis, in connection with the theory of water wheels. Early in the 20th century, the term Coriolis force began to be used in connection with meteorology.

In classical physics and special relativity, an inertial frame of reference is a frame of reference not undergoing any acceleration. It is a frame in which an isolated physical object—an object with zero net force acting on it—is perceived to move with a constant velocity or, equivalently, it is a frame of reference in which Newton's first law of motion holds. All inertial frames are in a state of constant, rectilinear motion with respect to one another; in other words, an accelerometer moving with any of them would detect zero acceleration. It has been observed that celestial objects which are far away from other objects and which are in uniform motion with respect to the cosmic microwave background radiation maintain such uniform motion.

In physics and astronomy, a frame of reference is an abstract coordinate system whose origin, orientation, and scale are specified by a set of reference points―geometric points whose position is identified both mathematically and physically.

In physics, the principle of relativity is the requirement that the equations describing the laws of physics have the same form in all admissible frames of reference.

In theoretical physics, particularly in discussions of gravitation theories, Mach's principle is the name given by Albert Einstein to an imprecise hypothesis often credited to the physicist and philosopher Ernst Mach. The hypothesis attempted to explain how rotating objects, such as gyroscopes and spinning celestial bodies, maintain a frame of reference.

Galilean invariance or Galilean relativity states that the laws of motion are the same in all inertial frames of reference. Galileo Galilei first described this principle in 1632 in his Dialogue Concerning the Two Chief World Systems using the example of a ship travelling at constant velocity, without rocking, on a smooth sea; any observer below the deck would not be able to tell whether the ship was moving or stationary.

<span class="mw-page-title-main">Absolute space and time</span> Theoretical foundation of Newtonian mechanics

Absolute space and time is a concept in physics and philosophy about the properties of the universe. In physics, absolute space and time may be a preferred frame.

A fictitious force is a force that appears to act on a mass whose motion is described using a non-inertial frame of reference, such as a linearly accelerating or rotating reference frame. Fictitious forces are invoked to maintain the validity and thus use of Newton's second law of motion, in frames of reference which are not inertial.

As described by the third of Newton's laws of motion of classical mechanics, all forces occur in pairs such that if one object exerts a force on another object, then the second object exerts an equal and opposite reaction force on the first. The third law is also more generally stated as: "To every action there is always opposed an equal reaction: or the mutual actions of two bodies upon each other are always equal, and directed to contrary parts." The attribution of which of the two forces is the action and which is the reaction is arbitrary. Either of the two can be considered the action, while the other is its associated reaction.

<span class="mw-page-title-main">Rotating reference frame</span> Concept in classical mechanics

A rotating frame of reference is a special case of a non-inertial reference frame that is rotating relative to an inertial reference frame. An everyday example of a rotating reference frame is the surface of the Earth.

In theoretical physics a Coriolis field is one of the apparent gravitational fields felt by a rotating or forcibly-accelerated body, together with the centrifugal field and the Euler field.

A non-inertial reference frame is a frame of reference that undergoes acceleration with respect to an inertial frame. An accelerometer at rest in a non-inertial frame will, in general, detect a non-zero acceleration. While the laws of motion are the same in all inertial frames, in non-inertial frames, they vary from frame to frame depending on the acceleration.

In classical mechanics, a reactive centrifugal force forms part of an action–reaction pair with a centripetal force.

<span class="mw-page-title-main">Centrifugal force</span> Type of inertial force

In Newtonian mechanics, the centrifugal force is an inertial force that appears to act on all objects when viewed in a rotating frame of reference. It is directed radially away from the axis of rotation. The magnitude of centrifugal force F on an object of mass m at the distance r from the axis of rotation of a frame of reference rotating with angular velocity ω is:

<span class="mw-page-title-main">Classical mechanics</span> Description of large objects physics

Classical mechanics is a physical theory describing the motion of macroscopic objects, from projectiles to parts of machinery and astronomical objects, such as spacecraft, planets, stars, and galaxies. The "classical" in "classical mechanics" does not refer to classical antiquity, as it might in, say, classical architecture. On the contrary, the development of classical mechanics involved substantial change in the methods and philosophy of physics. Instead, the qualifier distinguishes classical mechanics from physics developed after the revolutions of the early 20th century, which revealed limitations of classical mechanics.

Mechanics of planar particle motion is the analysis of the motion of particles gravitationally attracted to one another observed from non-inertial reference frames and the generalization of this problem to planetary motion. This type of analysis is closely related to centrifugal force, two-body problem, orbit and Kepler's laws of planetary motion. The mechanics of planar particle motion fall in the general field of analytical dynamics, and helps determining orbits from the given force laws. This article is focused more on the kinematic issues surrounding planar motion, which are the determination of the forces necessary to result in a certain trajectory given the particle trajectory.

Isaac Newton's rotating spheres argument attempts to demonstrate that true rotational motion can be defined by observing the tension in the string joining two identical spheres. The basis of the argument is that all observers make two observations: the tension in the string joining the bodies and the rate of rotation of the spheres. Only for the truly non-rotating observer will the tension in the string be explained using only the observed rate of rotation. For all other observers a "correction" is required that accounts for the tension calculated being different from the one expected using the observed rate of rotation. It is one of five arguments from the "properties, causes, and effects" of true motion and rest that support his contention that, in general, true motion and rest cannot be defined as special instances of motion or rest relative to other bodies, but instead can be defined only by reference to absolute space. Alternatively, these experiments provide an operational definition of what is meant by "absolute rotation", and do not pretend to address the question of "rotation relative to what?" General relativity dispenses with absolute space and with physics whose cause is external to the system, with the concept of geodesics of spacetime.

<span class="mw-page-title-main">Absolute rotation</span> Rotation independent of any external reference

In physics, the concept of absolute rotation—rotation independent of any external reference—is a topic of debate about relativity, cosmology, and the nature of physical laws.

References

  1. ↑ Domenico Bertoloni Meli (March 1990). "The Relativization of Centrifugal Force". Isis. The University of Chicago Press on behalf of The History of Science Society. 81 (1): 23–43. doi:10.1086/355247. JSTOR   234081. S2CID   144526407.
  2. ↑ Soshichi Uchii (October 9, 2001). "Inertia" . Retrieved 2008-05-25.
  3. ↑ "Anni Mirabiles". Lapham's Quarterly. Retrieved 2020-08-27.
  4. ↑ The Mathematical Papers of Isaac Newton. Vol. VI. Cambridge: University Press. 2008. ISBN   978-0-521-04585-8.
  5. ↑ Donald Gillies (1995). Revolutions in Mathematics. Oxford: University Press. p. 130. ISBN   978-0-19-851486-2.
  6. ↑ Herbert Goldstein (1980). Classical mechanics (2nd ed.). Addison-Wesley. p. 74. ISBN   978-0-201-02918-5.
  7. 1 2 3 Christopher M. Linton (2004). From Eudoxus to Einstein: a history of mathematical astronomy. Cambridge University Press. pp. 264–285. ISBN   978-0-521-82750-8.
  8. 1 2 3 Frank Swetz (1997). Learn from the masters!. MAA. pp. 268–269. ISBN   978-0-88385-703-8.
  9. 1 2 "Newton, Sir Isaac" . Retrieved 2008-05-25.
  10. ↑ A. R. Hall, Philosophers at War, 2002, pp 150-151
  11. ↑ M. Novello, Matt Visser & G. E. Volovik (2002). Artificial black holes. World Scientific. p. 200. ISBN   981-02-4807-5.
  12. ↑ Wilson (1994). "Newton's Orbit Problem: A Historian's Response". The College Mathematics Journal. Mathematical Association of America. 25 (3): 193–200. doi:10.2307/2687647. ISSN   0746-8342. JSTOR   2687647.
  13. 1 2 Meli 1990, "The Relativization of Centrifugal Force".
  14. ↑ René Dugas and J. R. Maddox (1988). A History of Mechanics. Courier Dover Publications. p. 387. ISBN   0-486-65632-2.
  15. ↑ Persson, Anders (July 1998). "How Do We Understand the Coriolis Force?". Bulletin of the American Meteorological Society79 (7): pp. 1373–1385. ISSN   0003-0007.
  16. ↑ Frederick Slate (1918). The Fundamental Equations of Dynamics and its Main Coordinate Systems Vectorially Treated and Illustrated from Rigid Dynamics. Berkeley, CA: University of California Press. p.  137. compound centrifugal force coriolis.
  17. ↑ Edward Robert Harrison (2000). Cosmology (2nd ed.). Cambridge University Press. p. 237. ISBN   0-521-66148-X.
  18. ↑ Ernst Mach (1915). The science of mechanics. The Open Court Publishing Co. p.  33. ISBN   0-87548-202-3. Try to fix Newton's bucket and rotate the heaven of fixed stars and then prove the absence of centrifugal forces
  19. ↑ J. F. Kiley, W. E. Carlo (1970). "The epistemology of Albert Einstein". Einstein and Aquinas. Springer. p. 27. ISBN   90-247-0081-7.
  20. ↑ Henning Genz (2001). Nothingness. Da Capo Press. p. 275. ISBN   0-7382-0610-5.[ permanent dead link ]
  21. ↑ J. Garcio-Bellido (2005). "The Paradigm of Inflation". In J. M. T. Thompson (ed.). Advances in Astronomy. Imperial College Press. p. 32, §9. ISBN   1-86094-577-5.
  22. ↑ Laurie M. Brown, Abraham Pais & A. B. Pippard (1995). Twentieth Century Physics. CRC Press. pp. 256–257. ISBN   0-7503-0310-7.
  23. ↑ The idea of transformation properties of physical laws under various transformations is a central topic in modern physics, related to such basic concepts as conservation laws like conservation of energy and momentum through Noether's theorem. See, for example, Harvey R. Brown (2005). Physical Relativity. Oxford University Press. p. 180. ISBN   0-19-927583-1., and Gennady Gorelik (2002). Yuri Balashov; Vladimir Pavlovich Vizgin (eds.). Einstein Studies in Russia. Birkhäuser. p. The problem of conservation laws and the Poincare quasigroup in general relativity; pp. 17 ff. ISBN   0-8176-4263-3. and Peter Mittelstaedt & Paul Weingartner (2005). Laws of Nature. Springer. p. 80. ISBN   3-540-24079-9.
  24. ↑ Milton A. Rothman (1989). Discovering the Natural Laws: The Experimental Basis of Physics . Courier Dover Publications. p.  23. ISBN   0-486-26178-6. reference laws of physics.
  25. ↑ Sidney Borowitz & Lawrence A. Bornstein (1968). A Contemporary View of Elementary Physics. McGraw-Hill. p. 138. ASIN   B000GQB02A.
  26. ↑ V. I. Arnol'd (1989). Mathematical Methods of Classical Mechanics. Springer. p. 129. ISBN   978-0-387-96890-2.
  27. ↑ Hans Christian Von Baeyer (2001). The Fermi Solution: Essays on science (Reprint of 1993 ed.). Courier Dover Publications. p. 78. ISBN   0-486-41707-7.
  28. ↑ Douglas C. Giancoli (2007). Physics for Scientists and Engineers with Modern Physics. Pearson Prentice Hall. p. 155. ISBN   978-0-13-149508-1.
  29. ↑ Charles Proteus Steinmetz (2005). Four Lectures on Relativity and Space. Kessinger Publishing. p. 49. ISBN   1-4179-2530-2.
  30. ↑ E. J. Aiton (1 March 1962). "The celestial mechanics of Leibniz in the light of Newtonian criticism". Annals of Science. Taylor & Francis. 18 (1): 31–41. doi:10.1080/00033796200202682.
  31. ↑ Bertrand Russell (1992). A Critical Exposition of the Philosophy of Leibniz (Reprint of 1937 2nd ed.). Routledge. p. 96. ISBN   0-415-08296-X.
  32. ↑ Wolfgang Lefèvre (2001). Between Leibniz, Newton, and Kant. Springer. p. 39. ISBN   0-7923-7198-4.
  33. ↑ Herbert Goldstein (2002). Classical Mechanics. San Francisco : Addison Wesley. pp. 74–77, 176. ISBN   0-201-31611-0.
  34. ↑ John Taylor (2005). Classical Mechanics. University Science Books. pp. 358–359. ISBN   1-891389-22-X.
  35. ↑ Whiting, J.S.S. (November 1983). "Motion in a central-force field" (PDF). Physics Education. 18 (6): 256–257. Bibcode:1983PhyEd..18..256W. doi:10.1088/0031-9120/18/6/102. ISSN   0031-9120. S2CID   250750602 . Retrieved May 7, 2009.
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