Centrifugal force 7532 226050640 2008-07-16T16:38:39Z Brews ohare 5680311 /* Moving objects and frames of reference */ link {{for|the ''real'' outward-acting force that can be found in circular motion|Reactive centrifugal force}} {{for|the external force required to make a body follow a curved path|Centripetal force}} {{for|general derivations and discussion of fictitious forces|Fictitious force}} In [[classical mechanics]], '''centrifugal force''' (from [[Latin]] ''centrum'' "center" and ''fugere'' "to flee") is an apparent force acting outward from the [[axis of rotation]] of a [[rotating reference frame]].<!-- REF --><ref>{{cite web|url=http://dlmcn.com/circle.html|title=Centrifugal and Coriolis Effects|author=David McNaughton|accessdate=2008-05-18}}</ref><ref>{{cite web|title=Centrifugal Effect|url=http://www.physics.sfsu.edu/~lwilliam/111/shm/tsld027.htm|author=Lynda Williams|accessdate=2008-05-18}}</ref> A centrifugal force is a particular kind of ''[[fictitious force]]''<!-- REF --><ref>{{cite web |url=http://scienceworld.wolfram.com/physics/CentrifugalForce.html |publisher=scienceworld.wolfram.com |title=Centrifugal Force |accessdate=2008-05-18 }}</ref><ref>{{cite web |url=http://www.britannica.com/eb/article-9022100/centrifugal-force |title=Centrifugal Force - Britannica online encyclopedia |accessdate=2008-05-18 }}</ref><!-- --> (also known as a ''pseudo force'', ''inertial force'' or ''d'Alembert force''), that exists for observers in a rotating reference frame.<!-- REF --><ref name=Born>{{cite book |title=Einstein's Theory of Relativity|author=Max Born & Günther Leibfried|url=http://books.google.com/books?id=Afeff9XNwgoC&pg=PA76&dq=%22inertial+forces%22&lr=&as_brr=0&sig=0kiN27BqUqHaZ9CkPdqLIjr-Nnw#PPA77,M1 |page=pp.76-78|isbn=0486607690|publisher=Courier Dover Publications|location=New York|year=1962}}</ref><ref>[http://www-istp.gsfc.nasa.gov/stargaze/Sframes2.htm NASA: ''Accelerated Frames of Reference: Inertial Forces'']</ref><ref name=regentsprep>[http://regentsprep.org/Regents/physics/phys06/bcentrif/centrif.htm Science Joy Wagon: ''Centrifugal force - the ''false'' force'']</ref><!-- --> Unlike real [[force]]s such as [[electromagnetic]] forces, fictitious forces do not originate from physical interactions between objects. In a frame that rotates about a fixed axis, the fictitious forces present are the ''centrifugal force'' and the ''[[Coriolis force]]'', to which is added the ''[[Euler force]]'' when angular velocity is time dependent. The centrifugal force appears to act on anything with [[mass]], depends only on the position and the mass of the object, and always is oriented outward from the axis of rotation of the rotating frame. The Coriolis force depends on both the velocity and mass of the object, and is directed perpendicular to its velocity, but is independent of its position. The Euler force depends on the mass and position of the object, and on the angular acceleration of the reference frame.<!-- REF --><ref name=Marsden>{{cite book |title=Introduction to Mechanics and Symmetry: A Basic Exposition of Classical Mechanical Systems |author=Jerrold E. Marsden, Tudor S. Ratiu |isbn=038798643X |year=1999 |publisher=Springer |page=p. 251 |url=http://books.google.com/books?id=I2gH9ZIs-3AC&pg=PA251&vq=Euler+force&dq=isbn:038798643X&source=gbs_search_s&sig=ACfU3U0DkJL1h3lGMMbXyKY15GtPpspHuQ}}</ref><ref name=Taylor1/><ref name=Marion/> ==Scope of article== [[Image:Gyroscope precession.gif|thumb|right|Figure 1: Precession of the frame of a gyroscope. The second and third [[Euler angles]] of the frame are constant]] The table of contents suggests the content of this article, which is not an exhaustive treatment of the subject. Some very common situations are not included, one being the simple gyroscope shown in Figure 1. In Figure 1 the gyroscope rotates about its own axis, while in turn its own axis rotates about a vertical fixed axis. Such compound motions lead to rather complicated fictitious forces, and are not treated here. The translation and rotation of a [[rigid body]] is governed by the ''[[Newton-Euler equations]]''.<!-- REF --><ref name=Hahn>The general case of a rigid body under translation of its center of mass and general rotation in orientation is beyond the scope of this article. See {{cite book |title=Rigid Body Dynamics of Mechanisms |author=Hubert Hahn |page=p. 143 |url=http://books.google.com/books?id=MqrN3KY7o6MC&pg=PA143&dq=EUler+equations+%22rigid+body%22&lr=&as_brr=0&sig=ACfU3U00jfE08smw1IqJt69QdcMSKvDIeA |isbn=3540423737 |publisher=Springer |year=2002}}, {{cite book |title=Computational Dynamics |author=Ahmed A. Shabana |page= p. 379 |url=http://books.google.com/books?id=dGfcbOsm2PwC&pg=PA379&dq=EUler+equations+%22rigid+body%22&lr=&as_brr=0&sig=ACfU3U01BZBb84es37aiHVpdE33IdGze-A |isbn=0471053260 |year=2001 |publisher=Wiley }}, {{cite book |title=Robot Analysis and Control |author=Haruhiko Asada, Jean-Jacques E. Slotine |page=§5.1.1, p. 94 |url=http://books.google.com/books?id=KUG1VGkL3loC&pg=PA94&dq=EUler+equations+%22rigid+body%22&lr=&as_brr=0&sig=ACfU3U3LiZyQRj0zYXQ8ON2zwuiiwQO7dA |isbn=0471830291 |publisher=Wiley/IEEE |year=1986 }}</ref> ==Analysis using fictitious forces== Fictitious forces do not appear in the [[Newton's laws of motion|equations of motion]] in an [[inertial frame of reference]]: in an inertial frame, the motion of an object is explained by the real impressed forces. In a non-inertial frame such as a rotating frame, however, Newton's first and second laws still can be used to make accurate physical predictions provided fictitious forces are included along with the real forces. For solving problems of mechanics in non-inertial reference frames, the advice given in textbooks is:<!-- REF --><ref name=Rao1>{{cite book |title=Classical Mechanics |author=K.S. Rao |page=p. 162 |url=http://books.google.com/books?id=Al7LRzoQhxsC&pg=PA268&vq=by+adding+the+terms&dq=real+%22inertial+forces%22&lr=&as_brr=0&source=gbs_search_s&sig=ACfU3U0uHeQp1Yb_zTlGk50Hj40vcMVkHg#PPA162,M1 |isbn=8173714363 |year=2003 |publisher=Orient Longman }}</ref><!-- --> <!-- REF --><ref name=Quote>{{cite book |title=Analytical Mechanics |author =Louis N. Hand, Janet D. Finch |page=p. 267 |url=http://books.google.com/books?id=1J2hzvX2Xh8C&pg=PA267&vq=fictitious+forces&dq=Hand+inauthor:Finch&lr=&as_brr=0&source=gbs_search_s&sig=ACfU3U33emV_6eJZihu3M6IZKurSt85_eg |isbn=0521575729 |publisher=Cambridge University Press |year=1998 }}</ref> {{Quotation|Treat the fictitious forces like real forces, and pretend you are in an inertial frame.|Louis N. Hand, Janet D. Finch ''Analytical Mechanics'', p. 267}} Because fictitious forces do not originate from other objects, there is no originating object to experience an associated reaction force: [[Newton's laws of motion#Newton's third law: law of reciprocal actions|Newton's third law]] does not apply to fictitious forces. ===Advantages of rotating frames=== A rotating reference frame can have advantages over an inertial reference frame.<!-- REF --><ref name=Taylor1>{{cite book |title=Classical Mechanics|author=John Robert Taylor|title=|title=Classical Mechanics |page=Chapter 9, pp. 327 ff |url=http://books.google.com/books?id=P1kCtNr-pJsC&pg=PP1&dq=isbn:189138922X&lr=&as_brr=0&sig=JVfFlMT5TvXh1I64JAFBFq7pA6s#PPA327,M1 |isbn=189138922X|publisher=University Science Books|location=Sausalito CA|year=2004}}</ref><ref name=Marion>{{cite book |title=Classical Dynamics of Particles and Systems |author=Stephen T. Thornton & Jerry B. Marion |page=Chapter 10 |year=2004 |isbn=0534408966 |publisher=Brook/Cole |location=Belmont CA |edition=5th Edition |url=http://worldcat.org/oclc/52806908&referer=brief_results}}</ref><!-- --> Sometimes the calculations are simpler (an example is [[Coriolis_force#Inertial_circles|inertial circles]]), and sometimes the intuitive picture coincides more closely with the rotational frame (an example is [[sedimentation]] in a centrifuge). With the addition of centrifugal force and other fictitious forces, Newton's first and second laws can be used to think about these systems, and to calculate motions within them. A specific example: centrifugal force is used in the FAA pilot's manual in describing turns.<!-- REF --><ref name=FAA>{{cite book |title=Pilot's Encyclopedia of Aeronautical Knowledge |author=Federal Aviation Administration |page=Figure 3-21 |url=http://books.google.com/books?id=m5V04SXE4zQC&pg=PT33&lpg=PT33&dq=+%22angle+of+bank%22&source=web&ots=iYTi_mZAra&sig=ytjcmr9RStdIdgZzaiBJJ-wxjts&hl=en |isbn=1602390347 |year=2007 |publisher=Skyhorse Publishing Inc. |location=Oklahoma City OK }}</ref><!-- --> Other examples are such systems as planets, centrifuges, carousels, turning cars, spinning buckets, and rotating space stations.<!-- REF --><ref name=Hubbard>{{cite book |title=Boater's Bowditch: The Small Craft American Practical Navigator |author=Richard Hubbard |page=p. 54 |url=http://books.google.com/books?id=nfWSxRr8VP4C&pg=PA54&dq=tides+centrifugal&lr=&as_brr=0&sig=ACfU3U2e_gEEDUG4mB1nO2GS21kCJwUJVQ |publisher=McGraw-Hill Professional|location=NY |year=2000 |isbn=0071361367}}</ref><!-- REF --><ref name=Wang>{{cite book |title=Handbook of Environmental Engineering: Air and Noise Pollution Control |author=Lawrence K. Wang, Norman C. Pereira |page=p. 63 |url=http://books.google.com/books?id=FkSnJZSbmxUC&pg=PA63&dq=cyclone+centrifugal&lr=&as_brr=0&sig=ACfU3U3ew6FQLiwvfqMzp8tYIUClK3LrtQ |isbn=0896030016 |year=1979 |publisher=Humana Press }}</ref><!-- <!-- REF --><ref name=Grenci>{{cite book |title=A World of Weather: Fundamentals of Meteorology |author=Lee M. Grenci, Jon M. Nese |page=p. 272 |url=http://books.google.com/books?id=oh8lqM5obuYC&pg=PA272&dq=meteorology+centrifugal&lr=&as_brr=0&sig=ACfU3U35-omMijGtQ7ASqruNiZYGdIuteA |isbn=0787277169 |year=2001 |publisher=Kendall Hunt }}</ref><!-- --> ==Choice of frame of reference== Centrifugal force can be confusing, which is nothing new to physics.<!-- REF --><ref name= Hestenes>{{cite journal |title=Force Concept Inventory |publisher=Physics Teacher, v30 n3 p141-58 Mar 1992 |url=http://blue.butler.edu/~phenders/STUFF/Concept%20Inventory/Force%20Concept%20Inventory.pdf |author=David Hestenes, Malcolm Wells, and Gregg Swackhamer}}</ref><!-- --> The ideas of Newtonian mechanics had to overcome numerous "common sense" perceptions, among them the belief that [[friction]] is an inherent property of motion rather than an externally applied force,<!-- REF --><ref name=Newton>{{cite book |title=From Clockwork to Crapshoot: A History of Physics|author=Roger G. Newton |page= p. 27 |location=Cambridge MA |url=http://books.google.com/books?id=KxlotI-7sWMC&pg=RA1-PA27&dq=Newton+friction+history&lr=&as_brr=0&sig=coDmt8UtlB8TsHr1E-hzXmy-g_M#PRA1-PA27,M1 |publisher=Belknap Press of Harvard University Press |year=2007 |isbn=0674023374}}</ref><ref name=Matthews>{{cite book |title=Time for science education: how teaching the history and philosophy of pendulum motion can contribute to science literacy |author=Michael R. Matthews |page= p. 84 |url=http://books.google.com/books?id=vCtYnEuW7TIC&pg=PA84&dq=Newton+friction+history&lr=&as_brr=0&sig=dWzVRIdwHgLnphrCLgh_7Vebkmk |publisher=Springer |location= New York |year=2000 |isbn=0306458802}}</ref><!-- --> and the failure to recognize that change in the ''direction of motion'' can be as important as change in speed (that is, the concept of velocity as a [[vector]] quantity).<!-- REF --><ref name=Dusek>{{cite book |title=The Holistic Inspirations of Physics: The Underground History of Electromagnetic Theory |url=http://books.google.com/books?id=UuEvFlgKkKIC&pg=PA187&dq=Newton+velocity+history&lr=&as_brr=0&sig=-80WxbfLgSECXn_pHzA36hRVV1k |author=Val Dusek |isbn=0813526353 |page=p. 187 |publisher=Rutgers University Press |location=New Brunswick NJ |year=1999}}</ref><!-- --> In particular, an object in circular motion must continuously change direction. This directional change requires an inward ''[[centripetal force]]'' that keeps the object on its circular path by changing the direction of its velocity; without the centripetal force, the object would follow a straight trajectory. However, intuition and common sense may disagree with this viewpoint. A source of confusion is the instinctive adoption of a reference frame, which can be unconscious, as in the example explored below. Consider a car going around a turn. A passenger may experience centrifugal force: as the car turns, the passenger feels pushed against the door by a force acting toward the outside of the curve. That interpretation of experience is the view from one reference frame: a non-inertial, rotating reference frame. Experience is interpreted in terms of the fictitious centrifugal force. Contrast this view with a rather more cerebral (but accurate) description of what the passenger feels as provided by Newton's laws in an inertial frame of reference; the passenger can imagine watching themselves in a kind of "[[out of body experience]]" or [[autoscopy]]. According to that description, the passenger tends to travel in a straight line, but because the car is going in a circle, it pushes the passenger inward (''not'' outward) to keep them turning. In accordance with Newton's third law, the passenger applies an outward reaction force to the car door, but no outward force acts on the passenger; the force the passenger feels pushing them outward against the door disappears in an inertial frame. Both descriptions are valid in the sense that either will lead to a successful design of the catch on the car door to retain the passenger inside the vehicle without popping open, and either description will lead to a correct banking of the curve on the road. Further discussion of this example can be found in the article on [[Reactive centrifugal force#Example: The turning car|reactive centrifugal force]]. As another illustration of the difference between reference frames, suppose we swing a ball around our head on a string. A natural viewpoint is that the ball is pulling on the string, and we have to resist that pull or the ball will fly away. That perspective puts us in a ''rotating frame of reference'' – we are reacting to the ball and have to fight centrifugal force. A less intuitive frame of mind is that we have to keep pulling on the ball, or else it will not change direction to stay in a circular path. That is, we are in an active frame of mind: we have to supply ''centripetal force''. That puts us in an inertial frame of reference. The centrifuge supplies another example, where often the rotating frame is preferred and centrifugal force is treated explicitly.<!-- REF --><ref name =Spinney2>{{cite book |url=http://books.google.com/books?id=5zgFAAAAMAAJ&pg=PA47&dq=%22circular+motion%22&lr=&as_brr=0#PPA52,M1|author=Louis Bevier Spinney|title=§47: The cream separator and centrifugal drier|page=p. 52|year=1911}}</ref><!-- --> This example can become more complicated than the ball on string, however, because there may be forces due to [[friction]], [[buoyancy]], and [[diffusion]]; not just the fictitious forces of rotational frames. The balance between dragging forces like friction and driving forces like the centrifugal force is called [[sedimentation]]. A complete description leads to the [[Lamm equation]].<!-- REF --><ref name=Rubinow>{{cite book |title=Introduction to mathematical biology |author = SI Rubinow |page=pp. 235-244 |url=http://books.google.com/books?id=3j0gu63QWmQC&pg=PA250&dq=Lamm+equation&sig=mH9T2kyIE0PIqMGzh6WF-KGyApc#PPA235,M1 |isbn=0486425320 |year=2002 (1975) |publisher=Courier/Dover Publications}}</ref><ref name=Mazumdar>{{cite book |title=An Introduction to Mathematical Physiology and Biology |url=http://books.google.com/books?id=mqD6qSYGM-QC&pg=PA33&dq=Lamm+equation&lr=&sig=qiZ-6RAfQsRqDAzyuOJsiz4gOlE |page= pp. 33 ff |author= Jagannath Mazumdar |year=1999 |publisher=Cambridge University Press |location=Cambridge UK |isbn=0521646758 }}</ref> Intuition can go either way, and we can become perplexed when we switch viewpoints unconsciously. Standard physics teaching is often ineffective in clarifying these intuitive perceptions,<!-- REF --><ref name= Hestenes/><ref name=Ploetzner>{{cite web |title=The Acquisition of Qualitative Physics Knowledge during Textbook-Based Physics Training |author=R. Ploetzner and K. Van Lehn |url=http://citeseer.ist.psu.edu/647586.html}}</ref><!-- --> and beliefs about centrifugal force (and other such forces) grounded in the rotating frame often remain fervently held as somehow real regardless of framework, despite the classical explanation that such descriptions always are framework dependent.<!-- REF --><ref name= Hestenes/><ref name=Ploetzner/><ref name=Glenbrook>{{cite web |url=http://www.glenbrook.k12.il.us/GBSSCI/PHYS/Class/circles/u6l1d.html |title=The Forbidden F-Word |author=Tom Henderson}}</ref> == Are centrifugal and Coriolis forces "real"? == {{seealso|Gravitron}} {{anchor|postulate}}According to the first postulate of the special theory of relativity:<!-- REF --><ref name=Einstein>{{cite book |title=The Principle of Relativity: a collection of original memoirs on the special and general theory of relativity |author=Einstein, A., Lorentz, H. A., Minkowski, H., & Weyl, H. |page=p. 111 |url=http://books.google.com/books?id=yECokhzsJYIC&pg=PA111&dq=postulate+%22Principle+of+Relativity%22&lr=&as_brr=0&sig=ACfU3U2GFu6cWo5WHzgumuNEqum_fBiTiw |isbn=0486600815 |publisher=Courier Dover Publications |year=1952 }}</ref><!-- -->{{Quotation|''Special principle of relativity'': If a system of coordinates K is chosen so that, in relation to it, physical laws hold good in their simplest form, the ''same'' laws hold good in relation to any other system of coordinates K' moving in uniform translation relatively to K.|Albert Einstein: ''The foundation of the general theory of relativity'', Section A, §1}} which postulate allows identification of what is termed an ''inertial frame of reference''. The centrifugal and Coriolis forces are called ''fictitious'' because they do not appear in an inertial frame of reference. In effect, one can identify an inertial frame by the observation that morphing fictitious forces are not present, forces that change form from one reference frame to another (and disappear altogether in an inertial frame). (An example is the [[#Rotating_identical_spheres|rotating spheres]] example below, where the fictitious forces are shown to disappear in an inertial frame and to take on a variety of forms in other frames.) Besides their variation from frame to frame, fictitious forces can be identified because, unlike real forces, they do not originate in objects in the environment.<!-- REF --><ref name=Taylor>{{cite book |title=Classical Mechanics |author=John Robert Taylor|page=pp. 343-344 |location=Sausalito CA |isbn=189138922X |year=2004 |url=http://books.google.com/books?id=P1kCtNr-pJsC&pg=PP1&dq=isbn:189138922X&lr=&as_brr=0&sig=JVfFlMT5TvXh1I64JAFBFq7pA6s#PPA343,M1 |publisher=University Science Books}}</ref> The distinction between real and fictitious forces is important in physics, where the study of [[Fundamental interaction|interactions]] between bodies is a major topic, for example, in the [[standard model]] and the [[theory of everything]]. However, apart from such fundamental issues of physics, for purposes of mechanics, fictitious forces are treated as real by observers within accelerating or rotating environments. An interesting exploration of the apparent reality of centrifugal forces is provided by the [[artificial gravity]] introduced into a space station by rotation.<!-- REF --><ref>[http://science.nasa.gov/headlines/y2003/07feb_stronggravity.htm The pull of hypergravity]</ref><!-- --> Such a form of gravity does have things in common with ordinary gravity. For example, playing catch, the ball must be thrown upward to counteract "gravity". Cream will rise to the top of milk (if it is not homogenized). There are differences from ordinary gravity: one is the rapid change in "gravity" with distance from the center of rotation, which would be very noticeable unless the space station were very large. More disconcerting is the associated [[Coriolis effect (perception)|Coriolis force]], which introduces signals in the ear that conflict with vision, leading to nausea.<!-- REF --><ref>For more detail see [http://www.spacefuture.com/archive/artificial_gravity_and_the_architecture_of_orbital_habitats.shtml Hall: ''Artificial gravity and the architecture of orbital habitats''].</ref><ref>[http://www.spacefuture.com/archive/inhabiting_artificial_gravity.shtml Hall: ''Inhabiting artificial gravity]</ref> Differences between artificial and real gravity can affect human health, and are a subject of study.<ref> See, for example, [http://dspace.mit.edu/bitstream/1721.1/35291/1/74276829.pdf Pouly] and [http://www.annalsnyas.org/cgi/content/abstract/871/1/367 Young].</ref><!-- --> In any event, the fictitious forces in this habitat would seem perfectly real to those living in the station. Although they readily could do experiments to reveal the space station was rotating, the inhabitants of the space station would find description of daily life remained more natural in terms of fictitious forces. From a time-honored viewpoint,<!-- REF --><ref name=Thorpe>Einstein wrote: "Things should be made as simple as possible, but not any simpler" {{cite book |title=How to think like Einstein: Simple ways to break the rules and discover your hidden genius |author=Scott Thorpe |url=http://books.google.com/books?id=9yrYQxBgIYEC&pg=PA35&dq=Einstein+simplest+theory&lr=&as_brr=0&sig=ACfU3U15NC3BwHgEid4gV0P_vpyPyJm5Cg |page=p. 35 |isbn=1570715858 |year=2000 |publisher=Sourcebooks }} </ref><!-- --> the simplest explanation is often to be preferred. The simplest explanation often involves fictitious forces. == Fictitious forces == {{main|Fictitious force}} An alternative to dealing with a rotating frame of reference from the inertial standpoint is to make Newton's laws of motion valid in the rotating frame by artificially adding pseudo forces to be the cause of the above acceleration terms, and then working directly in the rotating frame.<!-- REF --><ref name=Marsden/><ref name=Finch>{{cite book |title=Analytical Mechanics |author=Louis N. Hand & Janet D. Finch |page=p. 267 |url=http://books.google.com/books?id=1J2hzvX2Xh8C&pg=PA267&vq=centrifugal&dq=Hand+inauthor:Finch&lr=&as_brr=0&source=gbs_search_s&sig=ACfU3U33emV_6eJZihu3M6IZKurSt85_eg#PPA267,M1 Finch |publisher=Cambridge University Press|location=Cambridge UK |year=1998 |isbn=0521575729}}</ref><ref name=Taylor4>{{cite book |title=Classical Mechanics|author=John Robert Taylor|title=|title=Classical Mechanics |page=Chapter 9, p. 329 |url=http://books.google.com/books?id=P1kCtNr-pJsC&pg=PA300&vq=fictitious&dq=isbn:189138922X&lr=&as_brr=0&source=gbs_search_s&sig=ACfU3U3BpST4RE58iL2Hs5EPAoRha3zrfQ#PPA329,M1 |isbn=189138922X|publisher=University Science Books|location=Sausalito CA|year=2004}}</ref><!-- --> Here attention is restricted to frames rotating about a fixed axis. (For a discussion of complex rotations of a [[rigid body]], see [[Euler_angles#Relationship_with_physical_motions|Euler angles]].) In such frames, the centrifugal acceleration is added to the motion of every object, and attributed to a centrifugal force, given by: :{| |- |<math>\mathbf{F}_\mathrm{centrifugal} \,</math> |<math>= m \mathbf{a}_\mathrm{centrifugal} \,</math> |- | |<math>=m \omega^2 \mathbf{r}_\perp \,</math> |} where <math>m\,</math> is the mass of the object, and '''r'''<sub><math>\perp</math></sub> is the vector that locates the object relative to the center of rotation ('''r'''<sub><math>\perp</math></sub> is perpendicular to the axis of rotation and points outward to the location of the rotating object). This pseudo or fictitious centrifugal force is a sufficient correction to Newton's second law only if the body is stationary in the rotating frame. For bodies that move with respect to the rotating frame it must be supplemented with a second pseudo force, the "[[Coriolis force]]":<!-- REF --><ref name=Stommel>{{cite book |author=Henry M. Stommel & Dennis W. Moore|title=An Introduction to the Coriolis Force|url=http://books.google.com/books?id=-JQx_t3yGB4C&printsec=frontcover&dq=%22Coriolis+force%22&lr=&as_brr=0&sig=jvW_Oadj0Z8IgsA_wUnwwqkwWB0#PPA12,M1 |page=p. 12|isbn=0231066368|year=1989|publisher=Columbia University Press}}</ref> :<math>\mathbf{F}_\mathrm{coriolis} = -2 \, m \, \boldsymbol{\Omega} \times \boldsymbol v_{rot}\ ,</math> where '''v'''<sub>rot</sub> is the velocity ''as seen in the rotating frame of reference''. Here is an example. A body that is stationary relative to the ''non''-rotating inertial frame will be rotating when viewed from the rotating frame. Therefore, Newton's laws, as applied to what looks like circular motion in the rotating frame, requires an inward ''centripetal'' force of −''m'' ω<sup>2</sup> '''r'''<sub><math>\perp</math></sub> to account for the apparent circular motion. This centripetal force in the rotating frame is provided as the sum of the ''radially outward'' centrifugal pseudo force ''m'' ω<sup>2</sup> '''r'''<sub><math>\perp</math></sub> and the Coriolis force −2''m'' '''Ω <big>×</big> v'''.<!-- REF --><ref name=Joos>{{cite book |title=Theoretical Physics|author= Georg Joos & Ira M. Freeman|url=http://books.google.com/books?id=vIw5m2XuvpIC&pg=PA233&dq=inauthor:joos+coriolis&ei=EpgtSMitA4vcywSokozNAw&sig=wveOPKIvSGTCKQSpw-2jFQRe79M#PPA233,M1 |page=p. 233 |isbn=0486652270 |publisher=Courier Dover Publications |location=New York|year=1986}}</ref> <ref name=Taylor2>{{cite book |title=Classical Mechanics |author=John Robert Taylor|page=pp. 348-349 |location=Sausalito CA |isbn=189138922X |year=2004 |url=http://books.google.com/books?id=P1kCtNr-pJsC&pg=PP1&dq=isbn:189138922X&lr=&as_brr=0&sig=JVfFlMT5TvXh1I64JAFBFq7pA6s#PPA348,M1 |publisher=University Science Books}}</ref><!-- --> To evaluate the Coriolis force, we need the velocity ''as seen in the rotating frame''. Some pondering will show that this velocity is given by '''−Ω <big>×</big> r'''. Hence, the Coriolis force (in this example) is inward, in the opposite direction to the centrifugal force, and has the value −2''m'' ω<sup>2</sup> '''r'''<sub><math>\perp</math></sub>. The combination of the centrifugal and Coriolis force is then ''m'' ω<sup>2</sup> '''r'''<sub><math>\perp</math></sub>−2''m'' ω<sup>2</sup> '''r'''<sub><math>\perp</math></sub> = −''m'' ω<sup>2</sup> '''r'''<sub><math>\perp</math></sub>, exactly the centripetal force required by Newton's laws for circular motion.<!-- REF --><ref name=Spinney>{{cite book |title=A Text-book of Physics|author=Louis Bevier Spinney|page=pp. 47-49 |url=http://books.google.com/books?id=5zgFAAAAMAAJ&pg=PA47&dq=%22circular+motion%22&lr=&as_brr=0#PPA47,M1 |year=1911|publisher=Macmillan Co. }}</ref> <ref name=Schaum>{{cite book |title=Applied physics: Based on Schaum's Outline of Theory and Problems of Applied Physics (Third Edition)|url=http://books.google.com/books?id=6OKTgr-BXpIC&pg=PA37&dq=%22circular+motion%22&lr=&as_brr=0&sig=z2tyhG5FG4Nya41Intbn9EZ1GRI |page=p. 37 |author=Arthur Beiser & George J. Hademenos|isbn=0071398783|publisher=McGraw-Hill Professional|year=2003}}</ref><ref>{{cite journal |last=Burgel |first=B. |title=Centrifugal Force| journal=American Journal of Physics |volume=35 |pages=649 |year=1967 |doi=10.1119/1.1974204}}</ref> For further examples and discussion, see [[Centrifugal_force#Examples|below]], and see Taylor.<!-- REF --><ref name=Taylor3>{{cite book |title=pp. 349ff|author=John Robert Taylor |isbn=189138922X |year=2004 |url=http://books.google.com/books?id=P1kCtNr-pJsC&pg=PP1&dq=isbn:189138922X&lr=&as_brr=0&sig=JVfFlMT5TvXh1I64JAFBFq7pA6s#PPA349,M1 }}</ref> Because this centripetal force is combined from only pseudo forces, it is "fictitious" in the sense of having no apparent origin from physical sources (unlike electrical, magnetic or gravitational fields, which ''are'' produced by bodies in the environment), the combination of pseudo forces simply is posited as a "fact of life" in the rotating frame, it is just "there". It has to be included as a force in Newton's laws if calculations of trajectories in the rotating frame are to come out right. === Moving objects and frames of reference=== [[Image:Local unit vectors.PNG|thumb |315px|Figure 2: Local coordinate system for planar motion on a curve. Two different positions are shown for distances ''s'' and ''s'' + ''ds'' along the curve. At each position ''s'', unit vector '''u'''<sub>n</sub> points along the outward normal to the curve and unit vector '''u'''<sub>t</sub> is tangential to the path. The radius of curvature of the path is ρ as found from the rate of rotation of the tangent to the curve with respect to arc length, and is the radius of the [[osculating circle]] at position ''s''. The unit circle on the left shows the rotation of the unit vectors with ''s''.]] {{seealso|Generalized forces|Generalized force|Curvilinear coordinates|Generalized coordinates|Frenet-Serret formulas}} In discussion of a particle moving in a circular orbit,<ref name=particle>The term ''particle'' is used in mechanics to describe an object without reference to its orientation. The term [[rigid body]] is used when orientation is also a factor. Thus, the [[center of mass]] of a rigid body is a "particle". </ref> one can identify the centripetal and tangential forces. It then seems to be no problem to switch hats and talk about the fictitious centrifugal and Euler forces. But what underlies this switch is a change of frame of reference from the inertial frame where we started, where centripetal and tangential forces make sense, to a rotating frame of reference where the particle appears motionless and fictitious centrifugal and Euler forces have to be brought into play. That switch is unconscious, but real. Suppose we sit on an an particle in planar motion. What switch of hats leads to fictitious centrifugal and Euler forces? To explore that question, first consider a coordinate system commonly used in planar motion, the so-called [[Centripetal_force#Local_coordinates|''local'' coordinate system]]. See Figure 2. [Note: ''coordinate systems'' and ''frames of reference'' are independent. A ''frame of reference'' refers to the state of motion of the observer. A ''coordinate system'' refers to a choice of language used to describe observations. Any frame of reference can choose to employ any coordinate system (polar, elliptical, curvilinear, …) to describe observations in that system. A change in coordinate system does not mean a change in frame of reference, and ''vice versa''.] To introduce the unit vectors of the local coordinate system, one approach is to begin in Cartesian coordinates in an inertial framework and describe the local coordinates in terms of these Cartesian coordinates. In Figure 2, the [[arc length]] ''s'' is the distance the particle has traveled along its path in time ''t''. The path '''r''' (''t'') with components ''x''(''t''), ''y''(''t'') in Cartesian coordinates is described using arc length ''s''(''t'') as:<ref name=curvature>The article on [[curvature]] treats a more general case where the curve is parametrized by an arbitrary variable (denoted ''t''), rather than by the arc length ''s''.</ref> :<math>\mathbf{r}(s) = \left[ x(s),\ y(s) \right] \ . </math> One way to look at the use of ''s'' is to think of the path of the particle as sitting in space, like the trail left by a [[Skywriting|skywriter]], independent of time. Then an incremental displacement along the path ''ds'' is described by: :<math>d\mathbf{r}(s) = \left[ dx(s),\ dy(s) \right]=\left[ x'(s),\ y'(s) \right] ds \ , </math> where primes are introduced to denote derivatives with respect to ''s''. The magnitude of this displacement is ''ds'', showing that:<ref name=Shabana>{{cite book |title=Railroad Vehicle Dynamics: A Computational Approach |author=Ahmed A. Shabana, Khaled E. Zaazaa, Hiroyuki Sugiyama |page=p. 91 |url=http://books.google.com/books?id=YgIDSQT0FaUC&pg=RA1-PA207&dq=%22generalized+coordinate%22&lr=&as_brr=0&sig=ACfU3U2tosoLUEAUNkGu2x8TTtuxLfeLGQ#PRA1-PA91,M1 |isbn=1420045814 |publisher=CRC Press |year=2007 }}</ref> :<math>\left[ x'(s)^2 + y'(s)^2 \right] = 1 \ . </math>&ensp;&ensp;&ensp;&ensp;{{anchor|Eq. 1}}(Eq. 1) This displacement is necessarily tangent to the curve at ''s'', showing that the unit vector tangent to the curve is: :<math>\mathbf{u}_t(s) = \left[ x'(s), \ y'(s) \right] \ , </math> while the outward unit vector normal to the curve is :<math>\mathbf{u}_n(s) = \left[ y'(s),\ -x'(s) \right] \ , </math> Orthogonality can be verified by showing the vector [[dot product]] is zero. The unit magnitude of these vectors is a consequence of [[#Eq. 1|Eq. 1]]. As an aside, notice that the use of unit vectors that are not aligned along the Cartesian ''xy''-axes does not mean we are no longer in an inertial frame. All it means is that we are using unit vectors that vary with ''s'' to describe the path, but still observe the motion from the inertial frame. Using the tangent vector, the angle of the tangent to the curve, say θ, is given by: :<math>\sin \theta =\frac{y'(s)}{\sqrt{x'(s)^2+y'(s)^2}} = y'(s) \ ;</math> &ensp; and &ensp; <math>\cos \theta =\frac{x'(s)}{\sqrt{x'(s)^2+y'(s)^2}} = x'(s) \ .</math> The radius of curvature is introduced completely formally (without need for geometric interpretation) as: :<math>\frac{1}{\rho} = \frac{d\theta}{ds}\ . </math> The derivative of θ can be found from that for sin θ: :<math>\frac{d \sin\theta}{ds} = \cos \theta \frac {d\theta}{ds} = \frac{1}{\rho} \cos \theta \ </math> ::<math> = \frac{1}{\rho} x'(s)\ . </math> Now: :<math>\frac{d \sin \theta }{ds} = \frac{d}{ds} \frac{y'(s)}{\sqrt{x'(s)^2+y'(s)^2}}</math>&ensp;&ensp;<math>= \frac{y''(s)x'(s)^2-y'(s)x'(s)x''(s)} {\left(x'(s)^2+y'(s)^2\right)^{3/2}}\ , </math> in which the denominator is unity according to [[#Eq. 1|Eq. 1]]. With this formula for the derivative of the sine, the radius of curvature becomes: :<math>\frac {d\theta}{ds} = \frac{1}{\rho} = y''(s)x'(s) - y'(s)x''(s)\ </math>&ensp;<math>=\frac{y''(s)}{x'(s)} = -\frac{x''(s)}{y'(s)} \ ,</math> where the equivalence of the forms stems from differentiation of [[#Eq. 1|Eq. 1]]: :<math>x'(s)x''(s) + y'(s)y''(s) = 0 \ . </math> With these results, the acceleration in the inertial reference frame as described in terms of the components normal and tangential to the path of the particle can be found by introducing the rate at which the path is traveled, that is, by introducing the function ''s''(''t''): :<math>\mathbf{a}(s) = \frac{d}{dt}\mathbf{v}(s) </math>&ensp;&ensp;<math>= \frac{d}{dt}\left[\frac{ds}{dt} \left( x'(s), \ y'(s) \right) \right]\ </math> ::<math> = \left(\frac{d^2s}{dt^2}\right)\mathbf{u}_t(s) +\left(\frac{ds}{dt}\right) ^2 \left(x''(s),\ y''(s) \right) </math> ::<math> = \left(\frac{d^2s}{dt^2}\right)\mathbf{u}_t(s) - \left(\frac{ds}{dt}\right) ^2 \frac{1}{\rho} \mathbf{u}_n(s) \ , </math> as can be verified by taking the dot product with the unit vectors '''u'''<sub>t</sub>(''s'') and '''u'''<sub>n</sub>(''s''). This result for acceleration is the same as that for circular motion based on the radius ρ. Using this coordinate system in the inertial frame, it is easy to identify the force normal to the trajectory as the centripetal force and that parallel to the trajectory as the tangential force. Next, we change frames. Sitting on the particle, we adopt a non-inertial frame where the particle is at rest (zero velocity). This frame has a continuously changing origin, which at time ''t'' is the center of curvature of the path at time ''t'', and whose rate of rotation is the angular rate of motion of the particle about that origin at time ''t''. This non-inertial frame also employs unit vectors normal to the trajectory and parallel to it. The [[angular velocity]] of this frame is the angular velocity of the particle about the center of curvature at time ''t''. The centripetal force of the inertial frame becomes the force necessary to overcome the centrifugal force in the non-inertial frame where the body is at rest. Likewise, the force causing any acceleration of speed along the path seen in the inertial frame becomes the force necessary to overcome the Euler force in the non-inertial frame where the particle is at rest. There is zero Coriolis force in the frame, because the particle has zero velocity in this frame. For a pilot in an airplane, for example, these fictitious forces are a matter of direct experience.<ref name=Coriolis>However, the pilot also will experience Coriolis force, because the pilot is not a ''particle''. When the pilot's head moves, for example, the head has a velocity in the non-inertial frame, and becomes subject to Coriolis force. This force causes pilot disorientation in a turn. See [[Coriolis effect (perception)]], {{cite book |title=Space biology and medicine |author=Arnauld E. Nicogossian |page=p. 337 |url=http://books.google.com/books?id=aO6zut2K7lsC&pg=PA337&dq=Coriolis+effect+airplane+nausea&lr=&as_brr=0&sig=ACfU3U2ODlvCKri-JbJfB-OdyhXyhozbnw |isbn=1563471809 |year=1996 |publisher=American Institute of Aeronautics and Astronautics, Inc |location=Reston, VA}}, and {{cite book |title=Fundamentals of Space Medicine |author=Gilles Clément |page=p. 41 |url=http://books.google.com/books?id=Neura4O-taIC&pg=PA41&dq=Coriolis+effect+airplane+nausea&lr=&as_brr=0&sig=ACfU3U16LpzGILe3QVGIeOl5tyYFDAKLrA |isbn=1402015984 |year=2003 |publisher=Springer}}.</ref> However, these fictitious forces cannot be related to a simple observational frame of reference other than the particle itself, unless it is in a particularly simple path, like a circle. That said, from a qualitative standpoint, the path of an airplane can be approximated by an arc of a circle for a limited time, and for the limited time a particular radius of curvature applies, the centrifugal and Euler forces can be analyzed on the basis of circular motion with that radius. See article discussing [[Banked turn#Aviation| turning an airplane]]. Next, reference frames rotating about a fixed axis are discussed in more detail. == Uniformly rotating reference frames== {{seealso|Circular motion|Uniform circular motion|Inertial reference frame}} Rotating reference frames are used in physics, mechanics, or [[meteorology]] whenever they are the most convenient frame to use. The laws of physics are the same in all [[inertial frames]]. But a [[rotating reference frame]] is not an inertial frame, so the laws of physics are transformed from the inertial frame to the rotating frame. For example, assuming a ''constant'' rotation speed, transformation is achieved by adding to every object two ''coordinate accelerations'' that correct for the constant rotation of the coordinate axes. The [[vector equation]]s describing these accelerations are (see [[fictitious force]] for a derivation):<!-- REF --><ref name=Taylor/><ref name=Arnolʹd> {{cite book |title=Mathematical Methods of Classical Mechanics |url=http://books.google.com/books?id=Pd8-s6rOt_cC&printsec=frontcover&dq=intitle:Mathematical+intitle:Methods+intitle:of+intitle:Classical+intitle:Mechanics&lr=&as_brr=0&sig=T3ROdTzeYotJ0hPTNX-cgxicwI8#PPA129,M1 |author= Vladimir Igorevich Arnolʹd |page=§27 pp. 129 ff. |publisher=Springer |location=Berlin |isbn=0387968903 |year=1989}}</ref><ref name=Marion/> :{| |- |<math>\mathbf{a}_\mathrm{rot}\,</math> |<math>=\mathbf{a} - 2\mathbf{\Omega \times v_\mathrm{rot}} - \mathbf{\Omega \times (\Omega \times r)} \,</math> |- | |<math>=\mathbf{a + a_\mathrm{coriolis} + a_\mathrm{centrifugal}} \,</math> , |} where <math>\mathbf{a}_\mathrm{rot}\,</math> is the acceleration relative to the rotating frame, <math>\mathbf{a}\,</math> is the acceleration relative to the inertial frame, <math>\mathbf{\Omega}\,</math> is the [[angular velocity]] vector describing the rotation of the reference frame,<ref>This vector points along the axis of rotation with polarity determined by the [[right-hand rule]] and a magnitude |Ω| = ω = angular rate of rotation.</ref> <math>\mathbf{v_\mathrm{rot}}\,</math> is the velocity of the body relative to the rotating frame, and <math>\mathbf{r}\,</math> is the position vector of the body. The last term is the centrifugal acceleration: :<math> \mathbf{a}_\textrm{centrifugal} = - \mathbf{\Omega \times (\Omega \times r)} = \omega^2 \mathbf{r}_\perp </math>, where <math>\mathbf{r_\perp}</math> is the component of <math>\mathbf{r}\,</math> perpendicular to the axis of rotation. ==Non uniformly rotating reference frame== {{main|Accelerated reference frame}} Although changing coordinates from an [[inertial reference frame|inertial frame of reference]] to any rotating one alters the equations of motion to require the inclusion of two sources of fictitious force, the centrifugal force, and the Coriolis force,<ref name=Taylor1/><ref name=Marion/> a third term, the [[Euler force|Euler acceleration]] must be added if the rotation of the frame varies,<!-- REF --><ref>{{cite book |title=Introduction to Mechanics and Symmetry: A Basic Exposition of Classical Mechanical Systems |author=Jerrold E. Marsden, Tudor S. Ratiu |isbn=038798643X |year=1999 |publisher=Springer |page=p. 251 |url=http://books.google.com/books?id=I2gH9ZIs-3AC&pg=PP1&dq=isbn:038798643X&sig=tDWUiGpvGVpbRCCQcGK0Bx5Yk3g#PPA251,M1}}</ref><!-- --> and a fourth acceleration is needed if the frame is linearly accelerating.<!-- REF --><ref name=Arnolʹd> {{cite book |title=Mathematical Methods of Classical Mechanics |url=http://books.google.com/books?id=Pd8-s6rOt_cC&printsec=frontcover&dq=intitle:Mathematical+intitle:Methods+intitle:of+intitle:Classical+intitle:Mechanics&lr=&as_brr=0&sig=T3ROdTzeYotJ0hPTNX-cgxicwI8#PPA129,M1 |author= Vladimir Igorevich Arnolʹd |page=§27 pp. 129 ff. |publisher=Springer |location=Berlin |isbn=0387968903 |year=1989}}</ref> ==Examples== Below several examples illustrate both the inertial and rotating frames of reference, and the role of centrifugal force and its relation to Coriolis force in rotating frameworks. For more examples see [[Fictitious force#Crossing a carousel|Fictitious force]]. ===Whirling table=== [[Image:Whirling table.PNG|thumb|350px|Figure 3: The "whirling table". The rod is made to rotate about the axis and the centrifugal force of the sliding bead is balanced by the weight attached by a cord over two pulleys.]] Figure 3 shows a simplified version of an apparatus for studying centrifugal force called the "whirling table".<!-- REF --><ref name=Lardner>{{cite book |title=Mechanics |author=Dionysius Lardner |page=p. 150 |url=http://books.google.com/books?id=SDkDAAAAQAAJ&pg=PA134&dq=mechanics+%22tower+of+pisa%22&lr=&as_brr=0#PPA150,M1 |year=1877|publisher=Oxford University Press}}</ref><!-- --> The apparatus consists of a rod that can be whirled about an axis, causing a bead to slide on the rod under the influence of centrifugal force. A cord ties a weight to the sliding bead. By observing how the equilibrium balancing distance varies with the weight and the speed of rotation, the centrifugal force can be measured as a function of the rate of rotation and the distance of the bead from the center of rotation. From the viewpoint of an inertial frame of reference, equilibrium results when the bead is positioned to select the particular circular orbit for which the weight provides the correct centripetal force. The whirling table is a lab experiment, and standing there watching the table you have a detached viewpoint. It seems pretty much arbitrary whether to deal with centripetal force or centrifugal force. But if you were the bead, not the lab observer, and if you wanted to stay at a particular position on the rod, the centrifugal force would be how you looked at things. Centrifugal force would be pushing you around. Maybe the centripetal interpretation would come to you later, but not while you were coping with matters. Centrifugal force is not just mathematics. ===Rotating identical spheres=== [[Image:Rotating spheres.PNG|thumb|180px|Figure 4: Two spheres tied with a string and rotating at an angular rate ω. Because of the rotation, the string tying the spheres together is under tension.]] [[Image:Rotating-sphere forces.PNG|thumb|Figure 5: Exploded view of rotating spheres in an inertial frame of reference showing the centripetal forces on the spheres provided by the tension in the tying string.]] Figure 4 shows two identical spheres rotating about the center of the string joining them. This [[Bucket_argument#Rotating_spheres|sphere example]] is one used by Newton himself.<!-- REF --><ref name=Born1>{{cite book |title= Einstein's Theory of Relativity |author=Max Born |year=1962 |isbn=0486607690 |publisher=Courier Dover Publications |url=http://books.google.com/books?id=Afeff9XNwgoC&pg=PA76&dq=%22inertial+forces%22&lr=&as_brr=0&sig=0kiN27BqUqHaZ9CkPdqLIjr-Nnw#PPA79,M1 |page=Figure 43, p. 79}}</ref><!-- --> The axis of rotation is shown as a vector '''Ω''' with direction given by the [[right-hand rule]] and magnitude equal to the rate of rotation: '''|Ω|''' = ω. The angular rate of rotation ω is assumed independent of time ([[uniform circular motion]]). Because of the rotation, the string is under tension. (See [[reactive centrifugal force]].) The description of this system next is presented from the viewpoint of an inertial frame and from a rotating frame of reference. ====Inertial frame==== Adopt an inertial frame centered at the midpoint of the string. The balls move in a circle about the origin of our coordinate system. Look first at one of the two balls. To travel in a circular path, which is ''not'' uniform motion with constant velocity, but ''circular'' motion at constant speed, requires a force to act on the ball so as to continuously change the direction of its velocity. This force is directed inward, along the direction of the string, and is called a [[centripetal force]]. The other ball has the same requirement, but being on the opposite end of the string, requires a centripetal force of the same size, but opposite in direction. See Figure 5. These two forces are provided by the string, putting the string under tension, also shown in Figure 5. ====Rotating frame==== Adopt a rotating frame at the midpoint of the string. Suppose the frame rotates at the same angular rate as the balls, so the balls appear stationary in this rotating frame. Because the balls are not moving, observers say they are at rest. If they now apply Newton's law of inertia, they would say no force acts on the balls, so the string should be relaxed. However, they clearly see the string is under tension. (For example, they could split the string and put a spring in its center, which would stretch.) To account for this tension, they propose that in their frame a centrifugal force acts on the two balls, pulling them apart. This force originates from nowhere – it is just a "fact of life" in this rotating world, and acts on everything they observe, not just these spheres. In resisting this ubiquitous centrifugal force, the string is placed under tension, accounting for their observation, despite the fact that the spheres are at rest. ====Coriolis force==== {{main|Coriolis effect}} What if the spheres are ''not'' rotating in the inertial frame (string tension is zero)? Then string tension in the rotating frame also is zero. But how can that be? The spheres in the rotating frame now appear to be rotating, and should require an inward force to do that. According to the analysis of [[uniform circular motion]]:<ref name=Joos/> :<math> \mathbf{F}_{\mathrm{centripetal}} = -m \mathbf{\Omega \ \times} \left( \mathbf{\Omega \times x_B }\right) \ </math> :::::<math>= -m\omega^2 R\ \mathbf{u}_R \ ,</math> where '''u'''<sub>R</sub> is a unit vector pointing from the axis of rotation to one of the spheres, and '''Ω''' is a vector representing the angular rotation, with magnitude ω and direction normal to the plane of rotation given by the [[right-hand rule]], ''m'' is the mass of the ball, and ''R'' is the distance from the axis of rotation to the spheres (the magnitude of the displacement vector, |'''x'''<sub>B</sub>| = ''R'', locating one or the other of the spheres). According to the rotating observer, shouldn't the tension in the string be twice as big as before (the tension from the centrifugal force ''plus'' the extra tension needed to provide the centripetal force of rotation)? The reason the rotating observer sees zero tension is because of yet another fictitious force in the rotating world, the [[Coriolis force]], which depends on the velocity of a moving object. In this zero-tension case, according to the rotating observer the spheres now are moving, and the Coriolis force (which depends upon velocity) is activated. According to the article [[fictitious force]], the Coriolis force is:<ref name=Joos/> :<math> \mathbf{F}_{\mathrm{fict}} = - 2 m \boldsymbol\Omega \times \mathbf{v}_{B} \ </math> :::<math> = -2m \omega \left( \omega R \right)\ \mathbf{u}_R , </math> where ''R'' is the distance to the object from the center of rotation, and '''v'''<sub>B</sub> is the velocity of the object subject to the Coriolis force, |'''v'''<sub>B</sub>| = ω''R''. In the geometry of this example, this Coriolis force has twice the magnitude of the ubiquitous centrifugal force and is exactly opposite in direction. Therefore, it cancels out the ubiquitous centrifugal force found in the first example, and goes a step further to provide exactly the centripetal force demanded by uniform circular motion, so the rotating observer calculates there is no need for tension in the string − the Coriolis force looks after everything. ====General case==== What happens if the spheres rotate at one angular rate, say ω<sub>I</sub> (''I'' = inertial), and the frame rotates at a different rate ω<sub>R</sub> (''R'' = rotational)? The inertial observers see circular motion and the tension in the string exerts a centripetal inward force on the spheres of: :<math>\mathbf{T} = -m \omega_I^2 R \mathbf{u}_R \ . </math> This force also is the force due to tension seen by the rotating observers. The rotating observers see the spheres in circular motion with angular rate ω<sub>S</sub> = ω<sub>I</sub> − ω<sub>R</sub> (''S'' = spheres). That is, if the frame rotates more slowly than the spheres, ω<sub>S</sub> > 0 and the spheres advance counterclockwise around a circle, while for a more rapidly moving frame, ω<sub>S</sub> < 0, and the spheres appear to retreat clockwise around a circle. In either case, the rotating observers see circular motion and require a net inward centripetal force: :<math>\mathbf{F}_{\mathrm{Centripetal}} = -m \omega_S^2 R \mathbf{u}_R \ . </math> However, this force is not the tension in the string. So the rotational observers conclude that a force exists (which the inertial observers call a fictitious force) so that: :<math>\mathbf{F}_{\mathrm{Centripetal}} = \mathbf{T} + \mathbf{F}_{\mathrm{Fict}}\ , </math> or, :<math> \mathbf{F}_{\mathrm{Fict}} = -m \left( \omega_S^2 R -\omega_I^2 R \right) \mathbf{u}_R \ . </math> The fictitious force changes sign depending upon which of ω<sub>I</sub> and ω<sub>S</sub> is greater. The reason for the sign change is that when ω<sub>I</sub> > ω<sub>S</sub>, the spheres actually are moving faster than the rotating observers measure, so they measure a tension in the string that actually is larger than they expect; hence, the fictitious force must increase the tension (point outward). When ω<sub>I</sub> < ω<sub>S</sub>, things are reversed so the fictitious force has to decrease the tension, and therefore has the opposite sign (points inward). (Incidentally, checking the fictitious force needed to account for the tension in the string is one way for an observer to decide whether or not they are rotating – if the fictitious force is zero, they are not rotating. Of course, in an extreme case like the [[gravitron]] amusement ride, you do not need much convincing that you are rotating, but standing on the Earth's surface, the matter is more subtle.) =====Is the fictitious force ''ad hoc''?===== The introduction of '''F'''<sub>Fict</sub> allows the rotational observers and the inertial observers to agree on the tension in the string. However, we might ask: "Does this solution fit in with general experience with other situations, or is it simply a "cooked up" ''[[ad hoc]]'' solution?" That question is answered by seeing how this value for '''F'''<sub>Fict</sub> squares with the general result (derived in [[Fictitious force]]):<!-- REF --><ref name=Srivastava>Many sources are cited in [[Fictitious force]]. Here are two more: {{cite book |title=Mechanics |author=PF Srivastava |publisher=New Age International Publishers |year=2007 |isbn=978-81-224-1905-4 |location=New Delhi |url=http://books.google.com/books?id=yCw_Hq53ipsC&pg=PA46&dq=spheres+rotating++Coriolis&lr=&as_brr=0&sig=XJ1Xl2Qs1_j5RkLN70wxyUO6Vgc#PPA43,M1 |page=p. 43 }} and {{cite book |author=NC Rana and PS Joag |year=2004 |title=Mechanics |isbn= 0074603159 |publisher=Tata McGraw-Hill |location=New Delhi |page =p. 99ff |url=http://books.google.com/books?id=dptKVr-5LJAC&pg=PA28&dq=connected+rotating+%22two+spheres%22&lr=&as_brr=0&sig=Hu4brsD2Jkc25AOPwZLPuQ28uPE#PPA99,M1}}</ref> :<math> \mathbf{F}_{\mathrm{Fict}} = - 2 m \boldsymbol\Omega \times \mathbf{v}_{B} - m \boldsymbol\Omega \times (\boldsymbol\Omega \times \mathbf{x}_B ) </math>&ensp;<math>\ - m \frac{d \boldsymbol\Omega }{dt} \times \mathbf{x}_B \ . </math> For constant angular rate of rotation the last term is zero. To evaluate the other terms we need the position of one of the spheres: :<math> \mathbf{x}_B = R\mathbf{u}_R \ , </math> and the velocity of this sphere as seen in the rotating frame: :<math>\mathbf{v}_B = \omega_SR \mathbf{u}_{\theta} \ ,</math> where '''u'''<sub>θ</sub> is a unit vector perpendicular to '''u'''<sub>R</sub> pointing in the direction of motion. The vector of rotation '''Ω''' = ω<sub>R</sub> '''u'''<sub>z</sub> ('''u'''<sub>z</sub> a unit vector in the ''z''-direction), and '''Ω × u'''<sub>R</sub> = ω<sub>R</sub> ('''u'''<sub>z</sub> × '''u'''<sub>R</sub>) = ω<sub>R</sub> '''u'''<sub>θ</sub> ; '''Ω × u'''<sub>θ</sub> = −ω<sub>R</sub> '''u'''<sub>R</sub>. The centrifugal force is then: :<math>\mathbf{F}_\mathrm{Cfgl} = - m \boldsymbol\Omega \times (\boldsymbol\Omega \times \mathbf{x}_B ) =m\omega_R^2 R \mathbf{u}_R\ ,</math> which naturally depends only on the rate of rotation of the frame and is always outward. The Coriolis force is :<math>\mathbf{F}_\mathrm{Cor} = - 2 m \boldsymbol\Omega \times \mathbf{v}_{B} = 2m\omega_S \omega_R R \mathbf{u}_R </math> and has the ability to change sign, being outward when the spheres move faster than the frame ( ω<sub>S</sub> > 0 ) and being inward when the spheres move slower than the frame ( ω<sub>S</sub> < 0 ).<ref>The case ω<sub>S</sub> < 0 applies to the earlier example with [[#Coriolis force|spheres at rest]] in the inertial frame.</ref> Combining the terms: :<math> \mathbf{F}_{\mathrm{Fict}} = \mathbf{F}_\mathrm{Cfgl} + \mathbf{F}_\mathrm{Cor}</math>&ensp;<math> =\left( m\omega_R^2 R + 2m\omega_S \omega_R R\right) \mathbf{u}_R = m\omega_R \left( \omega_R + 2\omega_S \right) R \mathbf{u}_R </math> ::<math>=m(\omega_I-\omega_S)(\omega_I+\omega_S)\ R \mathbf{u}_R = -m \left(\omega_S^2-\omega_I^2\right)\ R \mathbf{u}_R . </math> Consequently, the fictitious force found above for this problem of rotating spheres is consistent with the general result and is not an ''[[ad hoc]]'' solution just "cooked up" to bring about agreement for this single example. Moreover, it is the Coriolis force that makes it possible for the fictitious force to change sign depending upon which of ω<sub>I</sub>, ω<sub>S</sub> is the greater, inasmuch as the centrifugal force contribution always is outward. ===Dropping ball=== [[Image:Falling object.PNG|thumb|450px|Figure 6: A ball moving vertically along the axis of rotation in an inertial frame appears to spiral downward in the rotating frame. The right panel shows a downward view in the rotating frame. The rate of rotation '''<nowiki>|</nowiki>Ω<nowiki>|</nowiki>''' = ω is assumed constant in time.]] [[Image:Coriolis force.PNG|thumb|250px|Figure 7: Vector cross product used to determine the Coriolis force. The vector '''Ω''' represents the rotation of the frame at angular rate ω; the vector '''v''' shows the velocity tangential to the circular motion as seen in the rotating frame. The vector '''Ω × v''' is found using the [[right-hand rule]] for [[vector cross product]]s. It is related to the negative of the Coriolis force (the Coriolis force is −2 ''m'' '''Ω × v'''). ]] Figure 6 {{anchor|Figure 6}}shows a ball dropping vertically (parallel to the axis of rotation '''Ω''' of the rotating frame). For simplicity, suppose it moves downward at a fixed speed in the inertial frame, occupying successively the vertically aligned positions numbered one, two, three. In the rotating frame it appears to spiral downward, and the right side of Figure 6 shows a top view of the circular trajectory of the ball in the rotating frame. Because it drops vertically at a constant speed, from this top view in the rotating frame the ball appears to move at a constant speed around its circular track. A description of the motion in the two frames is next. ====Inertial frame==== In the inertial frame the ball drops vertically at constant speed. It does not change direction, so the inertial observer says the acceleration is zero and there is no force acting upon the ball. ====Uniformly rotating frame==== In the rotating frame the ball drops vertically at a constant speed, so there is no vertical component of force upon the ball. However, in the horizontal plane perpendicular to the axis of rotation, the ball executes uniform circular motion as seen in the right panel of Figure 6. Applying Newton's law of motion, the rotating observer concludes that the ball must be subject to an inward force in order to follow a circular path. Therefore, the rotating observer believes the ball is subject to a force pointing radially inward toward the axis of rotation. According to the analysis of [[uniform circular motion]] :<math> \mathbf{F}_{\mathrm{fict}} = -m\omega^2 R \ ,</math> where ω is the angular rate of rotation, ''m'' is the mass of the ball, and ''R'' is the radius of the spiral in the horizontal plane. Because there is no apparent source for such a force (hence the label "fictitious"), the rotating observer concludes it is just "a fact of life" in the rotating world that there exists an inward force with this behavior. Inasmuch as the rotating observer already knows there is a ubiquitous ''outward'' centrifugal force in the rotating world, how can there be an inward force? The answer is again the Coriolis force: the component of velocity tangential to the circular motion seen in the right panel of Figure 6 activates the Coriolis force, which cancels the centrifugal force and, just as in the zero-tension case of the spheres, goes a step further to provide the centripetal force demanded by the calculations of the rotating observer. Some details of evaluation of the Coriolis force are shown in Figure 7. Because the Coriolis force and centrifugal forces combine to provide the centripetal force the rotating observer requires for the observed circular motion, the rotating observer does not need to apply any additional force to the object, in complete agreement with the inertial observer, who also says there is no force needed. One way to express the result: the fictitious forces look after the "fictitious" situation, so the ball needs no help to travel the perceived trajectory: all observers agree that nothing needs to be done to make the ball follow its path. ===Parachutist=== [[Image:Parachutist.PNG|thumb|450px|Figure 8: A parachutist moving vertically parallel to the axis of rotation in a rotating frame appears to spiral downward in the inertial frame. The parachutist begins the drop with a horizontal component of velocity the same as the target site. The left panel shows a downward view in the inertial frame. The rate of rotation '''<nowiki>|</nowiki>Ω<nowiki>|</nowiki>''' = ω is assumed constant in time.]] To show a different frame of reference, let's revisit the dropping ball example in [[#Figure 6|Figure 6]] from the viewpoint of a parachutist falling at constant speed to Earth (the rotating platform). The parachutist aims to land upon the point on the rotating ground directly below the drop-off point. Figure 8 shows the vertical path of descent seen in the rotating frame. The parachutist drops at constant speed, occupying successively the vertically aligned positions one, two, three. In the stationary frame, let us suppose the parachutist jumps from a helicopter hovering over the destination site on the rotating ground below, and therefore traveling at the same speed as the target below. The parachutist starts with the necessary speed tangential to his path (ω''R'') to track the destination site. If the parachutist is to land on target, the parachute must spiral downward on the path shown in Figure 8. The ''stationary'' observer sees a uniform circular motion of the parachutist when the motion is projected downward, as in the left panel of Figure 8. That is, in the horizontal plane, the stationary observer sees a centripetal force at work, -''m'' ω<sup>2</sup> ''R'', as is necessary to achieve the circular path. The parachutist needs a thruster to provide this force. Without thrust, the parachutist follows the dashed vertical path in the left panel of Figure 8, obeying Newton's law of inertia. The stationary observer and the observer on the rotating ground agree that there is no vertical force involved: the parachutist travels vertically at constant speed. However, the observer on the ground sees the parachutist simply drop vertically from the helicopter to the ground, following the vertically aligned positions one, two, three. There is no force necessary. So how come the parachutist needs a thruster? The ground observer has this view: there is always a centrifugal force in the rotating world. Without a thruster, the parachutist would be carried away by this centrifugal force and land far off the mark. From the parachutist's viewpoint, trying to keep the target directly below, the same appears true: a steady thrust radially inward is necessary, just to hold a position directly above target. Unlike the dropping ball case, where the fictitious forces conspired to produce no need for external agency, in this case they require intervention to achieve the trajectory. The basic rule is: if the inertial observer says a situation demands action or does not, the fictitious forces of the rotational frame will lead the rotational observer to the same conclusions, albeit by a different sequence. Notice that there is no Coriolis force in this discussion, because the parachutist has zero horizontal velocity from the viewpoint of the ground observer. == Potential energy == [[Image:Coriolis effect11.jpg|thumb|right|Figure 9: The interface of two [[Miscibility|immiscible]] liquids rotating around a vertical axis is an upward-opening circular paraboloid.]] {{seealso|Bucket argument|Coriolis effect}} In a uniformly rotating reference frame, the fictitious centrifugal force is [[conservative force|conservative]] and has a [[potential energy]] of the form :<math>E_p = -\frac{1}{2} m \omega^2 r^2 \ ,</math> where ''r'' is the radius from the axis of rotation. This result can be verified by taking the gradient of the potential to obtain the radially outward force: :<math> F_{Cfgl} = -\frac{\partial }{\partial r} E_p = m \omega^2 r \ . </math> The potential energy is useful, for example, in calculating the form of the water surface in a rotating bucket. Let the height of the water be <math>h(r)\,</math>: then the potential energy per unit mass contributed by gravity is <math>g h(r) \ </math> (''g'' = [[Gravitational acceleration|acceleration due to gravity]]) and the total potential energy per unit mass on the surface is <math>gh(r) - \frac{1}{2}\omega^2 r^2\,</math>. In a static situation (no motion of the fluid in the rotating frame), this energy is constant independent of position ''r''. Requiring the energy to be constant, we obtain the [[parabolic]] form: :<math>h(r) = \frac{\omega^2}{2g}r^2 + h(0) \ , </math> where <math>h(0)</math> is the height at ''r'' = 0 (the axis). See Figure 9. Similarly, the potential energy of the centrifugal force is often used in the calculation of the height of the [[tide]]s on the Earth (where the centrifugal force is included to account for the rotation of the Earth around the Earth-Moon center of mass). The principle of operation of the [[centrifuge]] also can be simply understood in terms of this expression for the potential energy, which shows that it is favorable energetically when the volume far from the axis of rotation is occupied by the heavier substance. The Coriolis force has no equivalent potential, as it acts perpendicular to the velocity vector and hence rotates the direction of motion, but does not change the energy of a body. == Development of the modern conception of centrifugal force == {{expert-subject|History of Science}} 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 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."<!-- REF --><ref>The Relativization of Centrifugal Force Author(s): Domenico Bertoloni Meli Source: Isis, Vol. 81, No. 1, (Mar., 1990), pp. 23-43 Published by: The University of Chicago Press on behalf of The History of Science Society.</ref> There is evidence that Sir [[Isaac Newton]] originally conceived circular motion as being caused a balance between an inward centripetal force and an outward centrifugal force.<!-- REF --><ref name="xnewton">{{cite web |url=http://www.phy.hr/~dpaar/fizicari/xnewton.html |title=Newton, Sir Isaac |accessdate=2008-05-25}}</ref> The modern conception of centrifugal force appears to have its origins in [[Christiaan Huygens]]' paper ''De Vi Centrifuga'', written in 1659.<!-- REF --><ref>{{cite web |url=http://www.bun.kyoto-u.ac.jp/~suchii/inertia.html |title=Inertia |author=Soshichi Uchii |date=October 9, 2001 |accessdate=2008-05-25}}</ref><!-- --> It has been suggested that the idea of circular motion as caused by a single force was introduced to Newton by [[Robert Hooke]].<ref name="xnewton"/> Newton described the role of centrifugal force upon the height of the oceans near the equator in the ''[http://www.archive.org/stream/newtonspmathema00newtrich/newtonspmathema00newtrich_djvu.txt Principia]'': {{Quotation|COR. 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}} ==Applications== The operations of numerous common rotating mechanical systems are most easily conceptualized in terms of centrifugal force. For example: {{multicol}} * A [[centrifugal governor]] regulates the speed of an engine by using spinning masses that move radially, adjusting the [[throttle]], as the engine changes speed. In the reference frame of the spinning masses, centrifugal force causes the radial movement. * A [[centrifugal clutch]] is used in small engine-powered devices such as chain saws, go-karts and model helicopters. It allows the engine to start and idle without driving the device but automatically and smoothly engages the drive as the engine speed rises. [[Self-locking device#Inertial drum brake ascender|Inertial drum brake ascenders]] used in [[rock climbing]] and the [[seat belt#Technologies|inertia reels]] used in many automobile seat belts operate on the same principle. * Centrifugal forces can be used to generate [[artificial gravity]], as in proposed designs for rotating space stations. The [[Mars Gravity Biosatellite]] will study the effects of [[Mars]]-level gravity on mice with gravity simulated in this way. *[[Spin casting]] and [[centrifugal casting]] are production methods that uses centrifugal force to disperse liquid metal or plastic throughout the negative space of a mold. {{multicol-break}} * [[Centrifuge]]s are used in science and industry to separate substances. In the reference frame spinning with the centrifuge, the centrifugal force induces a hydrostatic pressure gradient in fluid-filled tubes oriented perpendicular to the axis of rotation, giving rise to large [[buoyant force]]s which push low-density particles inward. Elements or particles denser than the fluid move outward under the influence of the centrifugal force. This is effectively [[Archimedes' principle]] as generated by centrifugal force as opposed to being generated by gravity. *Some [[amusement park]] [[amusement ride|ride]]s make use of centrifugal forces. For instance, a [[Gravitron]]’s spin forces riders against a wall and allows riders to be elevated above the machine’s floor in defiance of Earth’s gravity. {{multicol-end}} Nevertheless, all of these systems can also be described without requiring the concept of centrifugal force, in terms of motions and forces in an inertial frame, at the cost of taking somewhat more care in the consideration of forces and motions within the system. ==Notes and references== {{reflist|2}} ==Further reading== * [http://members.tripod.com/~gravitee/booki2.htm Newton's description in Principia] * [http://www.infoplease.com/ce6/sci/A0811114.html Centrifugal reaction force] - Columbia electronic encyclopedia * M. Alonso and E.J. Finn, ''Fundamental university physics'', Addison-Wesley * [http://regentsprep.org/Regents/physics/phys06/bcentrif/default.htm Centripetal force] vs. [http://regentsprep.org/Regents/physics/phys06/bcentrif/centrif.htm Centrifugal force] - from an online Regents Exam physics tutorial by the Oswego City School District * [http://www.npl.washington.edu/av/altvw55.html Centrifugal force acts inwards near a black hole] * [http://hyperphysics.phy-astr.gsu.edu/HBASE/corf.html#cent Centrifugal force] at the HyperPhysics concepts site * [http://www.netsurf.com/nss/misc/centrifugal.html A list of interesting links] ==External links== * [http://mensch.org/physlets/merry.html Motion over a flat surface] Java physlet by Brian Fiedler (from School of Meteorology at the University of Oklahoma) illustrating fictitious forces. The physlet shows both the perspective as seen from a rotating and from a non-rotating point of view. * [http://mensch.org/physlets/inosc.html Motion over a parabolic surface] Java physlet by Brian Fiedler (from School of Meteorology at the University of Oklahoma) illustrating fictitious forces. The physlet shows both the perspective as seen from a rotating and as seen from a non-rotating point of view. * [http://www.youtube.com/watch?v=49JwbrXcPjc Animation clip] showing scenes as viewed from both an inertial frame and a rotating frame of reference, visualizing the Coriolis and centrifugal forces. * [http://www.mathpages.com/home/kmath633/kmath633.htm Centripetal and Centrifugal Forces] at MathPages * [http://www.bbc.co.uk/dna/h2g2/A597152 Centrifugal Force] at h2g2 * [http://www.wisegeek.com/what-is-a-centrifuge.htm What is a centrifuge?] * [http://math.ucr.edu/home/baez/physics/General/Centrifugal/centri.html John Baez: ''Does centrifugal force hold the Moon up?'' ] * [http://xkcd.com/123/ XKCD demonstrates the life and death importance of centrifugal force] == See also == {{multicol}} * [[Circular motion]] * [[Coriolis effect|Coriolis force]] * [[Coriolis effect (perception)]] * [[Centripetal force]] * [[Equivalence principle]] * [[Bucket argument]] * [[Euler force]] - a force that appears when the frame angular rotation rate varies * [[Folk physics]] * [[Rotational motion]] * [[Reactive centrifugal force]] - a force that occurs as [[Newton's 3rd law|reaction]] due to a centripetal force * [[Lamm equation]] {{multicol-break}} * [[Lagrangian point]] - gravitationally stable points in rotating reference frames *[[Orthogonal coordinates]] * [[Frenet-Serret formulas]] *[[Statics]] *[[Kinetics (physics)]] *[[Kinematics]] *[[Applied mechanics]] *[[Analytical mechanics]] *[[Dynamics (physics)]] *[[Classical mechanics]] *[[D'Alembert's principle]] *[[Centrifuge]] {{multicol-end}} <br> {{Wiktionary|centrifugal}}{{Wiktionary|centrifugal force}} [[Category:Fictitious forces]] [[Category:Force]] [[Category:Mechanics]] [[Category:Rotation]] [[bg:Центробежна сила]] [[cs:Odstředivá síla]] [[da:Centrifugalkraft]] [[et:Tsentrifugaaljõud]] [[es:Fuerza centrífuga]] [[fr:Force centrifuge]] [[ko:원심력]] [[hr:Centrifugalna i centripetalna sila]] [[it:Forza centrifuga]] [[he:כוח צנטריפוגלי]] [[hu:Centrifugális erő]] [[nl:Middelpuntvliedende kracht]] [[ja:遠心力]] [[nn:Sentrifugalkraft]] [[pl:Siła odśrodkowa]] [[pt:Força centrífuga]] [[ru:Центробежная сила]] [[simple:Centrifugal force]] [[fi:Keskipakoisvoima]] [[sv:Centrifugalkraft]] [[vi:Lực ly tâm]] [[tr:Merkezkaç]] [[zh:離心力]]