cyanic-selkie/wikianc huggingface dataset:
Goal twoaie
Supernumerary rainbow
It is not possible to explain their existence using classical geometric optics. The alternating faint rainbows are caused by interference between rays of light following slightly different paths with slightly varying lengths within the raindrops. Some rays are in phase, reinforcing each other through constructive interference, creating a bright band; others are out of phase by up to half a wavelength, canceling each other out through destructive interference, and creating a gap. Given the different angles of refraction for rays of different colours, the patterns of interference are slightly different for rays of different colours, so each bright band is differentiated in colour, creating a miniature rainbow. Supernumerary rainbows are clearest when raindrops are small and of similar size. The very existence of supernumerary rainbows was historically a first indication of the wave nature of light, and the first explanation was provided by Thomas Young in 1804.
Reflected rainbow, reflection rainbow
When a rainbow appears above a body of water, two complementary mirror bows may be seen below and above the horizon, originating from different light paths. Their names are slightly different. A reflected rainbow will appear as a mirror image in the water surface below the horizon, if the surface is quiet (see photo above). The sunlight is first deflected by the raindrops, and then reflected off the body of water, before reaching the observer. The reflected rainbow is frequently visible, at least partially, even in small puddles.
Six (or even eight) bows may be distinguished if the reflection of the reflection bow, and the secondary bow with its reflections happen to appear simultaneously.
Circumhorizontal arc
The circumhorizontal arc is sometimes referred to by the misnomer "fire rainbow". As it originates in ice crystals, it is not a rainbow but a halo.
Rainbows on Titan
It has been suggested that rainbows might exist on Saturn's moon Titan, as it has a wet surface and humid clouds. The radius of a Titan rainbow would be about 49° instead of 42°, because the fluid in that cold environment is methane instead of water. A visitor might need infrared goggles to see the rainbow, as Titan's atmosphere is more transparent for those wavelengths.
-->
Caashey
Jannoo
T’ad jannoo caashey ass proteen as sahll vainney. She bainney ollee, ollee oaldey, goair as kirree t’ayn son y chooid smoo. T’ad cur er y vainney geayraghey as binjaghey, as t’eh rheynn ayns groo as meaig. Son y chooid smoo ta janteyr caashey cur stiagh stoo toshee gyle bacteyragh er son geayraghey as binjey er son binjaghey, agh ta shiartanse dy chaashey jeant lesh geayr hene; myr sampleyr, ta paneer jeant lesh feeyn geayr, soolagh limon ny geayr elley. T’ad tayrn magh y groo fondagh as eh y ‘aastey ayns faastan.
Ta tree soorçhyn mooarey dy chaashey ayn:
The Handmaid's Tale (straih hellveeish)
She straih hellveeish yee-launyssagh Americaanagh eh The Handmaid's Tale va crooit ec Bruce Miller as bunnit er y lioar fo'n ennym cheddin voish 1985 liorish yn ughtar Canadagh Margaret Atwood. Va'n straih currit ayns oardagh liorish y çhirveish stroo Hulu myr oardagh jeeragh-gys-straih va jeant ass 10 episode. Hoshee gientyn y straih ayns 2016 jeianagh. Ta'n skeeal goaill stiagh jee-launys lurg aa-chaggey theayagh ayns ny Steatyn Unnaneysit raad ta sheshaght ooilley smaghtagh smaghtaghey mraane troaragh, ta enmyssit "inneenyn meyl" ny Handmaiden, as cur ad ayns slaueaght torraghys.
Caashey
Sorçhyn
thumb|[[Feta ass Y Ghreag]]
The Handmaid's Tale (straih hellveeish)
Hie yn chied tree episode jeh'n straih magh er skeaylley er 26 Averil 2017; hie ny shiaght episode elley magh er skeaylley dagh Jecrean ny yei. Ayns Jerrey Souree 2019, hie yn straih er ny aanoaghey son y chiarroo straih, as hie yn chied episode jeh magh er 27 Averil 2021. Ayns Mean Fouyir 2019, v'eh fogrit magh dy row Hulu as MGM lhiassaghey straih eiyrtyssagh vees bunnit er noaskeeal Atwood voish 2019, The Testaments. Va'n straih aanoit ayns Mee ny Nollick 2020 roish y chied episode jeh'n chiarroo straih.
Caashey
Sorçhyn
Shimmey sorçh dy chaashey t’ayn. Ta mysh 500 sorçh goit rish ec yn International Dairy Federation. Ta ard-ynseyderyn elley cur enney er 400, 500 ny 1,000. Ta sorçhyn goll er rang-oardraghey rere slattyn towse, myr sampleyr eash appee, coyannoo, saase jannoo,cooid hahllagh,çheeroaylleeaght as reddyn elley. Dy cadjin, t’ad croghey er cooid ‘liugh, cooid hallagh as saase appee. Ta shiartanse dy ‘lieh er chur roue corssyn rang-oardree: ta corys Pieter Walstra croghey er bun- as fo-stooghyn toshee as er cooid ‘liugh, as ta corys Walter & Hargrove rere saase jannoo as aarheynn rere cooid ‘liugh.
The Handmaid's Tale (straih hellveeish)
Chossyn y chied straih jeh The Handmaid's Tale hoght aundyr Emmy ass 13 enmyssaght, goaill stiagh aundyr son y Straih Drama Ard-yindyssagh. T'eh ny chied chlaare liorish Hulu ren cosney aundyr mooar chammah's y chied straih er shirveish stroo ren cosney Emmy son Straihyn Ard-yindyssagh. Chammah's shen, ren eh cosney Aundyr Golden Globe son y Straih Hellveeish Share - Drama. Ren Elisabeth Moss cosney aundyr Primetime Emmy son yn Aghteyr Bwoirrin Toshee Share ayns Straih Drama as aundyr Golden Globe son yn Aghteyr Bwoirrin Share ayns Straih Drama Hellveeish.
Queiggey Zakarpattia
She queiggey 'syn Ookraan heear yass eh Queiggey Zakarpattia (Закарпатська область Zakarpats’ka oblast’). She Uzhhorod y balley reiragh.
Bnei Brak
She caayr er coose Meanvooiragh Israel, beggan çheu hiar veih Tel Aviv eh Bnei Brak ny Bene Beraq ( Bəne βəraq). T'eh soit ayns Gush Dan ayns Slyst Tel Aviv. She cree yn Ewaghys Cair-chredjuagh eh Bnei Brak.
Ta'n chaayr 7.3 km2 er eaghtyr. 'Sy vlein 2012, rere staydraa Ard-Rheynn Staydraa Israel, va 176,556 deiney cummal 'sy chaayr. Rere yn earroo shen, t'eh ny jeihoo caayr smoo 'sy çheer.
Baljyn as caayryn lannoonit
Ta Bnei Brak goll er lannooney rish:
X (lettyr)
She kiarroo lettyr as feed yn abbyrlhit Romanagh eh X ny x.
Shennaghys
Va ny shenn Ghreagee cliaghtey rish screeu y dhossan corocklagh /ks/ myr Chi (Greagish heear) ny Xi (Greagish hiar). Fy-yerrey, ren ad briwnys son Chi myr /kh/ (/x/ ayns Greagish yeianagh as Xi myr /ks/. Agh va ny h-Etruscee er ngoaill X jeh shenn Ghreagish heear, myr shen, va X cowraghey /ks/ ayns Etruscish as Ladjyn.
Cha nel fys ain my va Chi as Xi nyn ynlaghtyn Greagagh, ny my va bun Semetagh oc. Va Chi soit faggys da jerrey yn abbyrlhit Greagagh, as shen cowraghey dy row eh noa, foddee. Chammah's shen, cha row lettyr Semetagh erbee as eshyn çheet er /ks/. Ta eie cadjin ayn dy row Samekh Feaneeckagh (as eshyn çheet er /s/) bun Xi. Foddee dy row jalloo-ocklyn Egyptagh jeh colloo Djed bun Hamekh, agh cha nod shin gra dy shickyr.
Ayns Gaelg
Cha nel X mastey lettyryn abbyrlhit ny Gaelgey, agh t'eh ry-akin ny keayrtyn, ny yei shen ayns enmyn joarree, chammah's ayns maddaght as oaylleeaght.
Coadyn co-earrooder
Ayns Unicode, ta X mooar ec U+0058 as x beg ec U+0078.
Ayns ASCII, ta X mooar ec 88 as x beg ec 120. Ayns earrooyn jeesoil, shen 01011000 as 01111000.
Ayns EBCDIC, ta X mooar ec 231 as x beg ec 167.
Ayns HTML as XML, ta X mooar ec "<tt>&#88;</tt>" as x beg ec "<tt>&#120;</tt>".
Berrish ghoo
Ta'n verrish ghoo (Ribes nigrum) ny h-oltey jeh'n ghenus Ribes 'sy chynney Grossulariaceae (myrane lesh y verrish ghlass as y verrish yiarg). Ta'n verrish ny berrish ghooghyssagh jeh'n Oarpey veanagh as hwoaie, as yn Aishey hwoaie. T'ee ny rassan beg ta eddyr 1 as 2 m er yrjid. Ta ny duillagyn cor, neuchramp, eddyr 3 as 5 cm er lhiurid, as babbagh dy bassagh lesh oirr 'eeacklagh. Ta ny blaaghyn eddyr 4 as 6 cm er lhiurid, lesh queig petylyn ta glass ruyagh dys dhoanagh er daah; t'ad gientit ayns crouw-vlaaghyn eddyr 5 as 10 cm er lhiurid.
Traa nagh vel y rassan gymmyrkey messyn, t'eh cosoylagh rish rassan y verrish yiarg, as ad yn-chronnaghey lesh y chooraght lajer veih ny sthilk as ny duillagyn. Ta'n mess ny verrish yn-ee ta 1 cm er crantessen, as ad gorrym jiarg dorraghey, begnagh doo, er daah, lesh crackan gloasagh as seepal shassooagh er y vaare ta cummal rassyn ennagh ta gloo er beagheeyn. Foddee drine prowit gientyn seose dys 5 kg dy verrishyn car y tourey.
Richard Jebb
Foddee dy row shiu çheet er:
Alex Ferguson
Va Reejerey Alexander Chapman Ferguson ny chloieder as reireyder bluckan-coshey ren reirey Manchester United voish 1986 dys 2013. T'eh coontit myr nane jeh ny reireyderyn share as s'rahoil va rieau ayn.
Chloie Ferguson myr bwoalleyder son kuse mooar dy chlubyn bluckan-coshey Albinagh, goaill stiagh Dunfermline Athletic as Rangers. Tra v'eh cloie da Dunfermline, she eshyn va'n ard-scoareyder 'sy leeg Albinagh ayns 1965 dys 1966. Ec jerrey e vea-chloie, v'eh gobbraghey myr ynseyder bluckan-coshey as eisht ghow eh toshiaghyt er e choorse-bea myr reireyder lesh East Stirlingshire as St Mirren. Ghow Ferguson soylley jeh traa feer rahoil tra she reireyder Aberdeen v’eh as ren eh cosney yn Leeg Albinagh tree keayrtyn, quieg cappanyn Albinagh, as Cappan ny Chosneyderyn ayns 1983. Reir eh possan bluckan-coshey eddyr-ashoonagh Albinagh rish tammalt giare lurg hooar Jock Stein baase, as hug eh lesh ad dys Cappan y Theihll ayns 1986.
Georges de La Tour
Va Georges de La Tour (13 Mart 1593 - 30 Jerrey Geuree 1652) ny pheinteyr Frangagh. Rug eh ayns Vic-sur-Seille ayns 1593 as hooar eh baase ayns 1652 ayns Lunéville.
Cronk ny Marrey
Ta Cronk ny Marrey ny heeabin çhellveeish Gaelgagh ta jeant liorish ÇhellVeeish. Hie eh magh ec jerrey yn vlein 2013.
Soiaghey
Ta Cronk ny Marrey soit ayns y valley far-skeealagh Cronk ny Marrey.
São Paulo
Ta São Paulo (Portiugish son "Noo Paayl") ny chaayr smoo 'sy Vrasseel lesh ny smoo na 11 millioonyn cummaltee ayn. left|300px|thumb|Avenida Paulista - São Paulo
William Kennish
She feelee, yeshaghteyr, ronseyder, oayllee as ynlaghteyr Manninagh va William Kennish (r. ayns Maghal, Mannin 'sy vlein 1799, as hooar baase er 19 Mayrnt 1862 ayns Caayr York Noa). V'eh ny chied pheiagh va failt ec Reiltys ny Steatyn Unnaneyssit son bollagh da Ammyr y Phanamaa y gheddyn magh.
Rug Kennish ayns thie beg ayns Skylley Maghal, Mannin, er y raad eddyr Doolish as Rhumsaa. She Gaelgeyr v'ayn as cha row agh beggan dy Vaarle echey derrey dy jiagh eh stiagh ayns Lhuingys-marrey Hostyn tra v'eh 22 vleeaney d'eash. D'ynsee eh yn Vaarle as ren eh roshtyn rhenk yn Ard-Heyr Lhuingey tra v'eh 27 bleeaney d'eash. Ayns Mean Fouyir ny bleeaney 1826 ren eh poosey Mary Byford ass Gillingham, Chatham, Kent, Sostyn.
Eddyr ny bleeantyn 1827 as 1832, tra v'eh foast gobbraghey 'sy Lhuingys-marrey, ren eh ynlaghtey corys noa son dy chiartaghey aght jeeree ny gunnaghyn marrey er cam-leighaltys as corys son gunnaghyn marrey y scughey gys y thalloo. Ren eh jeshaghey cummey yn uillin towsheyder.
Tra d'irree anaase ec Lhuingys-marrey Hostyn 'sy ghaal-imman, eddyr ny bleeantyn 1832 as 1840, ren eh kiaddey as cummey shiartanse dy ghaal-ghreienyn as sampleyr leah jeh'n scrod-heiyeyder.
D'irree eh ass Lhuingys-marrey Hostyn 'sy vlein 1840 as hoshee eh screeu daanyn. Hie yn obbyr echey, Mona's Isle and Other Poems, er ny hoilshaghey magh ayns Lunnin 'sy vlein 1844. Blein ny s'anmey hoshee eh lessoonyn y choyrt.
'Sy vlein 1849 hie eh gys America as hoshee eh towshan thallooyn airhey 'sy Cholombey as ayns 1855 ren eh plannal bollagh da Ammyr ny Panamaa er son y reiltys Americaanagh. Va'n coontys echey goit stiagh 'syn obbyr The Practicality and Importance of a Ship Canal to Connect the Atlantic and Pacific Oceans va soilshit magh 'sy vlein 1855 liorish George F. Nesbitt & Co. ass York Noa.
McKayla Maroney
She aa-lheiltagh ellynagh as arraneyder Americaanagh ee McKayla Rose Maroney (ruggit er 9 Nollick 1995). She oltey wirran lheiltys mraane ny Steatyn Unnaneysit va enmyssit Fierce Five ec Cloiaghyn Olympagh y Touree, 2012 v'ee, raad chossyn ee boyn airhey 'sy chloie fwirranagh as boyn argid 'sy chloie lheimyragh. V;ee ny oltey jeh'n wirran Americaanagh chossyn boynyn airhey ec Feniaghtyssyn Lheiltys Ellynagh y Theihill, 2021, raad chossyn ee boynyn airhey ayns ny cloiaghyn fwirranagh as lheimyragh. Ren ee cosney as coadey yn enmys Dowanagh eck as chossyn ee boyn airhey ec y chloie lheimyragh ec Feniaghtyssyn Lheiltys Ellynagh y Theihill, 2013. She ish va'n chied lheiltagh bwoirrin Americaanagh fendeil enmys lheimyragh ec Feniaghtyssyn y Theihill.
Haink er fotograf jee as dreagh "neu-haasit" eck, va jeant lurg chossyn ee boyn airhey ec ny Cloiaghyn Olympagh ayns 2012, ve ny meme eddyr-voggylagh. Ayns 2016, dirree ee magh ass y lheiltys cohirreydagh.
Charlotte's Web (reddaghey)
Foddee dy vel Charlotte's Web çheet er:
Bossan gorrym
She lus 'sy chynney Campanulaceae eh bossan gorrym (Jasione montana). T'eh gaase ayns ardjyn sleitagh ny h-Oarpey as ny h-Aishey heear, er reeastaneyn ardjey ayns eayninyn as quarralyn raad ta'n thalloo thanney. Share da thalloo geyre, as cha nel eh gaase er clagh eayil.
Queiggey Zaporizhzhia
She queiggey 'syn Ookraan yiass eh Queiggey Zaporizhzhia (Запорізька область Zaporiz’ka oblast’) ny Zaporizhzhya (Ookraanish: Запоріжжя) myr t'eh enmyssit dy mennick. She Zaporizhzhia y balley reiragh.
Richard Jebb (leighder)
She leighder, briw, as politickeyr va'n Reejerey Richard Claverhouse Jebb (27 Jerrey Geuree 1807 — 7 Jerrey Geuree 1884 ). Rug eh ayns Divlyn, Nerin lesh Richard Jebb as Louisa Finlay, inneen lesh Oltey Pharlamaid Nerin John Finlay. Hooar Jebb e cheim veih Colaashtey ny Trinaid, Divlyn as v'eh symnit gys y Ving Hostynagh 'sy vlein 1832. V'eh pointit myr Briw Agglish er 16 Nollick 1861.
Jamys Peddyr MacMannys
She karacteyr 'sy chlaare çhellveeish Cronk ny Marrey eh Jamys Peddyr MacMannys.
Forse
right|300 px|thumb|Dy cadjin, t'ad soilshaghey forseyn myr seiy ny tayrn. Foddee ad gobbraghey liorish [[trimmid, magnaidys as phenomena elley ta cur er nhee siyraghey.]]
She parameter vaghtooragh eh forse, as ta paart mooar echey ayns obbrinaght chlassicagh. Foddee eh bieauaghey ny aachummey nheeghyn, as trooid eiyrtys 'orse, foddee oo jannoo obbyr ny caghlaa bree nhee ennagh. Ta mooadys as troa ec forse, as myr shen, she vaghtoor t'ayn.
raad ta:
F çheet er Forse (N)
m çheet er Glout (Kg)
a çheet er bieauaghey
Ren ny shenn 'allsoonee ymmyd jeh'n eie dy row forse ayn er son studeyrys er nheeghyn staddagh as gleashagh. Ren Aristotle ymmyd yn eie shen y veenaghey, agh va'n eie echey goaill stiagh far-hoiggalyn bunneydagh v'ayn foast rish ymmodee keeadyn. Haink Archimedes dy hoiggal eh ny share liorish cooilleeiney jeshaghtyn bunneydagh, agh ec y traa shen, shimmey peiagh va credjue eieyn Aristotle foast. Rish toshiaght Lhing ny Soilsheyraght, chiartee Isaac Newton ny far-hoiggalyn shoh lesh fys maddaghtoil va cummal magh rish tree keeadyn, faggys. Leah 'sy 19oo eash, doshil Einstein sheiltynys ny conastaght er lheh er son dy 'aaishnaghey dy kiart cre'n aght vees forseyn ardjaghey dy easponaantagh tra ta breneen çheet tayrn er-gerrey da bieauid hollys.
Pre-Newtonian concepts
Since antiquity, the concept of force has been recognized as integral to the functioning of each of the simple machines. The mechanical advantage given by a simple machine allowed for less force to be used in exchange for that force acting over a greater distance. Analysis of the characteristics of forces ultimately culminated in the work of Archimedes who was especially famous for formulating a treatment of buoyant forces inherent in fluids.
Aristotle provided a philosophical discussion of the concept of a force as an integral part of Aristotelian cosmology. In Aristotle's view, the natural world held four elements that existed in "natural states". Aristotle believed that it was the natural state of objects with mass on Earth, such as the elements water and earth, to be motionless on the ground and that they tended towards that state if left alone. He distinguished between the innate tendency of objects to find their "natural place" (e.g., for heavy bodies to fall), which led to "natural motion", and unnatural or forced motion, which required continued application of a force. This theory, based on the everyday experience of how objects move, such as the constant application of a force needed to keep a cart moving, had conceptual trouble accounting for the behavior of projectiles, such as the flight of arrows. The place where forces were applied to projectiles was only at the start of the flight, and while the projectile sailed through the air, no discernible force acts on it. Aristotle was aware of this problem and proposed that the air displaced through the projectile's path provided the needed force to continue the projectile moving. This explanation demands that air is needed for projectiles and that, for example, in a vacuum, no projectile would move after the initial push. Additional problems with the explanation include the fact that air resists the motion of the projectiles.
These shortcomings would not be fully explained and corrected until the seventeenth century work of Galileo Galilei, who was influenced by the late medieval idea that objects in forced motion carried an innate force of impetus. Galileo constructed an experiment in which stones and cannonballs were both rolled down an incline to disprove the Aristotelian theory of motion early in the seventeenth century. He showed that the bodies were accelerated by gravity to an extent which was independent of their mass and argued that objects retain their velocity unless acted on by a force, for example friction.
Newton's first law
Newton's first law of motion states that objects continue to move in a state of constant velocity unless acted upon by an external net force or resultant force. This law is an extension of Galileo's insight that constant velocity was associated with a lack of net force (see a more detailed description of this below). Newton proposed that every object with mass has an innate inertia that functions as the fundamental equilibrium "natural state" in place of the Aristotelian idea of the "natural state of rest". That is, the first law contradicts the intuitive Aristotelian belief that a net force is required to keep an object moving with constant velocity. By making rest physically indistinguishable from non-zero constant velocity, Newton's first law directly connects inertia with the concept of relative velocities. Specifically, in systems where objects are moving with different velocities, it is impossible to determine which object is "in motion" and which object is "at rest". In other words, to phrase matters more technically, the laws of physics are the same in every inertial frame of reference, that is, in all frames related by a Galilean transformation.
For example, while traveling in a moving vehicle at a constant velocity, the laws of physics do not change from being at rest. A person can throw a ball straight up in the air and catch it as it falls down without worrying about applying a force in the direction the vehicle is moving. This is true even though another person who is observing the moving vehicle pass by also observes the ball follow a curving parabolic path in the same direction as the motion of the vehicle. It is the inertia of the ball associated with its constant velocity in the direction of the vehicle's motion that ensures the ball continues to move forward even as it is thrown up and falls back down. From the perspective of the person in the car, the vehicle and every thing inside of it is at rest: It is the outside world that is moving with a constant speed in the opposite direction. Since there is no experiment that can distinguish whether it is the vehicle that is at rest or the outside world that is at rest, the two situations are considered to be physically indistinguishable. Inertia therefore applies equally well to constant velocity motion as it does to rest.
The concept of inertia can be further generalized to explain the tendency of objects to continue in many different forms of constant motion, even those that are not strictly constant velocity. The rotational inertia of planet Earth is what fixes the constancy of the length of a day and the length of a year. Albert Einstein extended the principle of inertia further when he explained that reference frames subject to constant acceleration, such as those free-falling toward a gravitating object, were physically equivalent to inertial reference frames. This is why, for example, astronauts experience weightlessness when in free-fall orbit around the Earth, and why Newton's Laws of Motion are more easily discernible in such environments. If an astronaut places an object with mass in mid-air next to herself, it will remain stationary with respect to the astronaut due to its inertia. This is the same thing that would occur if the astronaut and the object were in intergalactic space with no net force of gravity acting on their shared reference frame. This principle of equivalence was one of the foundational underpinnings for the development of the general theory of relativity.
[[Image:GodfreyKneller-IsaacNewton-1689.jpg|right|150 px|thumb| Though [[Sir Isaac Newton]]'s most famous equation is
, he actually wrote down a different form for his second law of motion that did not use differential calculus.]]
Newton's second law
A modern statement of Newton's second law is a vector differential equation: where is the momentum of the system, and  is the net (vector sum) force. In equilibrium, there is zero net force by definition, but (balanced) forces may be present nevertheless. In contrast, the second law states an unbalanced force acting on an object will result in the object's momentum changing over time.
By the definition of momentum, where m is the mass and is the velocity.
The product rule shows that
For closed systems (systems of constant total mass), the time derivative of mass is zero and the equation becomes
By substituting the definition of acceleration, the algebraic version of this common simplification of Newton's second law is derived:
It is sometimes called the "second most famous formula in physics". Newton never explicitly stated the formula in the reduced form above.
Newton's second law asserts the proportionality of acceleration and mass to force. Accelerations can be defined through kinematic measurements. However, while kinematics are well-described through reference frame analysis in advanced physics, there are still deep questions that remain as to what is the proper definition of mass. General relativity offers an equivalence between space-time and mass, but lacking a coherent theory of quantum gravity, it is unclear as to how or whether this connection is relevant on microscales. With some justification, Newton's second law can be taken as a quantitative definition of mass by writing the law as an equality; the relative units of force and mass then are fixed.
The use of Newton's second law as a definition of force has been disparaged in some of the more rigorous textbooks, because it is essentially a mathematical truism. The equality between the abstract idea of a "force" and the abstract idea of a "changing momentum vector" ultimately has no observational significance because one cannot be defined without simultaneously defining the other. What a "force" or "changing momentum" is must either be referred to an intuitive understanding of our direct perception, or be defined implicitly through a set of self-consistent mathematical formulas. Notable physicists, philosophers and mathematicians who have sought a more explicit definition of the concept of "force" include Ernst Mach, Clifford Truesdell and Walter Noll.
Newton's second law can be used to measure the strength of forces. For instance, knowledge of the masses of planets along with the accelerations of their orbits allows scientists to calculate the gravitational forces on planets.
Newton's third law
This law implies that forces always occur in action-and-reaction pairs. If object 1 and object 2 are considered to be in the same system, then the net force on the system due to the interactions between objects 1 and 2 is zero since
This means that in a closed system of particles, there are no internal forces that are unbalanced. That is, action-and-reaction pairs of forces shared between any two objects in a closed system will not cause the center of mass of the system to accelerate. The constituent objects only accelerate with respect to each other, the system itself remains unaccelerated. Alternatively, if an external force acts on the system, then the center of mass will experience an acceleration proportional to the magnitude of the external force divided by the mass of the system.
Combining Newton's second and third laws, it is possible to show that the linear momentum of a system is conserved. Using
is obtained. For a system which includes objects 1 and 2,
which is the conservation of linear momentum. Using the similar arguments, it is possible to generalizing this to a system of an arbitrary number of particles. This shows that exchanging momentum between constituent objects will not affect the net momentum of a system. In general, as long as all forces are due to the interaction of objects with mass, it is possible to define a system such that net momentum is never lost nor gained.
Descriptions
Since forces are perceived as pushes or pulls, this can provide an intuitive understanding for describing forces. As with other physical concepts (e.g. temperature), the intuitive understanding of forces is quantified using precise operational definitions that are consistent with direct observations and compared to a standard measurement scale. Through experimentation, it is determined that laboratory measurements of forces are fully consistent with the conceptual definition of force offered by Newtonian mechanics.
Forces act in a particular direction and have sizes dependent upon how strong the push or pull is. Because of these characteristics, forces are classified as "vector quantities". This means that forces follow a different set of mathematical rules than physical quantities that do not have direction (denoted scalar quantities). For example, when determining what happens when two forces act on the same object, it is necessary to know both the magnitude and the direction of both forces to calculate the result. If both of these pieces of information are not known for each force, the situation is ambiguous. For example, if you know that two people are pulling on the same rope with known magnitudes of force but you do not know which direction either person is pulling, it is impossible to determine what the acceleration of the rope will be. The two people could be pulling against each other as in tug of war or the two people could be pulling in the same direction. In this simple one-dimensional example, without knowing the direction of the forces it is impossible to decide whether the net force is the result of adding the two force magnitudes or subtracting one from the other. Associating forces with vectors avoids such problems.
Historically, forces were first quantitatively investigated in conditions of static equilibrium where several forces canceled each other out. Such experiments demonstrate the crucial properties that forces are additive vector quantities: they have magnitude and direction. When two forces act on an object, the resulting force, the resultant, can be determined by following the parallelogram rule of vector addition: the addition of two vectors represented by sides of a parallelogram, gives an equivalent resultant vector which is equal in magnitude and direction to the transversal of the parallelogram. . The magnitude of the resultant varies from the difference of the magnitudes of the two forces to their sum, depending on the angle between their lines of action.
Free-body diagrams can be used as a convenient way to keep track of forces acting on a system. Ideally, these diagrams are drawn with the angles and relative magnitudes of the force vectors preserved so that graphical vector addition can be done to determine the resultant.
As well as being added, forces can also be resolved into independent components at right angles to each other. A horizontal force pointing northeast can therefore be split into two forces, one pointing north, and one pointing east. Summing these component forces using vector addition yields the original force. Resolving force vectors into components of a set of basis vectors is often a more mathematically clean way to describe forces than using magnitudes and directions. This is because, for orthogonal components, the components of the vector sum are uniquely determined by the scalar addition of the components of the individual vectors. Orthogonal components are independent of each other because forces acting at ninety degrees to each other have no effect on the magnitude or direction of the other. Choosing a set of orthogonal basis vectors is often done by considering what set of basis vectors will make the mathematics most convenient. Choosing a basis vector that is in the same direction as one of the forces is desirable, since that force would then have only one non-zero component. Orthogonal force vectors can be three-dimensional with the third component being at right-angles to the other two.
Equilibria
Equilibrium occurs when the resultant force acting on a point particle is zero (that is, the vector sum of all forces is zero). When dealing with an extense body, it is also necessary that the net torque in it is 0.
There are two kinds of equilibrium: static equilibrium and dynamic equilibrium.
Static equilibrium
Static equilibrium was understood well before the invention of classical mechanics. Objects which are at rest have zero net force acting on them.
The simplest case of static equilibrium occurs when two forces are equal in magnitude but opposite in direction. For example, an object on a level surface is pulled (attracted) downward toward the center of the Earth by the force of gravity. At the same time, surface forces resist the downward force with equal upward force (called the normal force). The situation is one of zero net force and no acceleration.
Pushing against an object on a frictional surface can result in a situation where the object does not move because the applied force is opposed by static friction, generated between the object and the table surface. For a situation with no movement, the static friction force exactly balances the applied force resulting in no acceleration. The static friction increases or decreases in response to the applied force up to an upper limit determined by the characteristics of the contact between the surface and the object.
A static equilibrium between two forces is the most usual way of measuring forces, using simple devices such as weighing scales and spring balances. For example, an object suspended on a vertical spring scale experiences the force of gravity acting on the object balanced by a force applied by the "spring reaction force" which equals object's weight. Using such tools, some quantitative force laws were discovered: that the force of gravity is proportional to volume for objects of constant density (widely exploited for millennia to define standard weights); Archimedes' principle for buoyancy; Archimedes' analysis of the lever; Boyle's law for gas pressure; and Hooke's law for springs. These were all formulated and experimentally verified before Isaac Newton expounded his three laws of motion.
Dynamical equilibrium
Dynamical equilibrium was first described by Galileo who noticed that certain assumptions of Aristotelian physics were contradicted by observations and logic. Galileo realized that simple velocity addition demands that the concept of an "absolute rest frame" did not exist. Galileo concluded that motion in a constant velocity was completely equivalent to rest. This was contrary to Aristotle's notion of a "natural state" of rest that objects with mass naturally approached. Simple experiments showed that Galileo's understanding of the equivalence of constant velocity and rest to be correct. For example, if a mariner dropped a cannonball from the crow's nest of a ship moving at a constant velocity, Aristotelian physics would have the cannonball fall straight down while the ship moved beneath it. Thus, in an Aristotelian universe, the falling cannonball would land behind the foot of the mast of a moving ship. However, when this experiment is actually conducted, the cannonball always falls at the foot of the mast, as if the cannonball knows to travel with the ship despite being separated from it. Since there is no forward horizontal force being applied on the cannonball as it falls, the only conclusion left is that the cannonball continues to move with the same velocity as the boat as it falls. Thus, no force is required to keep the cannonball moving at the constant forward velocity.
Moreover, any object traveling at a constant velocity must be subject to zero net force (resultant force). This is the definition of dynamical equilibrium: when all the forces on an object balance but it still moves at a constant velocity.
A simple case of dynamical equilibrium occurs in constant velocity motion across a surface with kinetic friction. In such a situation, a force is applied in the direction of motion while the kinetic friction force exactly opposes the applied force. This results in a net zero force, but since the object started with a non-zero velocity, it continues to move with a non-zero velocity. Aristotle misinterpreted this motion as being caused by the applied force. However, when kinetic friction is taken into consideration it is clear that there is no net force causing constant velocity motion.
Special relativity
In the special theory of relativity mass and energy are equivalent (as can be seen by calculating the work required to accelerate an object). When an object's velocity increases so does its energy and hence its mass equivalent (inertia). It thus requires more force to accelerate it the same amount than it did at a lower velocity. Newton's second law
remains valid due to the fact that it is a mathematical definition. But in order to be conserved, relativistic momentum must be redefined as:
where the Lorentz factor
Relativistic force does not produce a constant acceleration, but an ever decreasing acceleration as the object approaches the speed of light. Note that \gamma is undefined for an object with a non zero rest mass at the speed of light, and the theory yields no prediction at that speed.
One can however restore the form of
Feynman diagrams
In modern particle physics, forces and the acceleration of particles are explained as the exchange of momentum-carrying gauge bosons. With the development of quantum field theory and general relativity, it was realized that "force" is a redundant concept arising from conservation of momentum (4-momentum in relativity and momentum of virtual particles in quantum electrodynamics). The conservation of momentum, from Noether's theorem, can be directly derived from the symmetry of space and so is usually considered more fundamental than the concept of a force. Thus the currently known fundamental forces are considered more accurately to be "fundamental interactions". When particle A emits (creates) or absorbs (annihilates) particle B, a force accelerates particle A in response to the momentum of particle B, thereby conserving momentum as a whole. This description applies for all forces arising from fundamental interactions. While sophisticated mathematical descriptions are needed to predict, in full detail, the nature of such interactions, there is a conceptually simple way to describe such interactions through the use of Feynman diagrams. In a Feynman diagram, each matter particle is represented as a straight line (see world line) traveling through time which normally increases up or to the right in the diagram. Matter and anti-matter particles are identical except for their direction of propagation through the Feynman diagram. World lines of particles intersect at interaction vertices, and the Feynman diagram represents any force arising from an interaction as occurring at the vertex with an associated instantaneous change in the direction of the particle world lines. Gauge bosons are emitted away from the vertex as wavy lines (similar to waves) and, in the case of virtual particle exchange, are absorbed at an adjacent vertex.
The utility of Feynman diagrams is that other types of physical phenomena that are part of the general picture of fundamental interactions but are conceptually separate from forces can also be described using the same rules. For example, a Feynman diagram can describe in succinct detail how a neutron decays into an electron, proton, and neutrino, an interaction mediated by the same gauge boson that is responsible for the weak nuclear force.
Fundamental models
All the forces in the universe are based on four fundamental forces. The strong and weak forces act only at very short distances, and are responsible for the interactions between subatomic particles including nucleons and compound nuclei. The electromagnetic force acts between electric charges and the gravitational force acts between masses. All other forces are based on the existence of the four fundamental interactions. For example, friction is a manifestation of the electromagnetic force acting between the atoms of two surfaces, and the Pauli Exclusion Principle, which does not allow atoms to pass through each other. The forces in springs, modeled by Hooke's law, are also the result of electromagnetic forces and the Exclusion Principle acting together to return the object to its equilibrium position. Centrifugal forces are acceleration forces which arise simply from the acceleration of rotating frames of reference.
The development of fundamental theories for forces proceeded along the lines of unification of disparate ideas. For example, Isaac Newton unified the force responsible for objects falling at the surface of the Earth with the force responsible for the orbits of celestial mechanics in his universal theory of gravitation. Michael Faraday and James Clerk Maxwell demonstrated that electric and magnetic forces were unified through one consistent theory of electromagnetism. In the twentieth century, the development of quantum mechanics led to a modern understanding that the first three fundamental forces (all except gravity) are manifestations of matter (fermions) interacting by exchanging virtual particles called gauge bosons. This standard model of particle physics posits a similarity between the forces and led scientists to predict the unification of the weak and electromagnetic forces in electroweak theory subsequently confirmed by observation. The complete formulation of the standard model predicts an as yet unobserved Higgs mechanism, but observations such as neutrino oscillations indicate that the standard model is incomplete. A grand unified theory allowing for the combination of the electroweak interaction with the strong force is held out as a possibility with candidate theories such as supersymmetry proposed to accommodate some of the outstanding unsolved problems in physics. Physicists are still attempting to develop self-consistent unification models that would combine all four fundamental interactions into a theory of everything. Einstein tried and failed at this endeavor, but currently the most popular approach to answering this question is string theory.
Gravity
What we now call gravity was not identified as a universal force until the work of Isaac Newton. Before Newton, the tendency for objects to fall towards the Earth was not understood to be related to the motions of celestial objects. Galileo was instrumental in describing the characteristics of falling objects by determining that the acceleration of every object in free-fall was constant and independent of the mass of the object. Today, this acceleration due to gravity towards the surface of the Earth is usually designated as \vec{g} and has a magnitude of about 9.81 meters per second squared (this measurement is taken from sea level and may vary depending on location), and points toward the center of the Earth. This observation means that the force of gravity on an object at the Earth's surface is directly proportional to the object's mass. Thus an object that has a mass of m will experience a force:
In free-fall, this force is unopposed and therefore the net force on the object is its weight. For objects not in free-fall, the force of gravity is opposed by the reactions of their supports. For example, a person standing on the ground experiences zero net force, since his weight is balanced by a normal force exerted by the ground.
Newton's contribution to gravitational theory was to unify the motions of heavenly bodies, which Aristotle had assumed were in a natural state of constant motion, with falling motion observed on the Earth. He proposed a law of gravity that could account for the celestial motions that had been described earlier using Kepler's Laws of Planetary Motion.
Newton came to realize that the effects of gravity might be observed in different ways at larger distances. In particular, Newton determined that the acceleration of the Moon around the Earth could be ascribed to the same force of gravity if the acceleration due to gravity decreased as an inverse square law. Further, Newton realized that the acceleration due to gravity is proportional to the mass of the attracting body. Combining these ideas gives a formula that relates the mass (M_\oplus) and the radius (R_\oplus) of the Earth to the gravitational acceleration:
where the vector direction is given by \hat{r}, the unit vector directed outward from the center of the Earth.
In this equation, a dimensional constant G is used to describe the relative strength of gravity. This constant has come to be known as Newton's Universal Gravitation Constant, though its value was unknown in Newton's lifetime. Not until 1798 was Henry Cavendish able to make the first measurement of G using a torsion balance; this was widely reported in the press as a measurement of the mass of the Earth since knowing the G could allow one to solve for the Earth's mass given the above equation. Newton, however, realized that since all celestial bodies followed the same laws of motion, his law of gravity had to be universal. Succinctly stated, Newton's Law of Gravitation states that the force on a spherical object of mass m_{1} due to the gravitational pull of mass m_2 is
where r is the distance between the two objects' centers of mass and \hat{r} is the unit vector pointed in the direction away from the center of the first object toward the center of the second object.
This formula was powerful enough to stand as the basis for all subsequent descriptions of motion within the solar system until the twentieth century. During that time, sophisticated methods of perturbation analysis were invented to calculate the deviations of orbits due to the influence of multiple bodies on a planet, moon, comet, or asteroid. The formalism was exact enough to allow mathematicians to predict the existence of the planet Neptune before it was observed.
It was only the orbit of the planet Mercury that Newton's Law of Gravitation seemed not to fully explain. Some astrophysicists predicted the existence of another planet (Vulcan) that would explain the discrepancies; however, despite some early indications, no such planet could be found. When Albert Einstein finally formulated his theory of general relativity (GR) he turned his attention to the problem of Mercury's orbit and found that his theory added a correction which could account for the discrepancy. This was the first time that Newton's Theory of Gravity had been shown to be less correct than an alternative.
Since then, and so far, general relativity has been acknowledged as the theory which best explains gravity. In GR, gravitation is not viewed as a force, but rather, objects moving freely in gravitational fields travel under their own inertia in straight lines through curved space-time – defined as the shortest space-time path between two space-time events. From the perspective of the object, all motion occurs as if there were no gravitation whatsoever. It is only when observing the motion in a global sense that the curvature of space-time can be observed and the force is inferred from the object's curved path. Thus, the straight line path in space-time is seen as a curved line in space, and it is called the ballistic trajectory of the object. For example, a basketball thrown from the ground moves in a parabola, as it is in a uniform gravitational field. Its space-time trajectory (when the extra ct dimension is added) is almost a straight line, slightly curved (with the radius of curvature of the order of few light-years). The time derivative of the changing momentum of the object is what we label as "gravitational force".
Electromagnetic forces
The electrostatic force was first described in 1784 by Coulomb as a force which existed intrinsically between two charges. The properties of the electrostatic force were that it varied as an inverse square law directed in the radial direction, was both attractive and repulsive (there was intrinsic polarity), was independent of the mass of the charged objects, and followed the law of superposition. Coulomb's Law unifies all these observations into one succinct statement.
Subsequent mathematicians and physicists found the construct of the electric field to be useful for determining the electrostatic force on an electric charge at any point in space. The electric field was based on using a hypothetical "test charge" anywhere in space and then using Coulomb's Law to determine the electrostatic force. Thus the electric field anywhere in space is defined as
where q is the magnitude of the hypothetical test charge.
where I is the magnitude of the hypothetical test current and \ell is the length of hypothetical wire through which the test current flows. The magnetic field exerts a force on all magnets including, for example, those used in compasses. The fact that the Earth's magnetic field is aligned closely with the orientation of the Earth's axis causes compass magnets to become oriented because of the magnetic force pulling on the needle.
Through combining the definition of electric current as the time rate of change of electric charge, a rule of vector multiplication called Lorentz's Law describes the force on a charge moving in an magnetic field. The connection between electricity and magnetism allows for the description of a unified electromagnetic force that acts on a charge. This force can be written as a sum of the electrostatic force (due to the electric field) and the magnetic force (due to the magnetic field). Fully stated, this is the law:
where \vec{F} is the electromagnetic force, q is the magnitude of the charge of the particle, \vec{E} is the electric field, \vec{v} is the velocity of the particle which is crossed with the magnetic field (\vec{B}).