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edu365.cat. Ciències de la naturalesa: exercicis i problemes. Segon cicle. Força gravitatòria

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Newton, en intentar explicar el moviment planetari, va establir que dos cossos, pel sol fet de tenir massa, s'atreien amb una força F. Aquesta força d'atracció era directament proporcional al producte de les masses i inversament proporcional al quadrat de la distància que separa els seus centres.

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edu365.cat. Ciències de la naturalesa: exercicis i problemes. Primer cicle. Unitats de mesura.

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UNITATS BÀSIQUES DEL SISTEMA INTERNACIONAL (SI)

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edu365.cat. Ciències de la naturalesa: exercicis i problemes. Primer cicle. Cinemàtica: MRU.

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Es diu que un mòbil té aquest moviment quan la seva trajectòria és una línia recta i la seva velocitat és constant. La velocitat mitjana i la instantània coincideixen. Cal establir un conveni de signes per resoldre problemes. Si el mòbil va d'esquerra a dreta, la velocitat és positiva; si va de dreta a esquerra, la velocitat és negativa.

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edu365.cat. Ciències de la naturalesa: exercicis i problemes. Segon cicle. Dinàmica.

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És la part de la física que estudia els efectes de les forces sobre el moviment dels cossos.

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No hi ha passatgers. Tots som tripulants.

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eix d'ordenades i el a l'eix d'abscisses . Ens informen sobre:

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En el fitxer adjunt, hi trobareu un document amb exercicis de pràctica de cinemàtica. LlistatExerciciscinematica

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Pràctica Cinemàtica 3: Treballem amb fòrmules!

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En aquesta pràctica aprendrem a treballar amb les fòrmules. Entre d'altres coses, alguns trucs que ens permetran no haver de memoritzar tantes fòrmules. Pràctica Cinemàtica 3

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Pràctica Cinemàtica 2: Creació i interpretació de gràfics

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En aquesta pràctica, a més de crear gràfics, aprendrem a interpretar-los, i a veure-hi coses que no veiem en els números. Pràctica Cinemàtica 2

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edu365.cat. Ciències de la naturalesa: exercicis i problemes. Primer cicle. Cinemàtica: MRU. Enunciats

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CONTINGUT DE LA PÀGINA Gràfiques del M.R.U.
L'estudi del moviment

La cinemàtica és la part de la mecànica clàssica que estudia els moviments, sense tenir en compte les causes que el produeixen. Podríem dir que la cinemàtica se limita a estudiar la trajectòria d'un mòbil en funció del temps.
Tot està en moviment... fins i tot, una roca clavada al terra.
El moviment és relatiu.

Què vol dir això? Doncs que necessitem un sistema de referència a l'hora de saber si un cos està en moviment o en repòs. Podem considerar que un cos es mou respecte a un punt de referència, i en canvi pot estar en repòs si prenem un altre punt.

Ho entendràs amb un exemple. Queda't en repòs, sense moure't.....

Ara et pregunto si et mous: Estàs en moviment?

Segurament respondràs que no. I és veritat. No et mous respecte del terra que tens als peus, o per algú que t'observi prop teu.... Però també estàs en moviment. Algú que t'observés des de la Lluna et voria donant voltes a gairebé 1600 km/h!... I si algú et pugues observar des d'una galàxia propera et veuria desplaçant-te amb la Terra a una velocitat aproximada de 106.000 km/h. Et mous, clar que ets mou, i a molta velocitat!!!

El mateix passa quan estem dins d'un tren i veiem que la persona que tenim asseguda al davant fa estona que dorm i no mou ni una cella. No es mou, està clar, però els que s'estan a l'estació i veuen passar aquest mateix tren diria que, tot i estar dormit, el passatger s'està movent...amb el tren. Clar que es mou!

Així doncs, en què quedem? Es mou o no es mou?

Doncs depèn, això del moviment és relatiu.

Amb això volem dir que el moviment depèn d'on està situat l'observador, és a dir, de quin sistema de referència prenem per observar el moviment. Si el sistema de referència és el propi tren, el seu interior, el passatger adormit no es mou. Si el sistema de referència està situat sobre l'estació, aleshores el passatger està en moviment...

Per definir un moviment cal fixar un sistema de referència

A partir d'ara nosaltres agafarem com a referència per saber si alguna cosa es mou o no, el nostre planeta, excepte si indiquem el contrari. Situarem a l'observador sobre el sòl i treballarem amb uns eixos de coordenades que ens donaran la posició exacta del cos que es mou. (Com quan juguem al joc d'enfonsar vaixells!)

Quan un cos canvia la seva posició diem que està en moviment . El cos que es troba en moviment respecte del sistema de referència s'anomena mòbil.

Nosaltres considerarem el mòbil com un punt, no ens importa la forma ni les dimensions que té per estudiar el seu moviment. Només volem analitzar les seves posicions, la seva velocitat, la seva trajectòria. La cinemàtica l'anomena mòbil puntual.

Un cos en moviment ocuparà diferents punts de l'espai o posicions x en cada instant de temps t

La trajectòria descriu el camí que fa un cos quan es mou per l'espai, i està formada per la línia que es forma unit les diferents posicions que va ocupant el mòbil al llarg del temps. Aquests camins poden tenir formes molt diverses, però es poden classificar fàcilment en dos grups:

* Trajectòries rectilínies
Aquelles que formen una recta. En aquestes trajectòries el mòbil es dirigeix sempre cap al mateix lloc, no canvia la seva direcció.

Totes aquelles en què el mòbil va canviant la seva direcció a mesura que avança.

Distància i desplaçament

En Física, distància i desplaçament també són conceptes diferents.

El moviment és un canvi de posició, un desplaçament. Una manera d'adonar-nos del moviment d'un cos és observant un canvi en la seva posició. Els mòbils realitzen un desplaçament. Des d'una posició inicial a una posició final:

El El signe ens indica en quin sentit és el desplaçament.desplaçament és una magnitud vectorial que en el SI (sistema internacional d'unitats) s'expressa en m . S'obté calculant la posició final menys la posició inicial i el seu resultat és un vector (té una direcció, un sentit i un mòdul o valor absolut). Això vol dir que la física descriu el moviment de manera precisa, indicant el sentit en què es dóna el moviment i que, per tant, no és el mateix un desplaçament de 40 m que un altre de -40 m (no és el mateix desplaçar-se de la posició 10 fins a la 50, que fer-ho al revés, de la 50 a la 10).

El desplaçament ens indica només el canvi de posició en el mòbil, és el vector que té l'origen en la posició inicial i l'extrem que porta la fletxa en la posició final. El signe +/- que acompanya el valor numèric ens indica el sentit del desplaçament . Per conveni, + quan es desplaça cap a la dreta respecte a l'origen, i - quan va en sentit contrari, cap a l'esquerra.

També podem utilitzar la lletra d per referir-nos al desplaçament.

Però què passa si un mòbil es mou en cercle i torna a la seua posició inicial? Aleshores el desplaçament = zero... oi?

Però sabem que s'ha mogut!

La distància recorreguda és la longitud de la trajectòria expressada en m. És una magnitud escalar que expressa una longitud, per tant sempre té un valor positiu.

Només quan la trajectòria és una recta, el desplaçament coincideix amb l'espai o distància recorreguda.

Per anar d'un punt a un altre triguem un temps. Així podem definir la Però no només és important la rapidesa com l'espai que hem recorregut en un temps determinat.
rapidesa del moviment, sinó també la seva direcció i sentit. Per això, en Física s'utilitza el terme ens indica la rapidesa, i també la direcció i el sentit del moviment. Rapidesa i velocitat són magnituds diferents

La velocitat es calcula dividint l'espai recorregut entre el temps que s'ha necessitat per a recórrer-lo. És una magnitud vectorial i en el SI s'expressa en (metres per segon).

La velocitat d'un mòbil pot ser constant tot el temps, mantenint-se igual tot el recorregut. Diem aleshores que porta una velocitat uniforme.

Normalment, quan viatgem d'un lloc a un altre no anem contínuament a la mateixa velocitat: frenem, accelerem....

Per aquest motiu podem definir una velocitat mitjana, simplement prenent el desplaçament total realitzat i dividint-lo entre el temps total que hem trigat a cobrir-lo.

Tanmateix, a cada instant de temps el mòbil porta una velocitat concreta: és la velocitat instantània.

Velocitat mitjana: desplaçament efectuat pel mòbil durant cada unitat de temps entre l'instant inicial i final del desplaçament.

Velocitat instantània: és la velocitat mitjana en un interval de temps infinitament curt, que anomenem instant.

Rapidesa: és el valor absolut de la velocitat del mòbil.

L'acceleració

És freqüent referir-nos a aquesta magnitud de manera errònia: quan un vehicle circula molt ràpid, solem dir que va molt accelerat. Això no és correcte.

L'acceleració no ens diu com és la velocitat d'un mòbil (si gran o petita), l'acceleració ens indica com canvia aquesta velocitat. Pot ser que la velocitat sigui molt gran però l'acceleració molt petita, o zero. Hem d'entendre aleshores que aquesta velocitat no està canviant, o ho fa molt poc a poc.

L'acceleració mitjana la calculem mitjançant aquesta fórmula: En el Sistema Internacional, la unitat d'acceleració és 1 (m/s)/s, és a dir 1 m/s²

Però l' També igual que per a la velocitat, existeix una acceleració, igual que la velocitat, no sempre és constant al llarg del temps, sinó que pot anar canviant a cada instant.
acceleració instantània, que inclou una direcció i un sentit, i una acceleració mitjana

El moviment rectilini uniforme

Dins la aquest curs estudiarem només els moviments que descriuen una trajectòria recta i que no varien la seva velocitat en tot el desplaçament, porten una velocitat uniforme: el moviment rectilini uniforme (MRU).

Les condicions del moviment rectilini uniforme són:

1) El mòbil descriu una trajectòria recta. Per tant el desplaçament i la distància recorreguda coincideixen.

2) La velocitat no varia, és constant, l'acceleració = 0 . La velocitat és uniforme. La velocitat mitjana i la instantània coincideixen.

Per descriure el moviment necessitarem conèixer la posició inicial, la posició final i el temps que s'ha empleat en desplaçar-se des de la posició inicial a la final.

L' equació del moviment és la fórmula que ens permet saber la posició d'un mòbil en un instant determinat. És l'equació de la posició en funció del temps. Si es donen valors a t s'obtenen valors de x.

- Vols una divertida lliçó amb moviment? Dins d'aquesta pàgina ves al primer capítol Cinemàtica.

-Una lliçó interactiva sobre el moviment i les forces amb gràfics animats (en castellà) .

Les gràfiques del moviment rectilini uniforme.

El moviment d'un mòbil també podem estudiar-lo a partir de les gràfiques del moviment: posició-temps i velocitat temps.

A l'eix d'ordenades ( y) es representa la posició o la velocitat, mentre que en l'eix d'abscisses ( x) es representa el temps.

- Una web que t'ajudarà amb les gràfiques del moviment

- Una lliçó virtual sobre la gràfica posició-temps que t'ajudarà a entendre-la.

PER RESOLDRE PROBLEMES DE CINEMÀTICA...

1. Llegeix atentament l'enunciat.
2. Escriu les dades que et dóna el problema: Posició inicial i final;Temps inicial i final; Velocitat ...
3. Determina què has de trobar.
4. Recorda les equacions del moviment: x(t) = x + vt
5. Construeix una gràfica, essent molt curós amb les línies i els punts.
6. Substitueix les dades a les equacions i no t'equivoquis en aïllar.
7. Comprova que les teues solucions són coherents, tenen sentit.

* Exercicis i problemes de cinemàtica . Podràs comprovar si els has fet bé perquè també venen les solucions i la manera de resoldre'ls. També hi pots trobar una autoavaluació per comprovar com portes tots els conceptes.

* Aquí tens un sobre el moviment dels cossos.

* Exercicis resolts de Cinemàtica i més problemes de cinemàtica i si encara vols seguir practicant... Exercicis de MRU amb solució

- Estudi interactiu dels moviments rectilinis

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CONTINGUT DE LA PÀGINA Gràfiques del M.R.U. La cinemàtica és la part de la mecànica clàssica que estudia els moviments, sense tenir en compte les causes que el produeixen. Podríem dir que la cinemàtica se limita a estudiar la trajectòria d'un mòbil en funció del temps. Tot està en moviment...

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Flying model rockets is a relatively safe and inexpensive way for students to learn the basics of forces and the response of a vehicle to external forces. A model rocket is subjected to four forces in flight; weight, thrust, and the aerodynamic forces, lift and drag. There are many different types of model rockets.

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Thrust is the force which moves the rocket through the air, and through space. Thrust is generated by the propulsion system of the rocket through the application of Newton's third law of motion; For every action there is an equal and opposite re-action.

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Forces are vector quantities having both a magnitude and a direction. When describing the action of forces, one must account for both the magnitude and the direction. In flight, a rocket is subjected to four forces; weight, thrust, and the aerodynamic forces, lift and drag.

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During the end of the 18th century and early into the 19th, rockets experienced a brief revival as a weapon of war. The success of Indian rocket barrages against the British in 1792 and again in 1799 caught the interest of an artillery expert, Colonel William Congreve.

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Sir Isaac Newton (25 December 1642 - 20 March 1726/27) was an English polymath active as a mathematician, physicist, astronomer, alchemist, theologian, and author who was described in his time as a natural philosopher. He was a key figure in the Scientific Revolution and the Enlightenment that followed. His pioneering book Philosophiæ Naturalis Principia Mathematica ( Mathematical Principles of Natural Philosophy), first published in 1687, consolidated many previous results and established classical mechanics. Newton also made seminal contributions to optics, and shares credit with German mathematician Gottfried Wilhelm Leibniz for developing infinitesimal calculus, though notably he developed calculus well before Leibniz. He is considered one of the greatest and most influential scientists in history.

In the Principia, Newton formulated the laws of motion and universal gravitation that formed the dominant scientific viewpoint for centuries until it was superseded by the theory of relativity. Newton used his mathematical description of gravity to derive Kepler's laws of planetary motion, account for tides, the trajectories of comets, the precession of the equinoxes and other phenomena, eradicating doubt about the Solar System's heliocentricity. He demonstrated that the motion of objects on Earth and celestial bodies could be accounted for by the same principles. Newton's inference that the Earth is an oblate spheroid was later confirmed by the geodetic measurements of Maupertuis, La Condamine, and others, convincing most European scientists of the superiority of Newtonian mechanics over earlier systems.

Newton built the first practical reflecting telescope and developed a sophisticated theory of colour based on the observation that a prism separates white light into the colours of the visible spectrum. His work on light was collected in his highly influential book , published in 1704. He also formulated an empirical law of cooling, made the first theoretical calculation of the speed of sound, and introduced the notion of a Newtonian fluid. In addition to his work on calculus, as a mathematician Newton contributed to the study of power series, generalised the binomial theorem to non-integer exponents, developed a method for approximating the roots of a function, and classified most of the cubic plane curves.

Newton was a fellow of Trinity College and the second Lucasian Professor of Mathematics at the University of Cambridge. He was a devout but unorthodox Christian who privately rejected the doctrine of the Trinity. He refused to take holy orders in the Church of England, unlike most members of the Cambridge faculty of the day. Beyond his work on the mathematical sciences, Newton dedicated much of his time to the study of alchemy and biblical chronology, but most of his work in those areas remained unpublished until long after his death. Politically and personally tied to the Whig party, Newton served two brief terms as Member of Parliament for the University of Cambridge, in 1689-1690 and 1701-1702. He was knighted by Queen Anne in 1705 and spent the last three decades of his life in London, serving as Warden (1696-1699) and Master (1699-1727) of the Royal Mint, as well as president of the Royal Society (1703-1727).

Isaac Newton was born (according to the Julian calendar in use in England at the time) on Christmas Day, 25 December 1642 ( NS 4 January 1643) at Woolsthorpe Manor in Woolsthorpe-by-Colsterworth, a hamlet in the county of Lincolnshire. His father, also named Isaac Newton, had died three months before. Born prematurely, Newton was a small child; his mother Hannah Ayscough reportedly said that he could have fit inside a quart mug. When Newton was three, his mother remarried and went to live with her new husband, the Reverend Barnabas Smith, leaving her son in the care of his maternal grandmother, Margery Ayscough (née Blythe). Newton disliked his stepfather and maintained some enmity towards his mother for marrying him, as revealed by this entry in a list of sins committed up to the age of 19: "Threatening my father and mother Smith to burn them and the house over them." Newton's mother had three children (Mary, Benjamin, and Hannah) from her second marriage.

The King's School

From the age of about twelve until he was seventeen, Newton was educated at The King's School in Grantham, which taught Latin and Ancient Greek and probably imparted a significant foundation of mathematics. He was removed from school and returned to Woolsthorpe-by-Colsterworth by October 1659. His mother, widowed for the second time, attempted to make him a farmer, an occupation he hated. Henry Stokes, master at The King's School, persuaded his mother to send him back to school. Motivated partly by a desire for revenge against a schoolyard bully, he became the top-ranked student, distinguishing himself mainly by building sundials and models of windmills.

University of Cambridge

In June 1661, Newton was admitted to Trinity College at the University of Cambridge. His uncle Reverend William Ayscough, who had studied at Cambridge, recommended him to the university. At Cambridge, Newton started as a subsizar, paying his way by performing valet duties until he was awarded a scholarship in 1664, which covered his university costs for four more years until the completion of his MA. At the time, Cambridge's teachings were based on those of Aristotle, whom Newton read along with then more modern philosophers, including Descartes and astronomers such as Galileo Galilei and Thomas Street. He set down in his notebook a series of "Quaestiones" about mechanical philosophy as he found it. In 1665, he discovered the generalised binomial theorem and began to develop a mathematical theory that later became calculus. Soon after Newton obtained his BA degree at Cambridge in August 1665, the university temporarily closed as a precaution against the Great Plague.

Although he had been undistinguished as a Cambridge student, Newton's private studies at his home in Woolsthorpe over the next two years saw the development of his theories on calculus, optics, and the law of gravitation.

In April 1667, Newton returned to the University of Cambridge, and in October he was elected as a fellow of Trinity. Fellows were required to take holy orders and be ordained as Anglican priests, although this was not enforced in the Restoration years, and an assertion of conformity to the Church of England was sufficient. He made the commitment that "I will either set Theology as the object of my studies and will take holy orders when the time prescribed by these statutes [7 years] arrives, or I will resign from the college." Up until this point he had not thought much about religion and had twice signed his agreement to the Thirty-nine Articles, the basis of Church of England doctrine. By 1675 the issue could not be avoided, and by then his unconventional views stood in the way.

His academic work impressed the Lucasian professor Isaac Barrow, who was anxious to develop his own religious and administrative potential (he became master of Trinity College two years later); in 1669, Newton succeeded him, only one year after receiving his MA. The terms of the Lucasian professorship required that the holder not be active in the church - presumably[weasel words ] to leave more time for science. Newton argued that this should exempt him from the ordination requirement, and King Charles II, whose permission was needed, accepted this argument; thus, a conflict between Newton's religious views and Anglican orthodoxy was averted.

Newton was elected a Fellow of the Royal Society (FRS) in 1672.

Newton's work has been said "to distinctly advance every branch of mathematics then studied". His work on the subject, usually referred to as fluxions or calculus, seen in a manuscript of October 1666, is now published among Newton's mathematical papers. His work De analysi per aequationes numero terminorum infinitas, sent by Isaac Barrow to John Collins in June 1669, was identified by Barrow in a letter sent to Collins that August as the work "of an extraordinary genius and proficiency in these things". Newton later became involved in a dispute with Leibniz over priority in the development of calculus (the Leibniz-Newton calculus controversy). Most modern historians believe that Newton and Leibniz developed calculus independently, although with very different mathematical notations. However, it is established that Newton came to develop calculus much earlier than Leibniz. Leibniz's notation and "differential Method", nowadays recognised as much more convenient notations, were adopted by continental European mathematicians, and after 1820 or so, also by British mathematicians.[citation needed ]

His work extensively uses calculus in geometric form based on limiting values of the ratios of vanishingly small quantities: in the Principia itself, Newton gave demonstration of this under the name of "the method of first and last ratios" and explained why he put his expositions in this form, remarking also that "hereby the same thing is performed as by the method of indivisibles." Because of this, the Principia has been called "a book dense with the theory and application of the infinitesimal calculus" in modern times and in Newton's time "nearly all of it is of this calculus." His use of methods involving "one or more orders of the infinitesimally small" is present in his De motu corporum in gyrum of 1684 and in his papers on motion "during the two decades preceding 1684".

Newton in 1702 by Godfrey Kneller

Newton had been reluctant to publish his calculus because he feared controversy and criticism. He was close to the Swiss mathematician Nicolas Fatio de Duillier. In 1691, Duillier started to write a new version of Newton's Principia, and corresponded with Leibniz. In 1693, the relationship between Duillier and Newton deteriorated and the book was never completed. Starting in 1699, other members[who? ] of the Royal Society accused Leibniz of plagiarism. The dispute then broke out in full force in 1711 when the Royal Society proclaimed in a study that it was Newton who was the true discoverer and labelled Leibniz a fraud; it was later found that Newton wrote the study's concluding remarks on Leibniz. Thus began the bitter controversy which marred the lives of both Newton and Leibniz until the latter's death in 1716.

Newton is generally credited with the generalised binomial theorem, valid for any exponent. He discovered Newton's identities, Newton's method, classified cubic plane curves ( polynomials of degree three in two variables), made substantial contributions to the theory of finite differences, and was the first to use fractional indices and to employ coordinate geometry to derive solutions to Diophantine equations. He approximated partial sums of the harmonic series by logarithms (a precursor to Euler's summation formula) and was the first to use power series with confidence and to revert power series. Newton's work on infinite series was inspired by Simon Stevin's decimals.

A replica of the reflecting telescope Newton presented to the Royal Society in 1672 (the first one he made in 1668 was loaned to an instrument maker but there is no further record of what happened to it).[52]

In 1666, Newton observed that the spectrum of colours exiting a prism in the position of minimum deviation is oblong, even when the light ray entering the prism is circular, which is to say, the prism refracts different colours by different angles. This led him to conclude that colour is a property intrinsic to light - a point which had, until then, been a matter of debate.

From 1670 to 1672, Newton lectured on optics. During this period he investigated the refraction of light, demonstrating that the multicoloured image produced by a prism, which he named a spectrum, could be recomposed into white light by a lens and a second prism. Modern scholarship has revealed that Newton's analysis and resynthesis of white light owes a debt to corpuscular alchemy.

He showed that coloured light does not change its properties by separating out a coloured beam and shining it on various objects, and that regardless of whether reflected, scattered, or transmitted, the light remains the same colour. Thus, he observed that colour is the result of objects interacting with already-coloured light rather than objects generating the colour themselves. This is known as Newton's theory of colour.

Illustration of a dispersive prism separating white light into the colours of the spectrum, as discovered by Newton

From this work, he concluded that the lens of any refracting telescope would suffer from the dispersion of light into colours ( chromatic aberration). As a proof of the concept, he constructed a telescope using reflective mirrors instead of lenses as the objective to bypass that problem. Building the design, the first known functional reflecting telescope, today known as a Newtonian telescope, involved solving the problem of a suitable mirror material and shaping technique. Newton ground his own mirrors out of a custom composition of highly reflective speculum metal, using Newton's rings to judge the quality of the optics for his telescopes. In late 1668, he was able to produce this first reflecting telescope. It was about eight inches long and it gave a clearer and larger image. In 1671, the Royal Society asked for a demonstration of his reflecting telescope. Their interest encouraged him to publish his notes, Of Colours, which he later expanded into the work . When Robert Hooke criticised some of Newton's ideas, Newton was so offended that he withdrew from public debate. Newton and Hooke had brief exchanges in 1679-80, when Hooke, appointed to manage the Royal Society's correspondence, opened up a correspondence intended to elicit contributions from Newton to Royal Society transactions, which had the effect of stimulating Newton to work out a proof that the elliptical form of planetary orbits would result from a centripetal force inversely proportional to the square of the radius vector. But the two men remained generally on poor terms until Hooke's death.

Facsimile of a 1682 letter from Newton to William Briggs, commenting on Briggs' A New Theory of Vision

Newton argued that light is composed of particles or corpuscles, which were refracted by accelerating into a denser medium. He verged on soundlike waves to explain the repeated pattern of reflection and transmission by thin films ( Opticks Bk. II, Props. 12), but still retained his theory of 'fits' that disposed corpuscles to be reflected or transmitted (Props.13). However, later physicists favoured a purely wavelike explanation of light to account for the interference patterns and the general phenomenon of diffraction. Today's quantum mechanics, photons, and the idea of wave-particle duality bear only a minor resemblance to Newton's understanding of light.

In his Hypothesis of Light of 1675, Newton posited the existence of the ether to transmit forces between particles. The contact with the Cambridge Platonist philosopher Henry More revived his interest in alchemy. He replaced the ether with occult forces based on Hermetic ideas of attraction and repulsion between particles. John Maynard Keynes, who acquired many of Newton's writings on alchemy, stated that "Newton was not the first of the age of reason: He was the last of the magicians." Newton's contributions to science cannot be isolated from his interest in alchemy. This was at a time when there was no clear distinction between alchemy and science, and had he not relied on the occult idea of action at a distance, across a vacuum, he might not have developed his theory of gravity.

In 1704, Newton published , in which he expounded his corpuscular theory of light. He considered light to be made up of extremely subtle corpuscles, that ordinary matter was made of grosser corpuscles and speculated that through a kind of alchemical transmutation "Are not gross Bodies and Light convertible into one another, ... and may not Bodies receive much of their Activity from the Particles of Light which enter their Composition?" Newton also constructed a primitive form of a frictional electrostatic generator, using a glass globe.

In his book Opticks, Newton was the first to show a diagram using a prism as a beam expander, and also the use of multiple-prism arrays. Some 278 years after Newton's discussion, multiple-prism beam expanders became central to the development of narrow-linewidth tunable lasers. Also, the use of these prismatic beam expanders led to the multiple-prism dispersion theory.

Subsequent to Newton, much has been amended. Young and Fresnel discarded Newton's particle theory in favour of Huygens' wave theory to show that colour is the visible manifestation of light's wavelength. Science also slowly came to realise the difference between perception of colour and mathematisable optics. The German poet and scientist, Goethe, could not shake the Newtonian foundation but "one hole Goethe did find in Newton's armour, ... Newton had committed himself to the doctrine that refraction without colour was impossible. He, therefore, thought that the object-glasses of telescopes must forever remain imperfect, achromatism and refraction being incompatible. This inference was proved by Dollond to be wrong."

Engraving of Portrait of Newton by John Vanderbank

Newton's own copy of Principia with Newton's hand-written corrections for the second edition, now housed in the Wren Library at Trinity College, Cambridge

Newton had been developing his theory of gravitation as far back as 1665. In 1679, Newton returned to his work on celestial mechanics by considering gravitation and its effect on the orbits of planets with reference to Kepler's laws of planetary motion. This followed stimulation by a brief exchange of letters in 1679-80 with Hooke, who had been appointed to manage the Royal Society's correspondence, and who opened a correspondence intended to elicit contributions from Newton to Royal Society transactions. Newton's reawakening interest in astronomical matters received further stimulus by the appearance of a comet in the winter of 1680-1681, on which he corresponded with John Flamsteed. After the exchanges with Hooke, Newton worked out a proof that the elliptical form of planetary orbits would result from a centripetal force inversely proportional to the square of the radius vector. Newton communicated his results to Edmond Halley and to the Royal Society in De motu corporum in gyrum, a tract written on about nine sheets which was copied into the Royal Society's Register Book in December 1684. This tract contained the nucleus that Newton developed and expanded to form the Principia.

The was published on 5 July 1687 with encouragement and financial help from Halley. In this work, Newton stated the three universal laws of motion. Together, these laws describe the relationship between any object, the forces acting upon it and the resulting motion, laying the foundation for classical mechanics. They contributed to many advances during the Industrial Revolution which soon followed and were not improved upon for more than 200 years. Many of these advances continue to be the underpinnings of non-relativistic technologies in the modern world. He used the Latin word gravitas (weight) for the effect that would become known as gravity, and defined the law of universal gravitation.

In the same work, Newton presented a calculus-like method of geometrical analysis using 'first and last ratios', gave the first analytical determination (based on Boyle's law) of the speed of sound in air, inferred the oblateness of Earth's spheroidal figure, accounted for the precession of the equinoxes as a result of the Moon's gravitational attraction on the Earth's oblateness, initiated the gravitational study of the irregularities in the motion of the Moon, provided a theory for the determination of the orbits of comets, and much more. Newton's biographer David Brewster reported that the complexity of applying his theory of gravity to the motion of the moon was so great it affected Newton's health: "[H]e was deprived of his appetite and sleep" during his work on the problem in 1692-93, and told the astronomer John Machin that "his head never ached but when he was studying the subject". According to Brewster, Edmund Halley also told John Conduitt that when pressed to complete his analysis Newton "always replied that it made his head ache, and kept him awake so often, that he would think of it no more". [Emphasis in original]

Newton made clear his heliocentric view of the Solar System-developed in a somewhat modern way because already in the mid-1680s he recognised the "deviation of the Sun" from the centre of gravity of the Solar System. For Newton, it was not precisely the centre of the Sun or any other body that could be considered at rest, but rather "the common centre of gravity of the Earth, the Sun and all the Planets is to be esteem'd the Centre of the World", and this centre of gravity "either is at rest or moves uniformly forward in a right line". (Newton adopted the "at rest" alternative in view of common consent that the centre, wherever it was, was at rest.)

Newton's postulate of an invisible force able to act over vast distances led to him being criticised for introducing " occult agencies" into science. Later, in the second edition of the Principia (1713), Newton firmly rejected such criticisms in a concluding General Scholium, writing that it was enough that the phenomena implied a gravitational attraction, as they did; but they did not so far indicate its cause, and it was both unnecessary and improper to frame hypotheses of things that were not implied by the phenomena. (Here Newton used what became his famous expression "hypotheses non-fingo".)

With the Principia, Newton became internationally recognised. He acquired a circle of admirers, including the Swiss-born mathematician Nicolas Fatio de Duillier.

In 1710, Newton found 72 of the 78 "species" of cubic curves and categorised them into four types. In 1717, and probably with Newton's help, James Stirling proved that every cubic was one of these four types. Newton also claimed that the four types could be obtained by plane projection from one of them, and this was proved in 1731, four years after his death.

Isaac Newton in old age in 1712, portrait by Sir James Thornhill

In the 1690s, Newton wrote a number of religious tracts dealing with the literal and symbolic interpretation of the Bible. A manuscript Newton sent to John Locke in which he disputed the fidelity of 1 John 5:7-the Johannine Comma -and its fidelity to the original manuscripts of the New Testament, remained unpublished until 1785.

Newton was also a member of the Parliament of England for Cambridge University in 1689 and 1701, but according to some accounts his only comments were to complain about a cold draught in the chamber and request that the window be closed. He was, however, noted by Cambridge diarist Abraham de la Pryme to have rebuked students who were frightening locals by claiming that a house was haunted.

Newton moved to London to take up the post of warden of the Royal Mint in 1696, a position that he had obtained through the patronage of Charles Montagu, 1st Earl of Halifax, then Chancellor of the Exchequer. He took charge of England's great recoining, trod on the toes of Lord Lucas, Governor of the Tower, and secured the job of deputy comptroller of the temporary Chester branch for Edmond Halley. Newton became perhaps the best-known Master of the Mint upon the death of Thomas Neale in 1699, a position Newton held for the last 30 years of his life. These appointments were intended as sinecures, but Newton took them seriously. He retired from his Cambridge duties in 1701, and exercised his authority to reform the currency and punish clippers and counterfeiters.

As Warden, and afterwards as Master, of the Royal Mint, Newton estimated that 20 percent of the coins taken in during the Great Recoinage of 1696 were counterfeit. Counterfeiting was high treason, punishable by the felon being hanged, drawn and quartered. Despite this, convicting even the most flagrant criminals could be extremely difficult, but Newton proved equal to the task.

Disguised as a habitué of bars and taverns, he gathered much of that evidence himself. For all the barriers placed to prosecution, and separating the branches of government, English law still had ancient and formidable customs of authority. Newton had himself made a justice of the peace in all the home counties. A draft letter regarding the matter is included in Newton's personal first edition of Philosophiæ Naturalis Principia Mathematica, which he must have been amending at the time. Then he conducted more than 100 cross-examinations of witnesses, informers, and suspects between June 1698 and Christmas 1699. Newton successfully prosecuted 28 coiners.

Coat of arms of the Newton family of Great Gonerby, Lincolnshire, afterwards used by Sir Isaac[93]

Newton was made president of the Royal Society in 1703 and an associate of the French Académie des Sciences. In his position at the Royal Society, Newton made an enemy of John Flamsteed, the Astronomer Royal, by prematurely publishing Flamsteed's Historia Coelestis Britannica, which Newton had used in his studies.

Knighthood

In April 1705, Queen Anne knighted Newton during a royal visit to Trinity College, Cambridge. The knighthood is likely to have been motivated by political considerations connected with the parliamentary election in May 1705, rather than any recognition of Newton's scientific work or services as Master of the Mint. Newton was the second scientist to be knighted, after Francis Bacon.

As a result of a report written by Newton on 21 September 1717 to the Lords Commissioners of His Majesty's Treasury, the bimetallic relationship between gold coins and silver coins was changed by royal proclamation on 22 December 1717, forbidding the exchange of gold guineas for more than 21 silver shillings. This inadvertently resulted in a silver shortage as silver coins were used to pay for imports, while exports were paid for in gold, effectively moving Britain from the silver standard to its first gold standard. It is a matter of debate as to whether he intended to do this or not. It has been argued that Newton conceived of his work at the Mint as a continuation of his alchemical work.

Newton was invested in the South Sea Company and lost some £20,000 (£4.4 million in 2020) when it collapsed in around 1720.

Toward the end of his life, Newton took up residence at Cranbury Park, near Winchester, with his niece and her husband, until his death. His half-niece, Catherine Barton, served as his hostess in social affairs at his house on Jermyn Street in London; he was her "very loving Uncle", according to his letter to her when she was recovering from smallpox.

Death mask of Newton, photographed c. 1906

Newton died in his sleep in London on 20 March 1727 ( OS 20 March 1726; NS 31 March 1727). He was given a ceremonial funeral, attended by nobles, scientists, and philosophers, and was buried in Westminster Abbey among kings and queens. He was the first scientist to be buried in the abbey. Voltaire may have been present at his funeral. A bachelor, he had divested much of his estate to relatives during his last years, and died intestate. His papers went to John Conduitt and Catherine Barton.

Shortly after his death, a plaster death mask was moulded of Newton. It was used by Flemish sculptor John Michael Rysbrack in making a sculpture of Newton. It is now held by the Royal Society, who created a 3D scan of it in 2012.

Newton's hair was posthumously examined and found to contain mercury, probably resulting from his alchemical pursuits. Mercury poisoning could explain Newton's eccentricity in late life.

Personality

Although it was claimed that he was once engaged, Newton never married. The French writer and philosopher Voltaire, who was in London at the time of Newton's funeral, said that he "was never sensible to any passion, was not subject to the common frailties of mankind, nor had any commerce with women-a circumstance which was assured me by the physician and surgeon who attended him in his last moments". There exists a widespread belief that Newton died a virgin, and writers as diverse as mathematician Charles Hutton, economist John Maynard Keynes, and physicist Carl Sagan have commented on it.

Newton had a close friendship with the Swiss mathematician Nicolas Fatio de Duillier, whom he met in London around 1689-some of their correspondence has survived. Their relationship came to an abrupt and unexplained end in 1693, and at the same time Newton suffered a nervous breakdown, which included sending wild accusatory letters to his friends Samuel Pepys and John Locke. His note to the latter included the charge that Locke "endeavoured to embroil me with woemen".

Newton was relatively modest about his achievements, writing in a letter to Robert Hooke in February 1676, "If I have seen further it is by standing on the shoulders of giants." Two writers think that the sentence, written at a time when Newton and Hooke were in dispute over optical discoveries, was an oblique attack on Hooke (said to have been short and hunchbacked), rather than-or in addition to-a statement of modesty. On the other hand, the widely known proverb about standing on the shoulders of giants, published among others by seventeenth-century poet George Herbert (a former orator of the University of Cambridge and fellow of Trinity College) in his Jacula Prudentum (1651), had as its main point that "a dwarf on a giant's shoulders sees farther of the two", and so its effect as an analogy would place Newton himself rather than Hooke as the 'dwarf'.

In a later memoir, Newton wrote, "I do not know what I may appear to the world, but to myself I seem to have been only like a boy playing on the sea-shore, and diverting myself in now and then finding a smoother pebble or a prettier shell than ordinary, whilst the great ocean of truth lay all undiscovered before me."

Religious views

Although born into an Anglican family, by his thirties Newton held a Christian faith that, had it been made public, would not have been considered orthodox by mainstream Christianity, with one historian labelling him a heretic.

By 1672, he had started to record his theological researches in notebooks which he showed to no one and which have only recently[when? ] been examined. They demonstrate an extensive knowledge of early Church writings and show that in the conflict between Athanasius and Arius which defined the Creed, he took the side of Arius, the loser, who rejected the conventional view of the Trinity. Newton "recognized Christ as a divine mediator between God and man, who was subordinate to the Father who created him." He was especially interested in prophecy, but for him, "the great apostasy was trinitarianism."

Newton tried unsuccessfully to obtain one of the two fellowships that exempted the holder from the ordination requirement. At the last moment in 1675 he received a dispensation from the government that excused him and all future holders of the Lucasian chair.

In Newton's eyes, worshipping Christ as God was idolatry, to him the fundamental sin. In 1999, historian Stephen D. Snobelen wrote, "Isaac Newton was a heretic. But ... he never made a public declaration of his private faith-which the orthodox would have deemed extremely radical. He hid his faith so well that scholars are still unraveling his personal beliefs." Snobelen concludes that Newton was at least a Socinian sympathiser (he owned and had thoroughly read at least eight Socinian books), possibly an Arian and almost certainly an anti-trinitarian.

Newton (1795, detail) by William Blake. Newton is depicted critically as a "divine geometer".[132]

Although the laws of motion and universal gravitation became Newton's best-known discoveries, he warned against using them to view the Universe as a mere machine, as if akin to a great clock. He said, "So then gravity may put the planets into motion, but without the Divine Power it could never put them into such a circulating motion, as they have about the sun".

Along with his scientific fame, Newton's studies of the Bible and of the early Church Fathers were also noteworthy. Newton wrote works on textual criticism, most notably An Historical Account of Two Notable Corruptions of Scripture and Observations upon the Prophecies of Daniel, and the Apocalypse of St. John. He placed the crucifixion of Jesus Christ at 3 April, AD 33, which agrees with one traditionally accepted date.

He believed in a rationally immanent world, but he rejected the hylozoism implicit in Leibniz and Baruch Spinoza. The ordered and dynamically informed Universe could be understood, and must be understood, by an active reason. In his correspondence, Newton claimed that in writing the Principia "I had an eye upon such Principles as might work with considering men for the belief of a Deity". He saw evidence of design in the system of the world: "Such a wonderful uniformity in the planetary system must be allowed the effect of choice". But Newton insisted that divine intervention would eventually be required to reform the system, due to the slow growth of instabilities. For this, Leibniz lampooned him: "God Almighty wants to wind up his watch from time to time: otherwise it would cease to move. He had not, it seems, sufficient foresight to make it a perpetual motion."

Newton's position was vigorously defended by his follower Samuel Clarke in a famous correspondence. A century later, Pierre-Simon Laplace's work Celestial Mechanics had a natural explanation for why the planet orbits do not require periodic divine intervention. The contrast between Laplace's mechanistic worldview and Newton's one is the most strident considering the famous answer which the French scientist gave Napoleon, who had criticised him for the absence of the Creator in the Mécanique céleste: "Sire, j'ai pu me passer de cette hypothèse" ("Sir, I didn't need this hypothesis").

Scholars long debated whether Newton disputed the doctrine of the Trinity. His first biographer, David Brewster, who compiled his manuscripts, interpreted Newton as questioning the veracity of some passages used to support the Trinity, but never denying the doctrine of the Trinity as such. In the twentieth century, encrypted manuscripts written by Newton and bought by John Maynard Keynes (among others) were deciphered and it became known that Newton did indeed reject Trinitarianism.

Religious thought

Newton and Robert Boyle's approach to the mechanical philosophy was promoted by rationalist pamphleteers as a viable alternative to the pantheists and enthusiasts, and was accepted hesitantly by orthodox preachers as well as dissident preachers like the latitudinarians. The clarity and simplicity of science was seen as a way to combat the emotional and metaphysical superlatives of both superstitious enthusiasm and the threat of atheism, and at the same time, the second wave of English deists used Newton's discoveries to demonstrate the possibility of a "Natural Religion".

The attacks made against pre- Enlightenment " magical thinking", and the mystical elements of Christianity, were given their foundation with Boyle's mechanical conception of the universe. Newton gave Boyle's ideas their completion through mathematical proofs and, perhaps more importantly, was very successful in popularising them.

Newton was not the first of the age of reason. He was the last of the magicians, the last of the Babylonians and Sumerians, the last great mind which looked out on the visible and intellectual world with the same eyes as those who began to build our intellectual inheritance rather less than 10,000 years ago. Isaac Newton, a posthumous child born with no father on Christmas Day, 1642, was the last wonderchild to whom the Magi could do sincere and appropriate homage.

Of an estimated ten million words of writing in Newton's papers, about one million deal with alchemy. Many of Newton's writings on alchemy are copies of other manuscripts, with his own annotations. Alchemical texts mix artisanal knowledge with philosophical speculation, often hidden behind layers of wordplay, allegory, and imagery to protect craft secrets. Some of the content contained in Newton's papers could have been considered heretical by the church.

In 1888, after spending sixteen years cataloguing Newton's papers, Cambridge University kept a small number and returned the rest to the Earl of Portsmouth. In 1936, a descendant offered the papers for sale at Sotheby's. The collection was broken up and sold for a total of about £9,000. John Maynard Keynes was one of about three dozen bidders who obtained part of the collection at auction. Keynes went on to reassemble an estimated half of Newton's collection of papers on alchemy before donating his collection to Cambridge University in 1946.

All of Newton's known writings on alchemy are currently being put online in a project undertaken by Indiana University: "The Chymistry of Isaac Newton" and summarised in a book.

Newton's fundamental contributions to science include the quantification of gravitational attraction, the discovery that white light is actually a mixture of immutable spectral colors, and the formulation of the calculus. Yet there is another, more mysterious side to Newton that is imperfectly known, a realm of activity that spanned some thirty years of his life, although he kept it largely hidden from his contemporaries and colleagues. We refer to Newton's involvement in the discipline of alchemy, or as it was often called in seventeenth-century England, "chymistry."

In June 2020, two unpublished pages of Newton's notes on Jan Baptist van Helmont's book on plague, De Peste, were being auctioned online by Bonhams. Newton's analysis of this book, which he made in Cambridge while protecting himself from London's 1665-1666 infection, is the most substantial written statement he is known to have made about the plague, according to Bonhams. As far as the therapy is concerned, Newton writes that "the best is a toad suspended by the legs in a chimney for three days, which at last vomited up earth with various insects in it, on to a dish of yellow wax, and shortly after died. Combining powdered toad with the excretions and serum made into lozenges and worn about the affected area drove away the contagion and drew out the poison".

Newton's tomb monument in Westminster Abbey by John Michael Rysbrack

The mathematician Joseph-Louis Lagrange said that Newton was the greatest genius who ever lived, and once added that Newton was also "the most fortunate, for we cannot find more than once a system of the world to establish." English poet Alexander Pope wrote the famous epitaph:

Nature, and Nature's laws lay hid in night.
God said, Let Newton be! and all was light.

But this was not allowed to be inscribed in the monument. The epitaph in the monument is as follows:

H. S. E. ISAACUS NEWTON Eques Auratus, / Qui, animi vi prope divinâ, / Planetarum Motus, Figuras, / Cometarum semitas, Oceanique Aestus. Suâ Mathesi facem praeferente / Primus demonstravit: / Radiorum Lucis dissimilitudines, / Colorumque inde nascentium proprietates, / Quas nemo antea vel suspicatus erat, pervestigavit. / Naturae, Antiquitatis, S. Scripturae, / Sedulus, sagax, fidus Interpres / Dei O. M. Majestatem Philosophiâ asseruit, / Evangelij Simplicitatem Moribus expressit. / Sibi gratulentur Mortales, / Tale tantumque exstitisse / HUMANI GENERIS DECUS. / NAT. XXV DEC. A.D. MDCXLII. OBIIT. XX. MAR. MDCCXXVI,

which can be translated as follows:

Here is buried Isaac Newton, Knight, who by a strength of mind almost divine, and mathematical principles peculiarly his own, explored the course and figures of the planets, the paths of comets, the tides of the sea, the dissimilarities in rays of light, and, what no other scholar has previously imagined, the properties of the colours thus produced. Diligent, sagacious and faithful, in his expositions of nature, antiquity and the holy Scriptures, he vindicated by his philosophy the majesty of God mighty and good, and expressed the simplicity of the Gospel in his manners. Mortals rejoice that there has existed such and so great an ornament of the human race! He was born on 25th December 1642, and died on 20th March 1726.

In a 2005 survey of members of Britain's Royal Society (formerly headed by Newton) asking who had the greater effect on the history of science, Newton or Albert Einstein, the members deemed Newton to have made the greater overall contribution. In 1999, an opinion poll of 100 of the day's leading physicists voted Einstein the "greatest physicist ever," with Newton the runner-up, while a parallel survey of rank-and-file physicists by the site PhysicsWeb gave the top spot to Newton. New Scientist called Newton "the supreme genius and most enigmatic character in the history of science". Einstein kept a picture of Newton on his study wall alongside ones of Michael Faraday and James Clerk Maxwell.

The SI derived unit of force is named the newton in his honour.

Woolsthorpe Manor is a Grade I listed building by Historic England through being his birthplace and "where he discovered gravity and developed his theories regarding the refraction of light".

In 1816, a tooth said to have belonged to Newton was sold for £730 in London to an aristocrat who had it set in a ring. Guinness World Records 2002 classified it as the most valuable tooth in the world, which would value approximately £25,000 (US$35,700) in late 2001. Who bought it and who currently has it has not been disclosed.

Apple incident

Newton himself often told the story that he was inspired to formulate his theory of gravitation by watching the fall of an apple from a tree. The story is believed to have passed into popular knowledge after being related by Catherine Barton, Newton's niece, to Voltaire. Voltaire then wrote in his Essay on Epic Poetry (1727), "Sir Isaac Newton walking in his gardens, had the first thought of his system of gravitation, upon seeing an apple falling from a tree."

Although it has been said that the apple story is a myth and that he did not arrive at his theory of gravity at any single moment, acquaintances of Newton (such as William Stukeley, whose manuscript account of 1752 has been made available by the Royal Society) do in fact confirm the incident, though not the apocryphal version that the apple actually hit Newton's head. Stukeley recorded in his Memoirs of Sir Isaac Newton's Life a conversation with Newton in Kensington on 15 April 1726:

we went into the garden, & drank thea under the shade of some appletrees, only he, & myself. amidst other discourse, he told me, he was just in the same situation, as when formerly, the notion of gravitation came into his mind. "why should that apple always descend perpendicularly to the ground," thought he to him self: occasion'd by the fall of an apple, as he sat in a comtemplative mood: "why should it not go sideways, or upwards? but constantly to the earths centre? assuredly, the reason is, that the earth draws it. there must be a drawing power in matter. & the sum of the drawing power in the matter of the earth must be in the earths center, not in any side of the earth. therefore dos this apple fall perpendicularly, or toward the center. if matter thus draws matter; it must be in proportion of its quantity. therefore the apple draws the earth, as well as the earth draws the apple."

John Conduitt, Newton's assistant at the Royal Mint and husband of Newton's niece, also described the event when he wrote about Newton's life:

In the year 1666 he retired again from Cambridge to his mother in Lincolnshire. Whilst he was pensively meandering in a garden it came into his thought that the power of gravity (which brought an apple from a tree to the ground) was not limited to a certain distance from earth, but that this power must extend much further than was usually thought. Why not as high as the Moon said he to himself & if so, that must influence her motion & perhaps retain her in her orbit, whereupon he fell a calculating what would be the effect of that supposition.

A wood engraving of Newton's famous steps under the apple tree

It is known from his notebooks that Newton was grappling in the late 1660s with the idea that terrestrial gravity extends, in an inverse-square proportion, to the Moon; however, it took him two decades to develop the full-fledged theory. The question was not whether gravity existed, but whether it extended so far from Earth that it could also be the force holding the Moon to its orbit. Newton showed that if the force decreased as the inverse square of the distance, one could indeed calculate the Moon's orbital period, and get good agreement. He guessed the same force was responsible for other orbital motions, and hence named it "universal gravitation".

Various trees are claimed to be "the" apple tree which Newton describes. The King's School, Grantham claims that the tree was purchased by the school, uprooted and transported to the headmaster's garden some years later. The staff of the (now) National Trust-owned Woolsthorpe Manor dispute this, and claim that a tree present in their gardens is the one described by Newton. A descendant of the original tree can be seen growing outside the main gate of Trinity College, Cambridge, below the room Newton lived in when he studied there. The National Fruit Collection at Brogdale in Kent can supply grafts from their tree, which appears identical to Flower of Kent, a coarse-fleshed cooking variety.

Commemorations

Newton statue on display at the Oxford University Museum of Natural History

Newton's monument (1731) can be seen in Westminster Abbey, at the north of the entrance to the choir against the choir screen, near his tomb. It was executed by the sculptor Michael Rysbrack (1694-1770) in white and grey marble with design by the architect William Kent. The monument features a figure of Newton reclining on top of a sarcophagus, his right elbow resting on several of his great books and his left hand pointing to a scroll with a mathematical design. Above him is a pyramid and a celestial globe showing the signs of the Zodiac and the path of the comet of 1680. A relief panel depicts putti using instruments such as a telescope and prism. The Latin inscription on the base translates as:

Here is buried Isaac Newton, Knight, who by a strength of mind almost divine, and mathematical principles peculiarly his own, explored the course and figures of the planets, the paths of comets, the tides of the sea, the dissimilarities in rays of light, and, what no other scholar has previously imagined, the properties of the colours thus produced. Diligent, sagacious and faithful, in his expositions of nature, antiquity and the holy Scriptures, he vindicated by his philosophy the majesty of God mighty and good, and expressed the simplicity of the Gospel in his manners. Mortals rejoice that there has existed such and so great an ornament of the human race! He was born on 25 December 1642, and died on 20 March 1726/7.

-Translation from G. L. Smyth, The Monuments and Genii of St. Paul's Cathedral, and of Westminster Abbey (1826), ii, 703-704.

From 1978 until 1988, an image of Newton designed by Harry Ecclestone appeared on Series D £1 banknotes issued by the Bank of England (the last £1 notes to be issued by the Bank of England). Newton was shown on the reverse of the notes holding a book and accompanied by a telescope, a prism and a map of the Solar System.

A statue of Isaac Newton, looking at an apple at his feet, can be seen at the Oxford University Museum of Natural History. A large bronze statue, Newton, after William Blake, by Eduardo Paolozzi, dated 1995 and inspired by Blake's etching, dominates the piazza of the British Library in London. A bronze statue of Newton was erected in 1858 in the centre of Grantham where he went to school, prominently standing in front of Grantham Guildhall.

The still-surviving farmhouse at Woolsthorpe By Colsterworth is a Grade I listed building by Historic England through being his birthplace and "where he discovered gravity and developed his theories regarding the refraction of light".

The Enlightenment

Enlightenment philosophers chose a short history of scientific predecessors-Galileo, Boyle, and Newton principally-as the guides and guarantors of their applications of the singular concept of nature and natural law to every physical and social field of the day. In this respect, the lessons of history and the social structures built upon it could be discarded.

It is held by European philosophers of the Enlightenment and by historians of the Enlightenment that Newton's publication of the Principia was a turning point in the Scientific Revolution and started the Enlightenment. It was Newton's conception of the universe based upon natural and rationally understandable laws that became one of the seeds for Enlightenment ideology. Locke and Voltaire applied concepts of natural law to political systems advocating intrinsic rights; the physiocrats and Adam Smith applied natural conceptions of psychology and self-interest to economic systems; and sociologists criticised the current social order for trying to fit history into natural models of progress. Monboddo and Samuel Clarke resisted elements of Newton's work, but eventually rationalised it to conform with their strong religious views of nature.

Published in his lifetime

Published posthumously

References

Bibliography

Further reading

Newton, Isaac. The Principia: Mathematical Principles of Natural Philosophy. University of California Press, (1999)
Brackenridge, J. Bruce. The Key to Newton's Dynamics: The Kepler Problem and the Principia: Containing an English Translation of Sections 1, 2, and 3 of Book One from the First (1687) Edition of Newton's Mathematical Principles of Natural Philosophy, University of California Press (1996)

Newton, Isaac. The Optical Papers of Isaac Newton. Vol. 1: The Optical Lectures, 1670-1672, Cambridge University Press (1984)

Newton, Isaac. Opticks (4th ed. 1730) online edition

Newton, I. (1952). Opticks, or A Treatise of the Reflections, Refractions, Inflections & Colours of Light. New York: Dover Publications.

Newton, I. Sir Isaac Newton's Mathematical Principles of Natural Philosophy and His System of the World, tr. A. Motte, rev. Florian Cajori. Berkeley: University of California Press (1934)

Newton, Isaac. The correspondence of Isaac Newton, ed. H.W. Turnbull and others, 7 vols (1959-77)

Newton's Philosophy of Nature: Selections from His Writings edited by H.S. Thayer (1953; online edition)

Isaac Newton, Sir; J Edleston; Roger Cotes, Correspondence of Sir Isaac Newton and Professor Cotes, including letters of other eminent men, London, John W. Parker, West Strand; Cambridge, John Deighton (1850, Google Books)

Maclaurin, C. (1748). An Account of Sir Isaac Newton's Philosophical Discoveries, in Four Books. London: A. Millar and J. Nourse

Newton, I. (1958). Isaac Newton's Papers and Letters on Natural Philosophy and Related Documents, eds. I.B. Cohen and R.E. Schofield. Cambridge: Harvard University Press

Newton, I. (1962). The Unpublished Scientific Papers of Isaac Newton: A Selection from the Portsmouth Collection in the University Library, Cambridge, ed. A.R. Hall and M.B. Hall. Cambridge: Cambridge University Press

Newton, I. (1975). Isaac Newton's 'Theory of the Moon's Motion' (1702). London: Dawson

Craig, John (1946). Newton at the Mint. Cambridge, England: Cambridge University Press.

Craig, John (1953). "XII. Isaac Newton". The Mint: A History of the London Mint from A.D. 287 to 1948. Cambridge, England: Cambridge University Press. pp. 198-222. ASIN B0000CIHG7.

de Villamil, Richard (1931). Newton, the Man. London: G. D. Knox. - Preface by Albert Einstein. Reprinted by Johnson Reprint Corporation, New York (1972)

Dobbs, B. J. T. (1975). The Foundations of Newton's Alchemy or "The Hunting of the Greene Lyon". Cambridge: Cambridge University Press.

Keynes, John Maynard (1963). Essays in Biography. W. W. Norton & Co. ISBN 978-0-393-00189-1. Keynes took a close interest in Newton and owned many of Newton's private papers.

Stukeley, W. (1936). Memoirs of Sir Isaac Newton's Life. London: Taylor and Francis. (edited by A.H. White; originally published in 1752)

Trabue, J. "Ann and Arthur Storer of Calvert County, Maryland, Friends of Sir Isaac Newton," The American Genealogist 79 (2004): 13-27.

Dobbs, Betty Jo Tetter. The Janus Faces of Genius: The Role of Alchemy in Newton's Thought. (1991), links the alchemy to Arianism

Force, James E., and Richard H. Popkin, eds. Newton and Religion: Context, Nature, and Influence. (1999), pp. xvii, 325.; 13 papers by scholars using newly opened manuscripts

Snobelen, Stephen D. (2001). "'God of Gods, and Lord of Lords': The Theology of Isaac Newton's General Scholium to the Principia". Osiris. 16: 169-208. Bibcode:2001Osir...16..169S. doi:10.1086/649344. JSTOR 301985. S2CID 170364912.

Berlinski, David. Newton's Gift: How Sir Isaac Newton Unlocked the System of the World. (2000); ISBN 0-684-84392-7

Cohen, I. Bernard and Smith, George E., ed. The Cambridge Companion to Newton. (2002). Focuses on philosophical issues only; excerpt and text search; complete edition online "The Cambridge Companion to Newton". Archived from the original on 8 October 2008.

Halley, E. (1687). "Review of Newton's Principia". Philosophical Transactions. 186: 291-97.

Hawking, Stephen, ed. On the Shoulders of Giants. ISBN 0-7624-1348-4 Places selections from Newton's Principia in the context of selected writings by Copernicus, Kepler, Galileo and Einstein

Herivel, J. W. (1965). The Background to Newton's Principia. A Study of Newton's Dynamical Researches in the Years 1664-84. Oxford: Clarendon Press.

Newton, Isaac. Papers and Letters in Natural Philosophy, edited by I. Bernard Cohen. Harvard University Press, 1958, 1978; ISBN 0-674-46853-8.

Pemberton, H. (1728). "A View of Sir Isaac Newton's Philosophy". The Physics Teacher. 4 (1): 8-9. Bibcode:1966PhTea...4....8M. doi:10.1119/1.2350900.

Shamos, Morris H. (1959). Great Experiments in Physics. New York: Henry Holt and Company, Inc. ISBN 978-0-486-25346-6.

External links

Writings by Newton

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Sir Isaac Newton (25 December 1642 - 20 March 1726/27) was an English polymath active as a mathematician, physicist, astronomer, alchemist, theologian, and author who was described in his time as a natural philosopher. He was a key figure in the Scientific Revolution and the Enlightenment that followed.

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Newton's laws of motion are three basic laws of classical mechanics that describe the relationship between the motion of an object and the forces acting on it. These laws can be paraphrased as follows:

1. A body remains at rest, or in motion at a constant speed in a straight line, unless acted upon by a force.

2. When a body is acted upon by a force, the time rate of change of its momentum equals the force.

3. If two bodies exert forces on each other, these forces have the same magnitude but opposite directions.

The three laws of motion were first stated by Isaac Newton in his Philosophiæ Naturalis Principia Mathematica ( Mathematical Principles of Natural Philosophy), originally published in 1687. Newton used them to investigate and explain the motion of many physical objects and systems, which laid the foundation for classical mechanics. In the time since Newton, the conceptual content of classical physics has been reformulated in alternative ways, involving different mathematical approaches that have yielded insights which were obscured in the original, Newtonian formulation. Limitations to Newton's laws have also been discovered; new theories are necessary when objects move at very high speeds ( special relativity), are very massive ( general relativity), or are very small ( quantum mechanics).

Prerequisites

Newton's laws are often stated in terms of point or particle masses, that is, bodies whose volume is negligible. This is a reasonable approximation for real bodies when the motion of internal parts can be neglected, and when the separation between bodies is much larger than the size of each. For instance, the Earth and the Sun can both be approximated as pointlike when considering the orbit of the former around the latter, but the Earth is not pointlike when considering activities on its surface.

The mathematical description of motion, or kinematics, is based on the idea of specifying positions using numerical coordinates. Movement is represented by these numbers changing over time: a body's trajectory is represented by a function that assigns to each value of a time variable the values of all the position coordinates. The simplest case is one-dimensional, that is, when a body is constrained to move only along a straight line. Its position can then be given by a single number, indicating where it is relative to some chosen reference point. For example, a body might be free to slide along a track that runs left to right, and so its location can be specified by its distance from a convenient zero point, or origin, with negative numbers indicating positions to the left and positive numbers indicating positions to the right. If the body's location as a function of time is , then its average velocity over the time interval from to is[6]

Here, the Greek letter ( delta) is used, per tradition, to mean "change in". A positive average velocity means that the position coordinate increases over the interval in question, a negative average velocity indicates a net decrease over that interval, and an average velocity of zero means that the body ends the time interval in the same place as it began. Calculus gives the means to define an instantaneous velocity, a measure of a body's speed and direction of movement at a single moment of time, rather than over an interval. One notation for the instantaneous velocity is to replace with the symbol , for example,

This denotes that the instantaneous velocity is the derivative of the position with respect to time. It can roughly be thought of as the ratio between an infinitesimally small change in position to the infinitesimally small time interval over which it occurs.[7] More carefully, the velocity and all other derivatives can be defined using the concept of a limit.[6] A function has a limit of at a given input value if the difference between and can be made arbitrarily small by choosing an input sufficiently close to . One writes,

Instantaneous velocity can be defined as the limit of the average velocity as the time interval shrinks to zero:

Acceleration is to velocity as velocity is to position: it is the derivative of the velocity with respect to time. Acceleration can likewise be defined as a limit:

Consequently, the acceleration is the second derivative of position,[7] often written .

Position, when thought of as a displacement from an origin point, is a vector: a quantity with both magnitude and direction.[9]: 1  Velocity and acceleration are vector quantities as well. The mathematical tools of vector algebra provide the means to describe motion in two, three or more dimensions. Vectors are often denoted with an arrow, as in , or in bold typeface, such as . Often, vectors are represented visually as arrows, with the direction of the vector being the direction of the arrow, and the magnitude of the vector indicated by the length of the arrow. Numerically, a vector can be represented as a list; for example, a body's velocity vector might be , indicating that it is moving at 3 metres per second along a horizontal axis and 4 metres per second along the vertical axis. The same motion described in a different coordinate system will be represented by different numbers, and vector algebra can be used to translate between these alternatives.[9]: 4

The physics concept of force makes quantitative the everyday idea of a push or a pull. Forces in Newtonian mechanics are often due to strings and ropes, friction, muscle effort, gravity, and so forth. Like displacement, velocity, and acceleration, force is a vector quantity.

Translated from the Latin, Newton's first law reads,

Every body continues in its state of rest, or of uniform motion in a straight line, unless it is compelled to change that state by forces impressed upon it.

Newton's first law expresses the principle of inertia: the natural behavior of a body is to move in a straight line at constant speed. In the absence of outside influences, a body's motion preserves the status quo.

The modern understanding of Newton's first law is that no inertial observer is privileged over any other. The concept of an inertial observer makes quantitative the everyday idea of feeling no effects of motion. For example, a person standing on the ground watching a train go past is an inertial observer. If the observer on the ground sees the train moving smoothly in a straight line at a constant speed, then a passenger sitting on the train will also be an inertial observer: the train passenger feels no motion. The principle expressed by Newton's first law is that there is no way to say which inertial observer is "really" moving and which is "really" standing still. One observer's state of rest is another observer's state of uniform motion in a straight line, and no experiment can deem either point of view to be correct or incorrect. There is no absolute standard of rest.

The change of motion of an object is proportional to the force impressed; and is made in the direction of the straight line in which the force is impressed.

By "motion", Newton meant the quantity now called momentum, which depends upon the amount of matter contained in a body, the speed at which that body is moving, and the direction in which it is moving. In modern notation, the momentum of a body is the product of its mass and its velocity:

Newton's second law, in modern form, states that the time derivative of the momentum is the force:

If the mass does not change with time, then the derivative acts only upon the velocity, and so the force equals the product of the mass and the time derivative of the velocity, which is the acceleration:

As the acceleration is the second derivative of position with respect to time, this can also be written

The forces acting on a body add as vectors, and so the total force on a body depends upon both the magnitudes and the directions of the individual forces. When the net force on a body is equal to zero, then by Newton's second law, the body does not accelerate, and it is said to be in mechanical equilibrium. A state of mechanical equilibrium is stable if, when the position of the body is changed slightly, the body remains near that equilibrium. Otherwise, the equilibrium is unstable.

A common visual representation of forces acting in concert is the free body diagram, which schematically portrays a body of interest and the forces applied to it by outside influences. For example, a free body diagram of a block sitting upon an inclined plane can illustrate the combination of gravitational force, "normal" force, friction, and string tension.

Newton's second law is sometimes presented as a definition of force, i.e., a force is that which exists when an inertial observer sees a body accelerating. In order for this to be more than a tautology - acceleration implies force, force implies acceleration - some other statement about force must also be made. For example, an equation detailing the force might be specified, like Newton's law of universal gravitation. By inserting such an expression for into Newton's second law, an equation with predictive power can be written.[note 6] Newton's second law has also been regarded as setting out a research program for physics, establishing that important goals of the subject are to identify the forces present in nature and to catalogue the constituents of matter.[note 7]

To every action, there is always opposed an equal reaction; or, the mutual actions of two bodies upon each other are always equal, and directed to contrary parts.

Overly brief paraphrases of the third law, like "action equals reaction" might have caused confusion among generations of students: the "action" and "reaction" apply to different bodies. For example, consider a book at rest on a table. The Earth's gravity pulls down upon the book. The "reaction" to that "action" is not the support force from the table holding up the book, but the gravitational pull of the book acting on the Earth.

Newton's third law relates to a more fundamental principle, the conservation of momentum. The latter remains true even in cases where Newton's statement does not, for instance when force fields as well as material bodies carry momentum, and when momentum is defined properly, in quantum mechanics as well.[note 9] In Newtonian mechanics, if two bodies have momenta and respectively, then the total momentum of the pair is , and the rate of change of is

By Newton's second law, the first term is the total force upon the first body, and the second term is the total force upon the second body. If the two bodies are isolated from outside influences, the only force upon the first body can be that from the second, and vice versa. By Newton's third law, these forces have equal magnitude but opposite direction, so they cancel when added, and is constant. Alternatively, if is known to be constant, it follows that the forces have equal magnitude and opposite direction.

Various sources have proposed elevating other ideas used in classical mechanics to the status of Newton's laws. For example, in Newtonian mechanics, the total mass of a body made by bringing together two smaller bodies is the sum of their individual masses. Frank Wilczek has suggested calling attention to this assumption by designating it "Newton's Zeroth Law". Another candidate for a "zeroth law" is the fact that at any instant, a body reacts to the forces applied to it at that instant. Likewise, the idea that forces add like vectors (or in other words obey the superposition principle), and the idea that forces change the energy of a body, have both been described as a "fourth law".

Work and energy

Physicists developed the concept of energy after Newton's time, but it has become an inseparable part of what is considered "Newtonian" physics. Energy can broadly be classified into kinetic, due to a body's motion, and potential, due to a body's position relative to others. Thermal energy, the energy carried by heat flow, is a type of kinetic energy not associated with the macroscopic motion of objects but instead with the movements of the atoms and molecules of which they are made. According to the work-energy theorem, when a force acts upon a body while that body moves along the line of the force, the force does work upon the body, and the amount of work done is equal to the change in the body's kinetic energy. In many cases of interest, the net work done by a force when a body moves in a closed loop - starting at a point, moving along some trajectory, and returning to the initial point - is zero. If this is the case, then the force can be written in terms of the gradient of a function called a scalar potential:

This is true for many forces including that of gravity, but not for friction; indeed, almost any problem in a mechanics textbook that does not involve friction can be expressed in this way. The fact that the force can be written in this way can be understood from the conservation of energy. Without friction to dissipate a body's energy into heat, the body's energy will trade between potential and (non-thermal) kinetic forms while the total amount remains constant. Any gain of kinetic energy, which occurs when the net force on the body accelerates it to a higher speed, must be accompanied by a loss of potential energy. So, the net force upon the body is determined by the manner in which the potential energy decreases.

Uniformly accelerated motion

If a body falls from rest near the surface of the Earth, then in the absence of air resistance, it will accelerate at a constant rate. This is known as free fall. The speed attained during free fall is proportional to the elapsed time, and the distance traveled is proportional to the square of the elapsed time. Importantly, the acceleration is the same for all bodies, independently of their mass. This follows from combining Newton's second law of motion with his law of universal gravitation. The latter states that the magnitude of the gravitational force from the Earth upon the body is

where is the mass of the falling body, is the mass of the Earth, is Newton's constant, and is the distance from the center of the Earth to the body's location, which is very nearly the radius of the Earth. Setting this equal to , the body's mass cancels from both sides of the equation, leaving an acceleration that depends upon , , and , and can be taken to be constant. This particular value of acceleration is typically denoted :

If the body is not released from rest but instead launched upwards and/or horizontally with nonzero velocity, then free fall becomes projectile motion.[37] When air resistance can be neglected, projectiles follow parabola-shaped trajectories, because gravity affects the body's vertical motion and not its horizontal. At the peak of the projectile's trajectory, its vertical velocity is zero, but its acceleration is downwards, as it is at all times. Setting the wrong vector equal to zero is a common confusion among physics students.[38]

Uniform circular motion

When a body is in uniform circular motion, the force on it changes the direction of its motion but not its speed. For a body moving in a circle of radius at a constant speed , its acceleration has a magnitude

and is directed toward the center of the circle.[note 12] The force required to sustain this acceleration, called the centripetal force, is therefore also directed toward the center of the circle and has magnitude . Many orbits, such as that of the Moon around the Earth, can be approximated by uniform circular motion. In such cases, the centripetal force is gravity, and by Newton's law of universal gravitation has magnitude , where is the mass of the larger body being orbited. Therefore, the mass of a body can be calculated from observations of another body orbiting around it.[39]: 130

Newton's cannonball is a thought experiment that interpolates between projectile motion and uniform circular motion. A cannonball that is lobbed weakly off the edge of a tall cliff will hit the ground in the same amount of time as if it were dropped from rest, because the force of gravity only affects the cannonball's momentum in the downward direction, and its effect is not diminished by horizontal movement. If the cannonball is launched with a greater initial horizontal velocity, then it will travel farther before it hits the ground, but it will still hit the ground in the same amount of time. However, if the cannonball is launched with an even larger initial velocity, then the curvature of the Earth becomes significant: the ground itself will curve away from the falling cannonball. A very fast cannonball will fall away from the inertial straight-line trajectory at the same rate that the Earth curves away beneath it; in other words, it will be in orbit (imagining that it is not slowed by air resistance or obstacles).

Harmonic motion

Consider a body of mass able to move along the axis, and suppose an equilibrium point exists at the position . That is, at , the net force upon the body is the zero vector, and by Newton's second law, the body will not accelerate. If the force upon the body is proportional to the displacement from the equilibrium point, and directed to the equilibrium point, then the body will perform simple harmonic motion. Writing the force as , Newton's second law becomes

This differential equation has the solution

where the frequency is equal to , and the constants and can be calculated knowing, for example, the position and velocity the body has at a given time, like .

One reason that the harmonic oscillator is a conceptually important example is that it is good approximation for many systems near a stable mechanical equilibrium. For example, a pendulum has a stable equilibrium in the vertical position: if motionless there, it will remain there, and if pushed slightly, it will swing back and forth. Neglecting air resistance and friction in the pivot, the force upon the pendulum is gravity, and Newton's second law becomes

where is the length of the pendulum and is its angle from the vertical. When the angle is small, the sine of is nearly equal to (see Taylor series), and so this expression simplifies to the equation for a simple harmonic oscillator with frequency .

A harmonic oscillator can be damped, often by friction or viscous drag, in which case energy bleeds out of the oscillator and the amplitude of the oscillations decreases over time. Also, a harmonic oscillator can be driven by an applied force, which can lead to the phenomenon of resonance.

Objects with variable mass

Newtonian physics treats matter as being neither created nor destroyed, though it may be rearranged. It can be the case that an object of interest gains or loses mass because matter is added to or removed from it. In such a situation, Newton's laws can be applied to the individual pieces of matter, keeping track of which pieces belong to the object of interest over time. For instance, if a rocket of mass , moving at velocity , ejects matter at a velocity relative to the rocket, then

where is the net external force (e.g., a planet's gravitational pull).[18]: 139

Rigid-body motion and rotation

A rigid body is an object whose size is too large to neglect and which maintains the same shape over time. In Newtonian mechanics, the motion of a rigid body is often understood by separating it into movement of the body's center of mass and movement around the center of mass.

Center of mass

Significant aspects of the motion of an extended body can be understood by imagining the mass of that body concentrated to a single point, known as the center of mass. The location of a body's center of mass depends upon how that body's material is distributed. For a collection of pointlike objects with masses at positions , the center of mass is located at

where is the total mass of the collection. In the absence of a net external force, the center of mass moves at a constant speed in a straight line. This applies, for example, to a collision between two bodies.[42] If the total external force is not zero, then the center of mass changes velocity as though it were a point body of mass . This follows from the fact that the internal forces within the collection, the forces that the objects exert upon each other, occur in balanced pairs by Newton's third law. In a system of two bodies with one much more massive than the other, the center of mass will approximately coincide with the location of the more massive body.[15]: 22-24

Rotational analogues of Newton's laws

When Newton's laws are applied to rotating extended bodies, they lead to new quantities that are analogous to those invoked in the original laws. The analogue of mass is the moment of inertia, the counterpart of momentum is angular momentum, and the counterpart of force is torque.

Angular momentum is calculated with respect to a reference point.[43] If the displacement vector from a reference point to a body is and the body has momentum , then the body's angular momentum with respect to that point is, using the vector cross product,

Taking the time derivative of the angular momentum gives

The first term vanishes because and point in the same direction. The remaining term is the torque,

When the torque is zero, the angular momentum is constant, just as when the force is zero, the momentum is constant.[15]: 14-15  The torque can vanish even when the force is non-zero, if the body is located at the reference point ( ) or if the force and the displacement vector are directed along the same line.

The angular momentum of a collection of point masses, and thus of an extended body, is found by adding the contributions from each of the points. This provides a means to characterize a body's rotation about an axis, by adding up the angular momenta of its individual pieces. The result depends on the chosen axis, the shape of the body, and the rate of rotation.

Multi-body gravitational system

Newton's law of universal gravitation states that any body attracts any other body along the straight line connecting them. The size of the attracting force is proportional to the product of their masses, and inversely proportional to the square of the distance between them. Finding the shape of the orbits that an inverse-square force law will produce is known as the Kepler problem. The Kepler problem can be solved in multiple ways, including by demonstrating that the Laplace-Runge-Lenz vector is constant, or by applying a duality transformation to a 2-dimensional harmonic oscillator. However it is solved, the result is that orbits will be conic sections, that is, ellipses (including circles), parabolas, or hyperbolas. The eccentricity of the orbit, and thus the type of conic section, is determined by the energy and the angular momentum of the orbiting body. Planets do not have sufficient energy to escape the Sun, and so their orbits are ellipses, to a good approximation; because the planets pull on one another, actual orbits are not exactly conic sections.

If a third mass is added, the Kepler problem becomes the three-body problem, which in general has no exact solution in closed form. That is, there is no way to start from the differential equations implied by Newton's laws and, after a finite sequence of standard mathematical operations, obtain equations that express the three bodies' motions over time. Numerical methods can be applied to obtain useful, albeit approximate, results for the three-body problem. The positions and velocities of the bodies can be stored in variables within a computer's memory; Newton's laws are used to calculate how the velocities will change over a short interval of time, and knowing the velocities, the changes of position over that time interval can be computed. This process is looped to calculate, approximately, the bodies' trajectories. Generally speaking, the shorter the time interval, the more accurate the approximation.

Chaos and unpredictability

Nonlinear dynamics

Newton's laws of motion allow the possibility of chaos. That is, qualitatively speaking, physical systems obeying Newton's laws can exhibit sensitive dependence upon their initial conditions: a slight change of the position or velocity of one part of a system can lead to the whole system behaving in a radically different way within a short time. Noteworthy examples include the three-body problem, the double pendulum, dynamical billiards, and the Fermi-Pasta-Ulam-Tsingou problem.

Newton's laws can be applied to fluids by considering a fluid as composed of infinitesimal pieces, each exerting forces upon neighboring pieces. The Euler momentum equation is an expression of Newton's second law adapted to fluid dynamics.[52][53] A fluid is described by a velocity field, i.e., a function that assigns a velocity vector to each point in space and time. A small object being carried along by the fluid flow can change velocity for two reasons: first, because the velocity field at its position is changing over time, and second, because it moves to a new location where the velocity field has a different value. Consequently, when Newton's second law is applied to an infinitesimal portion of fluid, the acceleration has two terms, a combination known as a total or material derivative. The mass of an infinitesimal portion depends upon the fluid density, and there is a net force upon it if the fluid pressure varies from one side of it to another. Accordingly, becomes

where is the density, is the pressure, and stands for an external influence like a gravitational pull. Incorporating the effect of viscosity turns the Euler equation into a Navier-Stokes equation:

where is the kinematic viscosity.[52]

Singularities

It is mathematically possible for a collection of point masses, moving in accord with Newton's laws, to launch some of themselves away so forcefully that they fly off to infinity in a finite time. This unphysical behavior, known as a "noncollision singularity", depends upon the masses being pointlike and able to approach one another arbitrarily closely, as well as the lack of a relativistic speed limit in Newtonian physics.

It is not yet known whether or not the Euler and Navier-Stokes equations exhibit the analogous behavior of initially smooth solutions "blowing up" in finite time. The question of existence and smoothness of Navier-Stokes solutions is one of the Millennium Prize Problems.

Relation to other formulations of classical physics

Classical mechanics can be mathematically formulated in multiple different ways, other than the "Newtonian" description (which itself, of course, incorporates contributions from others both before and after Newton). The physical content of these different formulations is the same as the Newtonian, but they provide different insights and facilitate different types of calculations. For example, Lagrangian mechanics helps make apparent the connection between symmetries and conservation laws, and it is useful when calculating the motion of constrained bodies, like a mass restricted to move along a curving track or on the surface of a sphere. Hamiltonian mechanics is convenient for statistical physics, leads to further insight about symmetry, and can be developed into sophisticated techniques for perturbation theory. Due to the breadth of these topics, the discussion here will be confined to concise treatments of how they reformulate Newton's laws of motion.

Lagrangian

Lagrangian mechanics differs from the Newtonian formulation by considering entire trajectories at once rather than predicting a body's motion at a single instant.[15]: 109  It is traditional in Lagrangian mechanics to denote position with and velocity with . The simplest example is a massive point particle, the Lagrangian for which can be written as the difference between its kinetic and potential energies:

where the kinetic energy is

and the potential energy is some function of the position, . The physical path that the particle will take between an initial point and a final point is the path for which the integral of the Lagrangian is "stationary". That is, the physical path has the property that small perturbations of it will, to a first approximation, not change the integral of the Lagrangian. Calculus of variations provides the mathematical tools for finding this path.[34]: 485  Applying the calculus of variations to the task of finding the path yields the Euler-Lagrange equation for the particle,

Evaluating the partial derivatives of the Lagrangian gives

which is a restatement of Newton's second law. The left-hand side is the time derivative of the momentum, and the right-hand side is the force, represented in terms of the potential energy.

Landau and Lifshitz argue that the Lagrangian formulation makes the conceptual content of classical mechanics more clear than starting with Newton's laws. Lagrangian mechanics provides a convenient framework in which to prove Noether's theorem, which relates symmetries and conservation laws. The conservation of momentum can be derived by applying Noether's theorem to a Lagrangian for a multi-particle system, and so, Newton's third law is a theorem rather than an assumption.

Hamiltonian

In Hamiltonian mechanics, the dynamics of a system are represented by a function called the Hamiltonian, which in many cases of interest is equal to the total energy of the system.[9]: 742  The Hamiltonian is a function of the positions and the momenta of all the bodies making up the system, and it may also depend explicitly upon time. The time derivatives of the position and momentum variables are given by partial derivatives of the Hamiltonian, via Hamilton's equations.[15]: 203  The simplest example is a point mass constrained to move in a straight line, under the effect of a potential. Writing for the position coordinate and for the body's momentum, the Hamiltonian is

In this example, Hamilton's equations are

and

Evaluating these partial derivatives, the former equation becomes

which reproduces the familiar statement that a body's momentum is the product of its mass and velocity. The time derivative of the momentum is

which, upon identifying the negative derivative of the potential with the force, is just Newton's second law once again.

As in the Lagrangian formulation, in Hamiltonian mechanics the conservation of momentum can be derived using Noether's theorem, making Newton's third law an idea that is deduced rather than assumed.

Among the proposals to reform the standard introductory-physics curriculum is one that teaches the concept of energy before that of force, essentially "introductory Hamiltonian mechanics".

Hamilton-Jacobi

The Hamilton-Jacobi equation provides yet another formulation of classical mechanics, one which makes it mathematically analogous to wave optics.[15]: 284  [62] This formulation also uses Hamiltonian functions, but in a different way than the formulation described above. The paths taken by bodies or collections of bodies are deduced from a function of positions and time . The Hamiltonian is incorporated into the Hamilton-Jacobi equation, a differential equation for . Bodies move over time in such a way that their trajectories are perpendicular to the surfaces of constant , analogously to how a light ray propagates in the direction perpendicular to its wavefront. This is simplest to express for the case of a single point mass, in which is a function , and the point mass moves in the direction along which changes most steeply. In other words, the momentum of the point mass is the gradient of :

The Hamilton-Jacobi equation for a point mass is

The relation to Newton's laws can be seen by considering a point mass moving in a time-independent potential , in which case the Hamilton-Jacobi equation becomes

Taking the gradient of both sides, this becomes

Interchanging the order of the partial derivatives on the left-hand side, and using the power and chain rules on the first term on the right-hand side,

Gathering together the terms that depend upon the gradient of ,

This is another re-expression of Newton's second law. The expression in brackets is a total or material derivative as mentioned above, in which the first term indicates how the function being differentiated changes over time at a fixed location, and the second term captures how a moving particle will see different values of that function as it travels from place to place:

Relation to other physical theories

Thermodynamics and statistical physics

In statistical physics, the kinetic theory of gases applies Newton's laws of motion to large numbers (typically on the order of the Avogadro number) of particles. Kinetic theory can explain, for example, the pressure that a gas exerts upon the container holding it as the aggregate of many impacts of atoms, each imparting a tiny amount of momentum.

The Langevin equation is a special case of Newton's second law, adapted for the case of describing a small object bombarded stochastically by even smaller ones. It can be written

where is a drag coefficient and is a force that varies randomly from instant to instant, representing the net effect of collisions with the surrounding particles. This is used to model Brownian motion.[66]

Electromagnetism

Newton's three laws can be applied to phenomena involving electricity and magnetism, though subtleties and caveats exist.

Coulomb's law for the electric force between two stationary, electrically charged bodies has much the same mathematical form as Newton's law of universal gravitation: the force is proportional to the product of the charges, inversely proportional to the square of the distance between them, and directed along the straight line between them. The Coulomb force that a charge exerts upon a charge is equal in magnitude to the force that exerts upon , and it points in the exact opposite direction. Coulomb's law is thus consistent with Newton's third law.[67]

Electromagnetism treats forces as produced by fields acting upon charges. The Lorentz force law provides an expression for the force upon a charged body that can be plugged into Newton's second law in order to calculate its acceleration.[68]: 85  According to the Lorentz force law, a charged body in an electric field experiences a force in the direction of that field, a force proportional to its charge and to the strength of the electric field. In addition, a moving charged body in a magnetic field experiences a force that is also proportional to its charge, in a direction perpendicular to both the field and the body's direction of motion. Using the vector cross product,

If the electric field vanishes ( ), then the force will be perpendicular to the charge's motion, just as in the case of uniform circular motion studied above, and the charge will circle (or more generally move in a helix) around the magnetic field lines at the cyclotron frequency .[65]: 222  Mass spectrometry works by applying electric and/or magnetic fields to moving charges and measuring the resulting acceleration, which by the Lorentz force law yields the mass-to-charge ratio.[69]

Collections of charged bodies do not always obey Newton's third law: there can be a change of one body's momentum without a compensatory change in the momentum of another. The discrepancy is accounted for by momentum carried by the electromagnetic field itself. The momentum per unit volume of the electromagnetic field is proportional to the Poynting vector.

There is subtle conceptual conflict between electromagnetism and Newton's first law: Maxwell's theory of electromagnetism predicts that electromagnetic waves will travel through empty space at a constant, definite speed. Thus, some inertial observers seemingly have a privileged status over the others, namely those who measure the speed of light and find it to be the value predicted by the Maxwell equations. In other words, light provides an absolute standard for speed, yet the principle of inertia holds that there should be no such standard. This tension is resolved in the theory of special relativity, which revises the notions of space and time in such a way that all inertial observers will agree upon the speed of light in vacuum.

Special relativity

In special relativity, the rule that Wilczek called "Newton's Zeroth Law" breaks down: the mass of a composite object is not merely the sum of the masses of the individual pieces.[74]: 33  Newton's first law, inertial motion, remains true. A form of Newton's second law, that force is the rate of change of momentum, also holds, as does the conservation of momentum. However, the definition of momentum is modified. Among the consequences of this is the fact that the more quickly a body moves, the harder it is to accelerate, and so, no matter how much force is applied, a body cannot be accelerated to the speed of light. Depending on the problem at hand, momentum in special relativity can be represented as a three-dimensional vector, , where is the body's rest mass and is the Lorentz factor, which depends upon the body's speed. Alternatively, momentum and force can be represented as four-vectors.[75]: 107

Newtonian mechanics is a good approximation to special relativity when the speeds involved are small compared to that of light.

General relativity

General relativity is theory of gravity that advances beyond that of Newton. In general relativity, gravitational force is reimagined as curvature of spacetime. A curved path like an orbit is not the result of a force deflecting a body from an ideal straight-line path, but rather the body's attempt to fall freely through a background that is itself curved by the presence of other masses. A remark by John Archibald Wheeler that has become proverbial among physicists summarizes the theory: "Spacetime tells matter how to move; matter tells spacetime how to curve." Wheeler himself thought of this reciprocal relationship as a modern, generalized form of Newton's third law. The relation between matter distribution and spacetime curvature is given by the Einstein field equations, which require tensor calculus to express.

The Newtonian theory of gravity is a good approximation to the predictions of general relativity when gravitational effects are weak and objects are moving slowly compared to the speed of light.

Quantum mechanics

Quantum mechanics is a theory of physics originally developed in order to understand microscopic phenomena: behavior at the scale of molecules, atoms or subatomic particles. Generally and loosely speaking, the smaller a system is, the more an adequate mathematical model will require understanding quantum effects. The conceptual underpinning of quantum physics is very different from that of classical physics. Instead of thinking about quantities like position, momentum, and energy as properties that an object has, one considers what result might appear when a measurement of a chosen type is performed. Quantum mechanics allows the physicist to calculate the probability that a chosen measurement will elicit a particular result. The expectation value for a measurement is the average of the possible results it might yield, weighted by their probabilities of occurrence.

The Ehrenfest theorem provides a connection between quantum expectation values and Newton's second law, a connection that is necessarily inexact, as quantum physics is fundamentally different from classical. In quantum physics, position and momentum are represented by mathematical entities known as Hermitian operators, and the Born rule is used to calculate the expectation values of a position measurement or a momentum measurement. These expectation values will generally change over time; that is, depending on the time at which (for example) a position measurement is performed, the probabilities for its different possible outcomes will vary. The Ehrenfest theorem says, roughly speaking, that the equations describing how these expectation values change over time have a form reminiscent of Newton's second law. However, the more pronounced quantum effects are in a given situation, the more difficult it is to derive meaningful conclusions from this resemblance.

The concepts invoked in Newton's laws of motion - mass, velocity, momentum, force - have predecessors in earlier work, and the content of Newtonian physics was further developed after Newton's time. Newton combined knowledge of celestial motions with the study of events on Earth and showed that one theory of mechanics could encompass both.

Antiquity and medieval background

The subject of physics is often traced back to Aristotle; however, the history of the concepts involved is obscured by multiple factors. An exact correspondence between Aristotelian and modern concepts is not simple to establish: Aristotle did not clearly distinguish what we would call speed and force, and he used the same term for density and viscosity; he conceived of motion as always through a medium, rather than through space. In addition, some concepts often termed "Aristotelian" might better be attributed to his followers and commentators upon him. These commentators found that Aristotelian physics had difficulty explaining projectile motion. Aristotle divided motion into two types: "natural" and "violent". The "natural" motion of terrestrial solid matter was to fall downwards, whereas a "violent" motion could push a body sideways. Moreover, in Aristotelian physics, a "violent" motion requires an immediate cause; separated from the cause of its "violent" motion, a body would revert to its "natural" behavior. Yet a javelin continues moving after it leaves the hand of its thrower. Aristotle concluded that the air around the javelin must be imparted with the ability to move the javelin forward. John Philoponus, a Byzantine Greek thinker active during the sixth century, found this absurd: the same medium, air, was somehow responsible both for sustaining motion and for impeding it. If Aristotle's idea were true, Philoponus said, armies would launch weapons by blowing upon them with bellows. Philoponus argued that setting a body into motion imparted a quality, impetus, that would be contained within the body itself. As long as its impetus was sustained, the body would continue to move. In the following centuries, versions of impetus theory were advanced by individuals including Nur ad-Din al-Bitruji, Avicenna, Abu'l-Barakāt al-Baghdādī, John Buridan, and Albert of Saxony. In retrospect, the idea of impetus can be seen as a forerunner of the modern concept of momentum. (The intuition that objects move according to some kind of impetus persists in many students of introductory physics.)

Inertia and the first law

The modern concept of inertia is credited to Galileo. Based on his experiments, Galileo concluded that the "natural" behavior of a moving body was to keep moving, until something else interfered with it. Galileo recognized that in projectile motion, the Earth's gravity affects vertical but not horizontal motion. However, Galileo's idea of inertia was not exactly the one that would be codified into Newton's first law. Galileo thought that a body moving a long distance inertially would follow the curve of the Earth. This idea was corrected by Isaac Beeckman, René Descartes, and Pierre Gassendi, who recognized that inertial motion should be motion in a straight line.

Force and the second law

Christiaan Huygens, in his Horologium Oscillatorium (1673), put forth the hypothesis that "By the action of gravity, whatever its sources, it happens that bodies are moved by a motion composed both of a uniform motion in one direction or another and of a motion downward due to gravity." Newton's second law generalized this hypothesis from gravity to all forces.

One important characteristic of Newtonian physics is that forces can act at a distance without requiring physical contact. For example, the Sun and the Earth pull on each other gravitationally, despite being separated by millions of kilometres. This contrasts with the idea, championed by Descartes among others, that the Sun's gravity held planets in orbit by swirling them in a vortex of transparent matter, . Newton considered aetherial explanations of force but ultimately rejected them. The study of magnetism by William Gilbert and others created a precedent for thinking of immaterial forces, and unable to find a quantitatively satisfactory explanation of his law of gravity in terms of an aetherial model, Newton eventually declared, " I feign no hypotheses": whether or not a model like Descartes's vortices could be found to underlie the Principia's theories of motion and gravity, the first grounds for judging them must be the successful predictions they made. And indeed, since Newton's time every attempt at such a model has failed.

Momentum conservation and the third law

Johannes Kepler suggested that gravitational attractions were reciprocal - that, for example, the Moon pulls on the Earth while the Earth pulls on the Moon - but he did not argue that such pairs are equal and opposite. In his Principles of Philosophy (1644), Descartes introduced the idea that during a collision between bodies, a "quantity of motion" remains unchanged. Descartes defined this quantity somewhat imprecisely by adding up the products of the speed and "size" of each body, where "size" for him incorporated both volume and surface area. Moreover, Descartes thought of the universe as a plenum, that is, filled with matter, so all motion required a body to displace a medium as it moved. During the 1650s, Huygens studied collisions between hard spheres and deduced a principle that is now identified as the conservation of momentum. Christopher Wren would later deduce the same rules for elastic collisions that Huygens had, and John Wallis would apply momentum conservation to study inelastic collisions. Newton cited the work of Huygens, Wren, and Wallis to support the validity of his third law.

Newton arrived at his set of three laws incrementally. In a 1684 manuscript written to Huygens, he listed four laws: the principle of inertia, the change of motion by force, a statement about relative motion that would today be called Galilean invariance, and the rule that interactions between bodies do not change the motion of their center of mass. In a later manuscript, Newton added a law of action and reaction, while saying that this law and the law regarding the center of mass implied one another. Newton probably settled on the presentation in the Principia, with three primary laws and then other statements reduced to corollaries, during 1685.

Newton expressed his second law by saying that the force on a body is proportional to its change of motion, or momentum. By the time he wrote the Principia, he had already developed calculus (which he called " the science of fluxions"), but in the Principia he made no explicit use of it, perhaps because he believed geometrical arguments in the tradition of Euclid to be more rigorous.[111]: 15  [112] Consequently, the Principia does not express acceleration as the second derivative of position, and so it does not give the second law as . This form of the second law was written (for the special case of constant force) at least as early as 1716, by Jakob Hermann; Leonhard Euler would employ it as a basic premise in the 1740s.[113] Euler pioneered the study of rigid bodies[114] and established the basic theory of fluid dynamics.[115] Pierre-Simon Laplace's five-volume Traité de mécanique céleste (1798-1825) forsook geometry and developed mechanics purely through algebraic expressions, while resolving questions that the Principia had left open, like a full theory of the tides.[116]

The concept of energy became a key part of Newtonian mechanics in the post-Newton period. Huygens' solution of the collision of hard spheres showed that in that case, not only is momentum conserved, but kinetic energy is as well (or, rather, a quantity that in retrospect we can identify as one-half the total kinetic energy). The question of what is conserved during all other processes, like inelastic collisions and motion slowed by friction, was not resolved until the 19th century. Debates on this topic overlapped with philosophical disputes between the metaphysical views of Newton and Leibniz, and variants of the term "force" were sometimes used to denote what we would call types of energy. For example, in 1742, Émilie du Châtelet wrote, "Dead force consists of a simple tendency to motion: such is that of a spring ready to relax; living force is that which a body has when it is in actual motion." In modern terminology, "dead force" and "living force" correspond to potential energy and kinetic energy respectively. Conservation of energy was not established as a universal principle until it was understood that the energy of mechanical work can be dissipated into heat. With the concept of energy given a solid grounding, Newton's laws could then be derived within formulations of classical mechanics that put energy first, as in the Lagrangian and Hamiltonian formulations described above.

Modern presentations of Newton's laws use the mathematics of vectors, a topic that was not developed until the late 19th and early 20th centuries. Vector algebra, pioneered by Josiah Willard Gibbs and Oliver Heaviside, stemmed from and largely supplanted the earlier system of quaternions invented by William Rowan Hamilton.

References

Further reading

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Newton's laws of motion are three basic laws of classical mechanics that describe the relationship between the motion of an object and the forces acting on it. These laws can be paraphrased as follows: A body remains at rest, or in motion at a constant speed in a straight line, unless acted upon by a force.

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Mars One aims to establish a permanent human settlement on Mars.

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Mars One's Journey (2011 - 2021)

Humanity has explored Mars since 1960, and we've never been closer. The next logical step for Mars exploration is permanent settlement, where crews that go to Mars stay and build a new society. Humans settling on Mars will inspire us all to make Earth a better place. Technology, scientific understanding, cooperation - is there anything we cannot achieve?

Mars One was founded in 2011 with the goal of sending humans to Mars to stay. Instead of trying to return them, more crews would be sent every two years, establishing a permanent, ever-growing settlement on the Red Planet.

Permanent settlement missions place even higher demands on the crew than a return mission. A first permanent settlement crew would be on Mars for two years before the second crew joins them. They would be able to communicate with friends and family on Earth, but only with time delays. Crew selection, training and testing on Earth would be necessary to make sure they can deal with this. Besides that, they would need to learn all the skills to survive on Mars without support from Earth, other than information. They would need to be able to fix every technical and medical problem, grow food and expand the settlement with hardware for upcoming crews. Crew selection is the biggest challenge of a permanent settlement mission to Mars.

Mars One had a business plan for such a mission to Mars, which can be summarized as monetizing the media value of the adventure of humans going to Mars. The Olympic Games are worth more than $4 billion dollars in broadcasting rights and sponsorships. The 1969 Moon landing is still the TV program with the highest viewership density ever. Imagine the value of a mission to Mars in the current media era. Mars One raised funds by having investors invest in a Mars One media company that held the rights to the mission.

In 2011, Mars One met with aerospace companies around the world to discuss the plan, including SpaceX, Hamilton Sundstrand, ILC Dover, MDA corporation, Paragon Space Development Corporation and Thales Alenia Space. With their feedback, Mars One developed a plan that was feasible, affordable and that had an acceptable risk. In 2012, the plan was announced, later followed by a job vacancy for Mars settlers. More than 200,000 people registered their interest on the Mars One website. More than 10,000 potentials finished the job application process, and Mars One narrowed down the applicants, first to 1058 round 2 candidates and then to 100 round three candidates.

In 2013 and 2014, Mars One had feasibility studies performed by Lockheed Martin, for a first robotic mission and with Paragon Space Development Corporation, for Mars suits and life support systems.

In 2016 Mars One ran out of funds and was unable to continue the selection program and the technical studies. In the years after that, several attempts were made to raise additional funds, but they were unsuccessful.

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Humanity has explored Mars since 1960, and we've never been closer. The next logical step for Mars exploration is permanent settlement, where crews that go to Mars stay and build a new society. Humans settling on Mars will inspire us all to make Earth a better place.