Cotes's spiral

In physics and in the mathematics of plane curves, a Cotes's spiral (also written Cotes' spiral and Cotes spiral) is one of a family of spirals classified by Roger Cotes.

Description

Cotes introduces his analysis of these curves as follows: “It is proposed to list the different types of trajectories which bodies can move along when acted on by centripetal forces in the inverse ratio of the cubes of their distances, proceeding from a given place, with given speed, and direction.” (N. b. he does not describe them as spirals).[1]

Cotes spirals corresponding to k equal to 2/3 (red), 1.0 (black), 1.5 (green), 3.0 (cyan) and 6.0 (blue)

The shape of spirals in the family depends on the parameters. The curves in polar coordinates, (r, θ), r > 0 are defined by one of the following five equations:

A > 0, k > 0 and ε are arbitrary real number constants. A determines the size, k determines the shape, and ε determines the angular position of the spiral.

Cotes referred to the different forms as "cases". The equations of the curves above correspond respectively to his 5 cases.[2]

The Diagram shows representative examples of the different curves. The centre is marked by ‘O’ and the radius from O to the curve is shown when θ is zero. The value of ε is zero unless shown.

The first and third forms are Poinsot's spirals; the second is the equiangular spiral; the fourth is the hyperbolic spiral; the fifth is the epispiral.

For more information about their properties, reference should be made to the individual curves.

Diagram of the Different Cotes Spirals

Classical mechanics

Cotes's spirals appear in classical mechanics, as the family of solutions for the motion of a particle moving under an inverse-cube central force. Consider a central force

where μ is the strength of attraction. Consider a particle moving under the influence of the central force, and let h be its specific angular momentum, then the particle moves along a Cotes's spiral, with the constant k of the spiral given by

when μ < h2 (cosine form of the spiral), or

when μ > h2, Poinsot form of the spiral. When μ = h2, the particle follows a hyperbolic spiral. The derivation can be found in the references.[3][4]

History

In the Harmonia Mensurarum (1722), Roger Cotes analysed a number of spirals and other curves, such as the Lituus. He described the possible trajectories of a particle in an inverse-cube central force field, which are the Cotes's spirals. The analysis is based on the method in the Principia Book 1, Proposition 42, where the path of a body is determined under an arbitrary central force, initial speed, and direction.

Depending on the initial speed and direction he determines that there are 5 different "cases" (excluding the trivial ones, the circle and straight line through the centre).

He notes that of the 5, "the first and the last are described by Newton, by means of the quadrature (i.e. integration) of the hyperbola and the ellipse".

Case 2 is the equiangular spiral, which is the spiral par excellence. This has great historical significance as in Proposition 9 of the Principia Book 1, Newton proves that if a body moves along an equiangular spiral, under the action of a central force, that force must be as the inverse of the cube of the radius (even before his proof, in Proposition 11, that motion in an ellipse directed to a focus requires an inverse-square force).

It has to be admitted that not all the curves conform to the usual definition of a spiral. For example, when the inverse-cube force is centrifugal (directed outwards), so that μ < 0, the curve does not even rotate once about the centre. This is represented by case 5, the first of the polar equations shown above, with k > 1 in this case.

Samuel Earnshaw in a book published in 1826 used the term “Cotes’ spirals”, so the terminology was in use at that time.[5] Earnshaw clearly describes Cotes's 5 cases and unnecessarily adds a 6th, which is when the force is centrifugal (repulsive). As noted above, Cotes's included this with case 5.

Following E. T. Whittaker, whose A Treatise on the Analytical Dynamics of Particles and Rigid Bodies (first published in 1904) only listed three of Cotes's spirals,[6] some subsequent authors have followed suit.[7]

See also

References

  1. ^ Roger Cotes (1722). Robert Smith (ed.). Harmonia Mensuarum. Cambridge: [publisher not identified]. p. 30.
  2. ^ Roger Cotes (1722). Robert Smith (ed.). Harmonia Mensuarum. Cambridge: [publisher not identified]. pp. 30–34, 98–101.
  3. ^ Nathaniel Grossman (1996). The sheer joy of celestial mechanics. Springer. p. 34. ISBN 978-0-8176-3832-0.
  4. ^ Whittaker, Edmund Taylor (1917). A treatise on the analytical dynamics of particles and rigid bodies; with an introduction to the problem of three bodies (Second ed.). Cambridge University Press. pp. 83.{{cite book}}: CS1 maint: location missing publisher (link)
  5. ^ Earnshaw, Samuel (1832). Dynamics, Or an Elementary Treatise On Motion; With a Great Variety of Examples Illustrative of the General Principles and Formulae: To Which Is Added a Short Treatise On Attractions. Cambridge: Printed by W. Metcalfe, for J. & J. J. Deighton. pp. 47.
  6. ^ Whittaker (1927).
  7. ^ Kelley & Leventhal (2016).

Bibliography

Read other articles:

Cristóbal MendozaPresiden Venezuela ke-1Masa jabatan5 Maret 1811 – 21 Maret 1812PenggantiSimón Bolívar Informasi pribadiLahir(1772-06-23)23 Juni 1772Trujillo, New GrenadaMeninggal8 Februari 1829(1829-02-08) (umur 56)Caracas, Kolombia RayaTanda tanganSunting kotak info • L • B Cristóbal Hurtado de Mendoza (23 Juni 1772 – 8 Februari 1829) adalah seorang politisi Venezuela. Cristóbal menjadi Presiden pertama Venezuela pada tahun 1811. Biografi I...

 

Pour les articles homonymes, voir Montréal (homonymie). Vue de la rivière des Mille-Îles et du Vieux-Terrebonne. La Rive-Nord, aussi appelée Couronne Nord, est l'appellation couramment donnée à l'ensemble de la banlieue nord de la région métropolitaine de Montréal, au Québec. Son territoire s'étend sur la rive nord de la rivière des Mille Îles et est constituée de 20 municipalités réparties entre les régions des Laurentides et de Lanaudière. Bien que la ville de Laval soit c...

 

Annapolis Spitzname: America’s Sailing Capital, Sailing Capital of the World, Naptown Maryland State House (2013) Siegel Flagge Lage im County und in Maryland Basisdaten Gründung: 1649 Staat: Vereinigte Staaten Bundesstaat: Maryland County: Anne Arundel County Koordinaten: 38° 58′ N, 76° 30′ W38.972944444444-76.50115833333312Koordinaten: 38° 58′ N, 76° 30′ W Zeitzone: Eastern (UTC−5/−4) Einwohner: 40.812 (Stand: 2020) Haushalte: ...

American investigative journalist (1864–1922) Nellie BlyCochran at 26 years old, circa 1890BornElizabeth Jane Cochran(1864-05-05)May 5, 1864Burrell Township, Pennsylvania, U.S.DiedJanuary 27, 1922(1922-01-27) (aged 57)New York City, U.S.NationalityAmericanOther namesElly Cochran, Elizabeth Jane Cochrane, and most commonly known as Nellie Bly as her pen-nameOccupations Journalist writer inventor Spouse Robert Seaman ​ ​(m. 1895; died 1904)&...

 

  此條目介紹的是元代行省。关于交趾的其它意思,请见「交趾」。 交趾行省(又称为安南行省),以安南國置,为元朝至元二十三年(1286年)二月设立的一个特殊的行中书省。与元朝其它的行中书省不同的是,其是元朝对越南作战时设置的一个军事化的特殊建置。 在蒙越战争中,元朝為一举发军征服安南、占城,在今越南设置荆湖占城行中书省(占城行省),。 1...

 

Hungarian piano composition This article is about song cycle for voice and piano. For collection of improvisations for piano solo, see Eight Improvisations on Hungarian Peasant Songs. Eight Hungarian Folksongsby Béla BartókBéla Bartók in 1903Native nameNyolc magyar népdalCatalogue Sz. 64 BB 47 Composed1907–1917Published1922 (1922): ViennaScoringHigh voice and piano Eight Hungarian Folksongs, Sz. 64, BB 47 (Hungarian: Nyolc magyar népdal) is a song cycle for high voice and piano b...

Weight control methodEast German cigarettes Cigarette smoking for weight loss is a weight control method whereby one consumes tobacco, often in the form of cigarettes, to decrease one's appetite. The practice dates to early knowledge of nicotine as an appetite suppressant. Tobacco smoking was associated with appetite suppression among Pre-Columbian indigenous Americans and Old World Europeans.[1] For decades, tobacco companies have employed these connections between slimness and smoki...

 

Traditional form of word-play in Finnic-speaking world Map showing the distribution of the Finnic languages, approximating the area where the Finnic riddles were found The corpus of traditional riddles from the Finnic-speaking world (including the modern Finland, Estonia, and parts of Western Russia) is fairly unitary, though eastern Finnish-speaking regions show particular influence of Russian Orthodox Christianity and Slavonic riddle culture.[1] The Finnish for 'riddle' is arvoitus ...

 

Board wargame Cover of 1974 edition, art by John Hill Bar-Lev, subtitled The Yom-Kippur War of 1973, is a board wargame published by Conflict Games in 1974, only months after the end of the Yom Kippur War. The game simulates battles on the two major fronts of the war: the Golan Heights and the Suez Canal. The game proved very popular, and a second edition was published by Game Designers' Workshop (GDW) in 1977. Background On 6 October 1973, a coalition of Arab nations jointly launched a surpr...

Paraguayan soup Vori voriChicken vorí voríTypeSoupPlace of originParaguayMain ingredientsCorn flour, cheese, stock or broth Vori vori is a thick, yellow soup with little balls made of corn flour, and cheese, and it's traditional of the Paraguayan cuisine. It is essentially of Cario-Guarani and Sephardic origins, and derives from one of the commemorative dishes of the Passover as it derives from the Matza balls, replacing the wheat semolina with the corn flour of the Carios. The name vorí v...

 

7 Seeds7SEEDS セブンシーズ(Sebun Shīzu) MangaPengarangYumi TamuraPenerbitShogakukanPenerbit bahasa IndonesiaElex Media KomputindoMajalahBetsucomi, FlowersDemografiShōjo, JoseiTerbitNovember 2001 – Juli 2017Volume35 (Daftar volume) Animasi web orisinalSutradaraYukio TakahashiSkenarioTouko MachidaStudioGonzoPelisensiNetflixTayangJuni 2019  Portal anime dan manga 7 Seeds (セブンシーズcode: ja is deprecated , Sebun Shīzu) adalah sebuah seri manga Jepang yang ditulis dan ...

 

Markus Braun Markus Braun (1969) è un imprenditore austriaco, dal gennaio 2002 fino alle sue dimissioni e arresto nel giugno 2020 CEO e CTO presso Wirecard, società di elaborazione dei pagamenti ora insolvente. Braun si è dimesso a causa delle accuse di frode, ma ha negato qualsiasi illecito. I processi sono pendenti a partire dal 2022. Indice 1 Biografia 1.1 Wirecard 1.2 Dimissioni ed arresto 1.3 Processo 2 Note 3 Altri progetti Biografia Braun si è laureato in informatica commerciale ed...

2008 American drama film DoubtTheatrical release posterDirected byJohn Patrick ShanleyScreenplay byJohn Patrick ShanleyBased onDoubt: A Parableby John Patrick ShanleyProduced byScott RudinStarring Meryl Streep Philip Seymour Hoffman Amy Adams Viola Davis CinematographyRoger DeakinsEdited byDylan TichenorMusic byHoward ShoreProductioncompaniesMiramax FilmsScott Rudin ProductionsDistributed byMiramax FilmsRelease dates October 30, 2008 (2008-10-30) (AFI Fest) December 12...

 

Historic district in Virginia, United States United States historic placeFolly Castle Historic DistrictU.S. National Register of Historic PlacesU.S. Historic districtVirginia Landmarks Register Folly Castle, December 2009Show map of VirginiaShow map of the United StatesLocationPerry and W. Washington Sts.; 235-618 Washington, 235-580 Hinton, 15-37 Guarantee, 18-115 Lafayette and 18-42 Perry Sts.; Roughly along South St. from Commerce St. to Farmer St., Petersburg, VirginiaCoordinates37°13′...

 

Duta Besar Indonesia untuk SingapuraAmbassador of Indonesia to SingaporeLambang Kementerian Luar Negeri Republik IndonesiaPetahanaSuryopratomosejak 14 September 2020Kementerian Luar Negeri Republik IndonesiaKedutaan Besar Republik Indonesia di SingapuraKantorSingapuraDitunjuk olehPresiden IndonesiaPejabat perdanaMohamad RazifDibentuk1950Situs webkemlu.go.id/singapore/id Berikut adalah daftar diplomat Indonesia yang pernah menjabat sebagai Duta Besar Republik Indonesia untuk Singapura No....

This article is about the Doctor Who serial. For the hypothetical planet beyond Neptune, see Planets beyond Neptune. For other subjects related to tenth planet, see Tenth planet (disambiguation). 1966 Doctor Who serial029 – The Tenth PlanetDoctor Who serialThe Cybermen take over the Snowcap base from General Cutler.CastDoctor William Hartnell – First Doctor Companions Anneke Wills – Polly Michael Craze – Ben Jackson Others Robert Beatty – General Cutler David Dodimead – Ba...

 

This article is about the Class II special and general elections to the United States Senate, for the completion of the unexpired term of Willis Smith and the open term beginning in 1955. For the simultaneous special Class III election to complete Clyde Hoey's unexpired term, see 1954 United States Senate special election in North Carolina. 1954 United States Senate election in North Carolina ← 1950 (special) November 2, 1954 1958 (special) →   Nominee W. Kerr Scot...

 

Bagi kesan tanggapan psikofizikal, lihat Kesan Coriolis (tanggapan). Dalam kerangka rujukan inersia (bahagian atas gambar), bola hitam bergerak dalam satu garisan lurus. Walau bagaimanapun, pemerhati (titik merah) yang berdiri di kerangka rujukan berputar/bukan inersia (bahagian bawahgamabr) melihat objek itu mengikuti laluan melengkung kerana daya-daya Coriolis dan memusat wujud dalam kerangka ini. Dalam fizik, kesan Coriolis ialah pesongan objek bergerak apabila dilihat dalam kerangka rujuk...

Hilario Sosa Hernández Osnovni podaci Država  Meksiko Savezna država Guanajuato Opština Comonfort Stanovništvo Stanovništvo (2014.) 17[1] Geografija Koordinate 20°45′09″N 100°57′26″W / 20.7525°N 100.95722°W / 20.7525; -100.95722 Vremenska zona UTC-6, leti UTC-5 Nadmorska visina 2012[1] m Hilario Sosa HernándezHilario Sosa Hernández na karti Meksika Hilario Sosa Hernández je naselje u Meksiku, u saveznoj državi Guanajuato, u o...

 

Sekolah Kebangsaan Long BohLokasiJln Tengku Budriah, Simpang Empat, 02700, Simpang Empat,, Perlis Indera Kayangan MalaysiaMaklumatJenis sekolahSekolah kebangsaan, Sekolah kerajaanNombor sekolah6049809219Kod sekolahRBA0077Faks6049809219 Sekolah Kebangsaan Long Boh atau nama ringkasnya SK Long Boh, merupakan sebuah Sekolah kebangsaan yang terletak di Jln Tengku Budriah, Simpang Empat. Pada 2009, Sekolah Kebangsaan Long Boh memiliki 101 pelajar lelaki dan 84 pelajar perempuan, menjadikan ju...

 

Strategi Solo vs Squad di Free Fire: Cara Menang Mudah!