Skip to main content
Erschienen in: Measurement Techniques 10/2021

21.01.2021

Dimensions of Plane and Solid Angles and Their Units in the International System of Units (SI)

verfasst von: M. I. Kalinin, L. K. Isaev, F. V. Bulygin

Erschienen in: Measurement Techniques | Ausgabe 10/2021

Einloggen

Aktivieren Sie unsere intelligente Suche um passende Fachinhalte oder Patente zu finden.

search-config
loading …

Abstract

The situation was analyzed that has been established in the International System of Units (SI) as a result of adoption of the recommendation of the International Committee of Weights and Measures (CIPM) in 1980, which proposed to consider plane and solid angles as dimensionless derived quantities. It was shown that such decision was based on a misunderstanding of the mathematical formula, relating the arc length of a circle with its radius and corresponding central angle, as well as series expansions of trigonometric functions. As follows from the conducted analysis, the plane angle value does not depend on any of the SI quantities and should be treated as a base quantity, while its unit (radian) should be added to the population of base SI quantities. Therefore, the value of a solid angle becomes a derived quantity from a plane angle, and its unit (steradian) represents a coherent derived unit equal to a square radian.

Sie haben noch keine Lizenz? Dann Informieren Sie sich jetzt über unsere Produkte:

Springer Professional "Wirtschaft+Technik"

Online-Abonnement

Mit Springer Professional "Wirtschaft+Technik" erhalten Sie Zugriff auf:

  • über 102.000 Bücher
  • über 537 Zeitschriften

aus folgenden Fachgebieten:

  • Automobil + Motoren
  • Bauwesen + Immobilien
  • Business IT + Informatik
  • Elektrotechnik + Elektronik
  • Energie + Nachhaltigkeit
  • Finance + Banking
  • Management + Führung
  • Marketing + Vertrieb
  • Maschinenbau + Werkstoffe
  • Versicherung + Risiko

Jetzt Wissensvorsprung sichern!

Springer Professional "Technik"

Online-Abonnement

Mit Springer Professional "Technik" erhalten Sie Zugriff auf:

  • über 67.000 Bücher
  • über 390 Zeitschriften

aus folgenden Fachgebieten:

  • Automobil + Motoren
  • Bauwesen + Immobilien
  • Business IT + Informatik
  • Elektrotechnik + Elektronik
  • Energie + Nachhaltigkeit
  • Maschinenbau + Werkstoffe




 

Jetzt Wissensvorsprung sichern!

Fußnoten
1
Expressional (6) and (7) are provided in all mathematical reference books and encyclopedias (see, for example, [11 , p. 15]).
 
2
In the physical literature, there is a definition of dimensional and dimensionless values [12]. Quantities whose numerical values depend on the selection of measurement units are called dimensional quantities. Quantities whose numerical values do not depend on the selection of measurement units are called dimensionless quantities. For example, mass, energy, and velocity are dimensional quantities. At the same time, the fine structure constant is a dimensionless quantity, for which even the type of a physical formula depends on the choice of measurement units. Nevertheless, it retains its numerical value in all systems of units. According to this definition, a plane angle is a dimensional quantity, same as mass and length.
 
Literatur
1.
Zurück zum Zitat CIPM, 58 Session: Recommendation 1 (1969), p. 30, https://www.bipm.org/en/CIPM/db/1969/1/, acc. Aug. 31, 2020. CIPM, 58 Session: Recommendation 1 (1969), p. 30, https://​www.​bipm.​org/​en/​CIPM/​db/​1969/​1/​, acc. Aug. 31, 2020.
3.
Zurück zum Zitat CIPM, 69 Session: Recommendation 1 (CI-1980) (1980), p. 24, https://www.bipm.org/utils/common/pdf/CIPM-PVOCR/CIPM1980.pdf, acc. Aug. 31, 2020. CIPM, 69 Session: Recommendation 1 (CI-1980) (1980), p. 24, https://​www.​bipm.​org/​utils/​common/​pdf/​CIPM-PVOCR/​CIPM1980.​pdf, acc. Aug. 31, 2020.
9.
Zurück zum Zitat D. K. Maxwell, A Treatise on Electricity and Magnetism [Russian translation], Ser. Classics of Science, Nauka, Moscow (1989), Vol. 1. D. K. Maxwell, A Treatise on Electricity and Magnetism [Russian translation], Ser. Classics of Science, Nauka, Moscow (1989), Vol. 1.
10.
Zurück zum Zitat Standard ISO 80000-1, Quantities and Units, Part 1: General (2009). Standard ISO 80000-1, Quantities and Units, Part 1: General (2009).
11.
Zurück zum Zitat I. M. Vinogradov et al., Mathematical Encyclopedia, Sov. Entsiklopediya, Moscow (1984), Vol. 4. I. M. Vinogradov et al., Mathematical Encyclopedia, Sov. Entsiklopediya, Moscow (1984), Vol. 4.
12.
Zurück zum Zitat L. I. Sedov, Similarity and Dimensional Methods in Mechanics, Nauka, Moscow (1977). L. I. Sedov, Similarity and Dimensional Methods in Mechanics, Nauka, Moscow (1977).
15.
Zurück zum Zitat V. A. Ilyin and E. G. Poznyak, Fundamentals of Mathematical Analysis, Part 1, Fizmatlit, Moscow (2005). V. A. Ilyin and E. G. Poznyak, Fundamentals of Mathematical Analysis, Part 1, Fizmatlit, Moscow (2005).
16.
Zurück zum Zitat A. B. Torrens, Metrologia, 22, 1, 1–7 (1986), 10.1088/0026-1394/22/1/002. A. B. Torrens, Metrologia, 22, 1, 1–7 (1986), 10.1088/0026-1394/22/1/002.
17.
Zurück zum Zitat K. R. Brownstein, Am. J. Phys., 65, 7, 605–614 (1997), 10.1119/1.18616. K. R. Brownstein, Am. J. Phys., 65, 7, 605–614 (1997), 10.1119/1.18616.
21.
Zurück zum Zitat I. M. Vinogradov et al., Mathematical Encyclopedia, Sov. Entsiklopediya, Moscow (1985), Vol. 5. I. M. Vinogradov et al., Mathematical Encyclopedia, Sov. Entsiklopediya, Moscow (1985), Vol. 5.
25.
Zurück zum Zitat M. I. Kalinin, “Plain and solid angles in the International System of Units,” Zakonodat. Prikl. Metrol., No. 6, 12–16 (2018). M. I. Kalinin, “Plain and solid angles in the International System of Units,” Zakonodat. Prikl. Metrol., No. 6, 12–16 (2018).
Metadaten
Titel
Dimensions of Plane and Solid Angles and Their Units in the International System of Units (SI)
verfasst von
M. I. Kalinin
L. K. Isaev
F. V. Bulygin
Publikationsdatum
21.01.2021
Verlag
Springer US
Erschienen in
Measurement Techniques / Ausgabe 10/2021
Print ISSN: 0543-1972
Elektronische ISSN: 1573-8906
DOI
https://doi.org/10.1007/s11018-021-01855-y

Weitere Artikel der Ausgabe 10/2021

Measurement Techniques 10/2021 Zur Ausgabe