View source for Moment (physics)

{{short description|Product of a distance and physical quantity}}
{{Distinguish|Momentum}}
{{About||the mathematical concept|Moment (mathematics)|the moment of a force, sometimes shortened to "moment"|Torque}}
{{Classical mechanics|cTopic=Fundamental concepts}}

In [[physics]], a '''moment''' is a mathematical expression involving the product of a force and perpendicular [[distance]] from the point of action of the force. Moments are usually defined with respect to a fixed reference point and refer to physical quantities located some distance from the reference point. In this way, the moment accounts for the quantity's location or arrangement. For example, the moment of [[force]], often called [[torque]], is the product of a force on an object and the distance from the reference point to the object. In principle, any physical quantity can be multiplied by a distance to produce a moment. Commonly used quantities include forces, [[mass]]es, and [[electric charge]] distributions.

== Elaboration ==

In its most basic form, a moment is the [[product (mathematics)|product]] of the distance to a point, raised to a power, and a physical quantity 
(such as force or electrical charge) at that point:

: <math>\mu_n = r^n\,Q,</math>

where <math>Q</math> is the physical quantity such as a force applied at a point, or a point charge, or a point mass, etc. If the quantity is not concentrated solely at a single point, the moment is the [[integral]] of that quantity's density over space:

:<math>\mu_n = \int r^n \rho(r)\,dr</math>

where <math>\rho</math> is the distribution of the density of charge, mass, or whatever quantity is being considered.

More complex forms take into account the angular relationships between the distance and the physical quantity, but the above equations capture the essential feature of a moment, namely the existence of an underlying <math>r^n \rho(r)</math> or equivalent term.  This implies that there are multiple moments (one for each value of ''n'') and that the moment generally depends on the reference point from which the distance <math>r</math> is measured, although for certain moments (technically, the lowest non-zero moment) this dependence vanishes and the moment becomes independent of the reference point.

Each value of ''n'' corresponds to a different moment: the 1st moment corresponds to ''n''&nbsp;=&nbsp;1; the 2nd moment to ''n''&nbsp;=&nbsp;2, etc.  The 0th moment (''n''&nbsp;=&nbsp;0) is sometimes called the ''monopole moment''; the 1st moment (''n''&nbsp;=&nbsp;1) is sometimes called the ''dipole moment'', and the 2nd moment (''n''&nbsp;=&nbsp;2) is sometimes called the ''[[quadrupole moment]]'', especially in the context of electric charge distributions.

=== Examples ===
* The ''moment of force'', or ''[[torque]]'', is a first moment: <math>\mathbf{\tau} =  rF</math>, or, more generally, <math>\mathbf{r} \times \mathbf{F}</math>.
* Similarly, ''[[angular momentum]]'' is the 1st moment of [[momentum]]: <math>\mathbf{L} = \mathbf{r} \times \mathbf{p}</math>.  Momentum itself is ''not'' a moment.  
* The ''[[electric dipole moment]]'' is also a 1st moment: <math>\mathbf{p} = q\,\mathbf{d}</math> for two opposite point charges or <math display="inline">\int \mathbf{r}\,\rho(\mathbf{r})\,d^3r</math> for a distributed charge with charge density <math>\rho(\mathbf{r})</math>.

Moments of mass:
* The ''total [[mass]]'' is the zeroth moment of mass.
* The ''[[center of mass]]'' is the 1st moment of mass normalized by total mass: <math display="inline">\mathbf{R} = \frac 1M \sum_i \mathbf{r}_i m_i</math> for a collection of point masses, or <math display="inline"> \frac 1M \int \mathbf{r} \rho(\mathbf{r}) \, d^3r</math> for an object with mass distribution <math>\rho(\mathbf{r})</math>.
* The ''[[moment of inertia]]'' is the 2nd moment of mass: <math>I = r^2 m</math> for a point mass, <math display="inline">\sum_i r_i^2 m_i</math> for a collection of point masses, or <math display="inline">\int r^2\rho(\mathbf{r}) \, d^3r</math> for an object with mass distribution <math>\rho(\mathbf{r})</math>.  The center of mass is often (but not always) taken as the reference point.

== Multipole moments ==
Assuming a density function that is finite and localized to a particular region, outside that region a 1/''r'' [[Scalar potential|potential]] may be expressed as a series of [[spherical harmonics]]:
:<math>
\Phi(\mathbf{r}) = 
\int \frac{\rho(\mathbf{r'})}{|\mathbf{r}-\mathbf{r'}|} \, d^3r' =
\sum_{\ell=0}^\infty
\sum_{m=-\ell}^\ell
\left( \frac{4\pi}{2\ell+1} \right)
q_{\ell m}\,
\frac{Y_{\ell m}(\theta, \varphi)}{r^{\ell+1}}
</math>

The coefficients <math>q_{\ell m}</math> are known as ''multipole moments'', and take the form:
:<math>
q_{\ell m} = \int 
(r')^\ell\,
\rho(\mathbf{r'})\,
Y^*_{\ell m}(\theta',\varphi')\,
d^3r'
</math>

where <math>\mathbf{r}'</math> expressed in spherical coordinates <math>\left(r', \varphi', \theta'\right)</math> is a variable of integration.  A more complete treatment may be found in pages describing [[multipole expansion]] or [[spherical multipole moments]].  (The convention in the above equations was taken from Jackson<ref>J. D. Jackson, ''Classical Electrodynamics'', 2nd edition, Wiley, New York, (1975). p. 137</ref> – the conventions used in the referenced pages may be slightly different.)

When <math>\rho</math> represents an electric charge density, the <math>q_{lm}</math> are, in a sense, projections of the moments of electric charge: <math>q_{00}</math> is the monopole moment; the <math>q_{1m}</math> are projections of the dipole moment, the <math>q_{2m}</math> are projections of the quadrupole moment, etc.

== Applications of multipole moments ==
The multipole expansion applies to 1/''r'' scalar potentials, examples of which include the  [[electric potential]] and the [[gravitational potential]].  For these potentials, the expression can be used to approximate the strength of a field produced by a localized distribution of charges (or mass) by calculating the first few moments.  For sufficiently large ''r'', a reasonable approximation can be obtained from just the monopole and dipole moments.  Higher fidelity can be achieved by calculating higher order moments. Extensions of the technique can be used to [[Spherical multipole moments#Interaction energies of spherical multipoles|calculate interaction energies]] and intermolecular forces.

The technique can also be used to determine the properties of an unknown distribution <math>\rho</math>.  Measurements pertaining to multipole moments may be taken and used to infer properties of the underlying distribution.  This technique applies to small objects such as molecules,<ref>{{Cite journal|doi=10.1021/cr00016a005|title=Molecular electric moments from x-ray diffraction data|year=1992|last1=Spackman|first1=M. A.|journal=Chemical Reviews|volume=92|issue=8|pages=1769–1797}}</ref><ref>
Dittrich and Jayatilaka, ''Reliable Measurements of Dipole Moments from Single-Crystal Diffraction Data and Assessment of an In-Crystal Enhancement'', Electron Density and Chemical Bonding II, Theoretical Charge Density Studies, Stalke, D. (Ed), 2012, https://www.springer.com/978-3-642-30807-9</ref>
but has also been applied to the universe itself,<ref>{{cite arXiv |eprint=0907.5424|last1=Baumann|first1=Daniel|title=TASI Lectures on Inflation|year=2009|class=hep-th}}</ref> being for example the technique employed by the [[WMAP]] and [[Planck (spacecraft)|Planck]] experiments to analyze the [[cosmic microwave background]] radiation.

==History==

[[File:Lever Principle 3D.png|thumb|right|A lever in balance]]
In works believed to stem from [[Ancient Greece]], the concept of a moment is alluded to by the word [[wikt:ῥοπή|ῥοπή]] (''rhopḗ'', {{lit.}} "inclination") and composites like [[wikt:ἰσόρροπα|ἰσόρροπα]] (''isorropa'', {{lit.}} "of equal inclinations").<ref name=mersenne>{{cite book
 |last1=Mersenne
 |first1=Marin
 |author-link=Marin Mersenne
 |title=Les Méchaniques de Galilée
 |date=1634
 |location=Paris
 |url=https://books.google.com/books?id=X-UPAAAAQAAJ&pg=PA7
 |pages=[https://books.google.com/books?id=X-UPAAAAQAAJ&pg=PA7 7–8]
}}</ref>{{r|clagett2}}<ref name=lsj1>{{LSJ|r(oph/|ῥοπή|ref}}</ref> The context of these works is [[mechanics]] and [[geometry]] involving the [[lever]].<ref name=clagett1>{{cite book
 |last1=Clagett
 |first1=Marshall
 |author-link=Marshall Clagett
 |title=The Science of Mechanics in the Middle Ages
 |date=1959
 |location=Madison, WI
 |publisher=University of Wisconsin Press
}}</ref> In particular, in extant works attributed to [[Archimedes]], the moment is pointed out in phrasings like:

:"[[Commensurability (mathematics)|Commensurable]] magnitudes ({{wikt-lang|grc|σύμμετρα}} {{wikt-lang|grc|μεγέθεα}}) [A and B] are equally balanced ({{wikt-lang|grc|ἰσορροπέοντι}}){{efn|An alternative translation is "have equal moments" as used by [[Francesco Maurolico]] in the 1500s.{{r|clagett2}} A literal translation is "have equal inclinations".}} if their distances [to the center Γ, i.e., ΑΓ and ΓΒ] are [[Proportionality (mathematics)#Inverse proportionality|inversely proportional]] ({{wikt-lang|grc|ἀντιπεπονθότως}}) to their weights ({{wikt-lang|grc|βάρεσιν}})."<ref name=clagett2>{{cite book
 |last1=Clagett
 |first1=Marshall
 |author-link=Marshall Clagett
 |title=Archimedes in the Middle Ages
 |type=5 vols in 10 tomes
 |date=1964–84
 |publisher=Madison, WI: University of Wisconsin Press, 1964; Philadelphia: American Philosophical Society, 1967–1984.
}}</ref><ref>{{cite book
 |last1=Dijksterhuis
 |first1=E. J.
 |author-link=Eduard Jan Dijksterhuis
 |title=Archimedes
 |date=1956
 |publisher=E. Munksgaard
 |location=Copenhagen
 |page=[https://archive.org/details/archimedes0000dijk/page/288 288]
 |url=https://archive.org/details/archimedes0000dijk/page/288
}}</ref>

Moreover, in extant texts such as ''[[The Method of Mechanical Theorems]]'', moments are used to infer the [[center of gravity]], area, and volume of geometric figures.

In 1269, [[William of Moerbeke]] translates various works of [[Archimedes]] and [[Eutocius of Ascalon|Eutocious]] into [[Latin]]. The term ῥοπή is [[transliteration|transliterated]] into ''ropen''.{{r|clagett2}}

Around 1450, [[Iacopo da San Cassiano|Jacobus Cremonensis]] translates ῥοπή in similar texts into the [[Latin]] term ''momentum''{{Citation needed|reason=dubious claim that Cremonensis initiated the transition to momentum|date=November 2023}} ({{lit.}} "movement"<ref>{{cite encyclopedia
 |title=moment
 |encyclopedia=Oxford English Dictionary
 |year=1933
 |url=https://archive.org/details/in.ernet.dli.2015.271836/page/n1125
}}</ref>). The same term is kept in a 1501 translation by [[Giorgio Valla]], and subsequently by [[Francesco Maurolico]], [[Federico Commandino]], [[Guidobaldo del Monte]], [[Adriaan van Roomen]], [[Florence Rivault]], [[Francesco Buonamici (philosopher)|Francesco Buonamici]], [[Marin Mersenne]]{{r|mersenne}}, and [[Galileo Galilei]]. That said, why was the word ''momentum'' chosen for the translation? One clue, according to [[Treccani]], is that ''momento'' in Medieval Italy, the place the early translators lived, in a transferred sense meant both a "moment of time" and a "moment of weight" (a small amount of weight that turns the [[Steelyard balance|scale]]).{{efn|[[Treccani]] writes in its entry on [https://www.treccani.it/vocabolario/momento moménto]: "[...] alla tradizione medievale, nella quale momentum significava, per lo più, minima porzione di tempo, la più piccola parte dell’ora (precisamente, 1/40 di ora, un minuto e mezzo), ma anche minima quantità di peso, e quindi l’ago della bilancia (basta l’applicazione di un momento di peso perché si rompa l’equilibrio e la bilancia tracolli in un momento);"}}

In 1554, [[Francesco Maurolico]] clarifies the Latin term ''momentum'' in the work ''Prologi sive sermones''. Here is a Latin to English translation as given by [[Marshall Clagett]]:{{r|clagett2}}

<blockquote>
"[...] equal weights at unequal distances do not weigh equally, but unequal weights [at these unequal distances may] weigh equally. For a weight suspended at a greater distance is heavier, as is obvious in a [[Steelyard balance|balance]]. Therefore, there exists a certain third kind of power or third difference of magnitude—one that differs from both body and weight—and this they call '''moment'''.{{efn|In Latin: ''momentum''.}} Therefore, a body acquires weight from both quantity [i.e., size] and quality [i.e., material], but a weight receives its moment from the distance at which it is suspended. Therefore, when distances are reciprocally proportional to weights, the moments [of the weights] are equal, as [[Archimedes]] demonstrated in ''[[On the Equilibrium of Planes|The Book on Equal Moments]]''.{{efn|The modern translation of this book is "on the equilibrium of planes". The translation "on equal moments (of planes)" as used by Maurolico is also echoed in his four-volume book called ''De momentis aequalibus'' ("about equal moments") where he applies Archimedes' ideas to solid bodies.}} Therefore, weights or [rather] moments like other continuous quantities, are joined at some common terminus, that is, at something common to both of them like the center of weight, or at a point of equilibrium. Now the [[center of gravity]] in any weight is that point which, no matter how often or whenever the body is suspended, always inclines perpendicularly toward the universal center.

In addition to body, weight, and moment, there is a certain fourth power, which can be called impetus or force.{{efn|In Latin: ''impetus'' or ''vis''. This fourth power was the intellectual precursor to the English [[Latinism]] ''[[momentum]]'', also called ''quantity of motion''.}} [[Aristotle]] investigates it in ''On Mechanical Questions'', and it is completely different from [the] three aforesaid [powers or magnitudes]. [...]"
</blockquote>

in 1586, [[Simon Stevin]] uses the [[Dutch language|Dutch]] term ''staltwicht'' ("parked weight") for momentum in ''[[De Beghinselen Der Weeghconst]]''.

In 1632, [[Galileo Galilei]] publishes ''[[Dialogue Concerning the Two Chief World Systems]]'' and uses the [[Italian language|Italian]] ''momento'' with many meanings, including the one of his predecessors.<ref>{{cite book
 |last1=Galluzzi
 |first1=Paolo
 |title=Momento. Studi Galileiani
 |date=1979
 |location=Rome
 |publisher=Edizioni dell' Ateneo & Bizarri
}}</ref>

In 1643, Thomas Salusbury translates some of Galilei's works into [[English language|English]]. Salusbury translates Latin ''momentum'' and Italian ''momento'' into the English term ''moment''.{{efn|This is very much in line with other Latin ''-entum'' words such as ''documentum'', ''monumentum'', or ''argumentum'' which turned into ''document'', ''monument'', and ''argument'' in [[French language|French]] and English.}}

In 1765, the [[Latin]] term ''momentum inertiae'' ([[English language|English]]: ''[[moment of inertia]]'') is used by [[Leonhard Euler]] to refer to one of [[Christiaan Huygens]]'s quantities in ''[[Horologium Oscillatorium]]''.<ref name=euler>{{cite book
 |last=Euler
 |first=Leonhard
 |author-link=Leonhard Euler
 |title=Theoria motus corporum solidorum seu rigidorum: Ex primis nostrae cognitionis principiis stabilita et ad omnes motus, qui in huiusmodi corpora cadere possunt, accommodata [The theory of motion of solid or rigid bodies: established from first principles of our knowledge and appropriate for all motions which can occur in such bodies.]
 |publisher= A. F. Röse
 |location= Rostock and Greifswald (Germany)
 |date=1765
 |page=[https://archive.org/details/theoriamotuscor00eulegoog/page/n202 166]|url= https://archive.org/details/theoriamotuscor00eulegoog
 |language=latin
 |isbn=978-1-4297-4281-8
}} From page 166:  ''"Definitio 7.  422.  Momentum inertiae corporis respectu eujuspiam axis est summa omnium productorum, quae oriuntur, si singula corporis elementa per quadrata distantiarum suarum ab axe multiplicentur."'' (Definition 7.  422.  A body's moment of inertia with respect to any axis is the sum of all of the products, which arise, if the individual elements of the body are multiplied by the square of their distances from the axis.)</ref> Huygens 1673 work involving finding the [[center of oscillation]] had been stimulated by [[Marin Mersenne]], who suggested it to him in 1646.<ref>{{cite book
 |last1=Huygens
 |first1=Christiaan
 |title=Horologium oscillatorium, sive de Motu pendulorum ad horologia aptato demonstrationes geometricae
 |date=1673
 |page=[https://gallica.bnf.fr/ark:/12148/bpt6k1523597w/f111.item 91]
 |url=https://gallica.bnf.fr/ark:/12148/bpt6k1523597w/f111.item
 |language=latin
}}</ref><ref>{{cite web
 |last1=Huygens
 |first1=Christiaan
 |author-link=Christiaan Huygens
 |title=Center of Oscillation (translation)
 |date=1977–1995
 |translator-last1=Mahoney
 |translator-first1=Michael S.
 |url=https://www.princeton.edu/~hos/mike/texts/huygens/centosc/huyosc.htm
 |access-date=22 May 2022
}}</ref>

In 1811, the French term ''moment d'une force'' ([[English language|English]]: ''moment of force'') with respect to a point and plane is used by [[Siméon Denis Poisson]] in ''Traité de mécanique''.<ref>{{cite book
 |last1=Poisson
 |first1=Siméon-Denis
 |author-link=Siméon Denis Poisson
 |title=Traité de mécanique, tome premier
 |date=1811
 |url=https://gallica.bnf.fr/ark:/12148/bpt6k903370/f99.item
 |page=[https://gallica.bnf.fr/ark:/12148/bpt6k903370/f99.item 67]
}}</ref> [https://babel.hathitrust.org/cgi/pt?id=mdp.39015067064249 An English translation] appears in 1842.

In 1884, the term ''[[torque]]'' is suggested by [[James Thomson (engineer)|James Thomson]] in the context of measuring rotational forces of [[machine]]s (with [[propeller]]s and [[rotor (electric)|rotor]]s).<ref>{{cite book
 |last1=Thompson
 |first1=Silvanus Phillips
 |title=Dynamo-electric machinery: A Manual For Students Of Electrotechnics
 |url=https://archive.org/details/dynamoelectricm17thomgoog/page/n125
 |publisher=New York, Harvard publishing co 
 |date=1893
 |edition=4th
 |page=[https://archive.org/details/dynamoelectricm17thomgoog/page/n125 108]
}}</ref><ref>{{cite book
 |url= https://archive.org/details/collectedpapers00larmgoog/page/n110
 |title= Collected Papers in Physics and Engineering
 |page=civ
 |first1= James
 |last1= Thomson
 |first2= Joseph
 |last2= Larmor
 |publisher= University Press
 |date= 1912
}}</ref> Today, a [[dynamometer]] is used to measure the torque of machines.

In 1893, [[Karl Pearson]] uses the term ''n-th moment'' and <math>\mu_n</math> in the context of [[curve fitting|curve-fitting]] scientific measurements.<ref>{{cite journal
 |last1=Pearson
 |first1=Karl
 |title=Asymmetrical Frequency Curves
 |journal=Nature
 |date=October 1893
 |volume=48
 |issue=1252
 |pages=615–616
 |doi=10.1038/048615a0
|bibcode=1893Natur..48..615P
 |s2cid=4057772
 }}</ref> Pearson wrote in response to [[John Venn]], who, some years earlier, observed a peculiar pattern involving [[Meteorology|meteorological]] data and asked for an explanation of its cause.<ref>{{cite journal
 |last1=Venn
 |first1=J.
 |title=The Law of Error
 |journal=Nature
 |date=September 1887
 |volume=36
 |issue=931
 |pages=411–412
 |doi=10.1038/036411c0
|bibcode=1887Natur..36..411V
 |s2cid=4098315
 |url=https://zenodo.org/record/1696687
 }}</ref> In Pearson's response, this analogy is used: the mechanical "center of gravity" is the [[mean]] and the "distance" is the [[Deviation (statistics)|deviation]] from the mean. This later evolved into [[Moment (mathematics)|moments in mathematics]]. The analogy between the mechanical concept of a moment and the [[statistics|statistical]] function involving the sum of the {{math|n}}th powers of deviations was noticed by several earlier, including [[Pierre-Simon Laplace|Laplace]], [[Christian Kramp|Kramp]], [[Carl Friedrich Gauss|Gauss]], [[Johann Franz Encke|Encke]], [[Emanuel Czuber|Czuber]], [[Adolphe Quetelet|Quetelet]], and [[Erastus L. De Forest|De Forest]].<ref>{{cite book
 |last1=Walker
 |first1=Helen M.
 |author-link=Helen M. Walker
 |title=Studies in the history of statistical method, with special reference to certain educational problems
 |page=[https://archive.org/details/studiesinhistory00walk/page/71 71]
 |date=1929
 |publisher=Baltimore, Williams & Wilkins Co.
 |url=https://archive.org/details/studiesinhistory00walk/page/71
}}</ref>

== See also ==
*[[Torque]] (or ''moment of force''), see also the article [[couple (mechanics)]]<!-- Please keep this first in list due to similarity of "Torque" and "Moment". Also refer to Talk page at both articles. -->
*[[Moment (mathematics)]]
*[[Mechanical equilibrium]], applies when an object is balanced so that the sum of the clockwise moments about a pivot is equal to the sum of the anticlockwise moments about the same pivot
*[[Moment of inertia]] <math>\left(I = \Sigma m r^2\right)</math>, analogous to [[mass]] in discussions of rotational motion. It is a measure of an object's resistance to changes in its rotation rate
*[[Angular momentum|Moment of momentum]] <math>(\mathbf{L} = \mathbf{r} \times m\mathbf{v})</math>, the rotational analog of linear [[momentum]].
*[[Magnetic moment]] <math>\left(\mathbf{\mu} = I\mathbf{A}\right)</math>, a [[dipole]] moment measuring the strength and direction of a magnetic source.
*[[Electric dipole moment]], a dipole moment measuring the charge difference and direction between two or more charges.  For example, the electric dipole moment between a charge of –''q'' and ''q'' separated by a distance of '''d''' is <math>(\mathbf{p} = q \mathbf{d})</math>
*[[Bending moment]], a moment that results in the bending of a structural element
*[[First moment of area]], a property of an object related to its resistance to shear stress
*[[Second moment of area]], a property of an object related to its resistance to bending and deflection
*[[Polar moment of inertia]], a property of an object related to its resistance to torsion
*[[Image moments]], statistical properties of an image
*[[Seismic moment]], quantity used to measure the size of an earthquake
*[[Vlasov equation#Moment equations|Plasma moments]], fluid description of plasma in terms of density, velocity and pressure
*[[List of area moments of inertia]]
*[[List of moments of inertia]]
*[[Multipole expansion]]
*[[Spherical multipole moments]]

==Notes==
{{notelist}}

== References ==
{{Reflist}}

==External links==
{{Wiktionary|moment}}
*{{Commons category-inline|Moment (physics)}}
*[http://www.collinsdictionary.com/dictionary/english/moment] A dictionary definition of moment.

[[Category:Length]]
[[Category:Moment (physics)| ]]
[[Category:Physical quantities]]
[[Category:Multiplication]]

[[el:Ροπή]]
[[sq:Momenti]]