Not to tát be confused with Momentum.

For the mathematical concept, see Moment (mathematics). For the moment of a force, sometimes shortened to tát "moment", see Torque.

In physics, a **moment** is a mathematical expression involving the product of a distance and physical quantity. Moments are usually defined with respect to tát a fixed reference point and refer to tát 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 tát the object. In principle, any physical quantity can be multiplied by a distance to tát produce a moment. Commonly used quantities include forces, masses, and electric charge distributions.

## Elaboration[edit]

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

where 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 mật độ trùng lặp từ khóa over space:

where is the distribution of the mật độ trùng lặp từ khóa of charge, mass, or whatever quantity is being considered.

More complex forms take into tài khoản 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 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 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 tát a different moment: the 1st moment corresponds to tát *n* = 1; the 2nd moment to tát *n* = 2, etc. The 0th moment (*n* = 0) is sometimes called the *monopole moment*; the 1st moment (*n* = 1) is sometimes called the *dipole moment*, and the 2nd moment (*n* = 2) is sometimes called the *quadrupole moment*, especially in the context of electric charge distributions.

### Examples[edit]

Moments of mass:

## Multipole moments[edit]

Assuming a mật độ trùng lặp từ khóa function that is finite and localized to tát a particular region, outside that region a 1/*r* potential may be expressed as a series of spherical harmonics:

The coefficients are known as *multipole moments*, and take the form:

where expressed in spherical coordinates is a variable of integration. A more complete treatment may be found in pages describing multipole expansion or spherical multipole moments. (Note: the convention in the above equations was taken from Jackson^{[1]} – the conventions used in the referenced pages may be slightly different.)

When represents an electric charge mật độ trùng lặp từ khóa, the are, in a sense, projections of the moments of electric charge: is the monopole moment; the are projections of the dipole moment, the are projections of the quadrupole moment, etc.

## Applications of multipole moments[edit]

The multipole expansion applies to tát 1/*r* scalar potentials, examples of which include the electric potential and the gravitational potential. For these potentials, the expression can be used to tát 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 tát calculate interaction energies and intermolecular forces.

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The technique can also be used to tát determine the properties of an unknown distribution . Measurements pertaining to tát multipole moments may be taken and used to tát infer properties of the underlying distribution. This technique applies to tát small objects such as molecules,^{[2]}^{[3]}
but has also been applied to tát the universe itself,^{[4]} being for example the technique employed by the WMAP and Planck experiments to tát analyze the cosmic microwave background radiation.

## History[edit]

In works believed to tát stem from Ancient Greece, the concept of a moment is alluded to tát by the word ῥοπή (*rhopḗ*, lit. "inclination") and composites lượt thích ἰσόρροπα (*isorropa*, lit. "of equal inclinations").^{}[5]^{[6]}^{[7]} The context of these works is mechanics and geometry involving the lever.^{[8]} In particular, in extant works attributed to tát Archimedes, the moment is pointed out in phrasings like:

- "Commensurable magnitudes (σύμμετρα μεγέθεα) [A and B] are equally balanced (ἰσορροπέοντι)
^{[a]}if their distances [to the center Γ, i.e., ΑΓ and ΓΒ] are inversely proportional (ἀντιπεπονθότως) to tát their weights (βάρεσιν)."^{[6]}^{[9]}

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

In 1269, William of Moerbeke translates various works of Archimedes and Eutocious into Latin. The term ῥοπή is transliterated into *ropen*.^{[6]}

Around 1450, Jacobus Cremonensis translates ῥοπή in similar texts into the Latin term *momentum* (lit. "movement"^{[10]}). The same term is kept in a 1501 translation by Giorgio Valla, and subsequently by Francesco Maurolico, Federico Commandino, Guidobaldo del Monte, Adriaan cầu xin Roomen, Florence Rivault, Francesco Buonamici, Marin Mersenne^{[5]}, and Galileo Galilei. That said, why was the word *momentum* chosen for the translation? One clue, according to tát 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 scale).^{[b]}

In 1554, Francesco Maurolico clarifies the Latin term *momentum* in the work *Prologi sive sermones*. Here is a Latin to tát English translation as given by Marshall Clagett:^{[6]}

"[...] equal weights at unequal distances vì thế 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 balance. Therefore, there exists a certain third kind of power or third difference of magnitude—one that differs from both body toàn thân and weight—and this they Gọi

moment.^{[c]}Therefore, a body toàn thân 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 tát weights, the moments [of the weights] are equal, as Archimedes demonstrated inThe Book on Equal Moments.^{[d]}Therefore, weights or [rather] moments lượt thích other continuous quantities, are joined at some common terminus, that is, at something common to tát both of them lượt thích 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 toàn thân is suspended, always inclines perpendicularly toward the universal center.In addition to tát body toàn thân, weight, and moment, there is a certain fourth power, which can be called impetus or force.

^{[e]}Aristotle investigates it inOn Mechanical Questions, and it is completely different from [the] three aforesaid [powers or magnitudes]. [...]"

in 1586, Simon Stevin uses the 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 *momento* with many meanings, including the one of his predecessors.^{[11]}

In 1643, Thomas Salusbury translates some of Galilei's works into English. Salusbury translates Latin *momentum* and Italian *momento* into the English term *moment*.^{[f]}

In 1765, the Latin term *momentum inertiae* (English: *moment of inertia*) is used by Leonhard Euler to tát refer to tát one of Christiaan Huygens's quantities in *Horologium Oscillatorium*.^{[12]} Huygens 1673 work involving finding the center of oscillation had been stimulated by Marin Mersenne, who suggested it to tát him in 1646.^{[13]}^{[14]}

In 1811, the French term *moment d'une force* (English: *moment of force*) with respect to tát a point and plane is used by Siméon Denis Poisson in *Traité de mécanique*.^{[15]} An English translation appears in 1842.

In 1884, the term *torque* is suggested by James Thomson in the context of measuring rotational forces of machines (with propellers and rotors).^{[16]}^{[17]} Today, a dynamometer is used to tát measure the torque of machines.

In 1893, Karl Pearson uses the term *n-th moment* and in the context of curve-fitting scientific measurements.^{[18]} Pearson wrote in response to tát John Venn, who, some years earlier, observed a peculiar pattern involving meteorological data and asked for an explanation of its cause.^{[19]} In Pearson's response, this analogy is used: the mechanical "center of gravity" is the mean and the "distance" is the deviation from the mean. This later evolved into moments in mathematics. The analogy between the mechanical concept of a moment and the statistical function involving the sum of the nth powers of deviations was noticed by several earlier, including Laplace, Kramp, Gauss, Encke, Czuber, Quetelet, and De Forest.^{[20]}

## See also[edit]

- Torque (or
*moment of force*), see also the article couple (mechanics) - Moment (mathematics)
- Mechanical equilibrium, applies when an object is balanced ví that the sum of the clockwise moments about a pivot is equal to tát the sum of the anticlockwise moments about the same pivot
- Moment of inertia , analogous to tát mass in discussions of rotational motion. It is a measure of an object's resistance to tát changes in its rotation rate
- Moment of momentum , the rotational analog of linear momentum.
- Magnetic moment , 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 - Bending moment, a moment that results in the bending of a structural element
- First moment of area, a property of an object related to tát its resistance to tát shear stress
- Second moment of area, a property of an object related to tát its resistance to tát bending and deflection
- Polar moment of inertia, a property of an object related to tát its resistance to tát torsion
- Image moments, statistical properties of an image
- Seismic moment, quantity used to tát measure the size of an earthquake
- Plasma moments, fluid mô tả tìm kiếm of plasma in terms of mật độ trùng lặp từ khóa, velocity and pressure
- List of area moments of inertia
- List of moments of inertia
- Multipole expansion
- Spherical multipole moments

## Notes[edit]

**^**An alternative translation is "have equal moments" as used by Francesco Maurolico in the 1500s.^{[6]}A literal translation is "have equal inclinations".**^**Treccani writes in its entry on moménto: "[...] alla tradizione medievale, nella quale momentum significava, per bồn chồn più, minima porzione di tempo, la più piccola parte dell’ora (precisamente, 1/40 di ora, un minuto e mezzo), yêu tinh 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 Latin:*momentum*.**^**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 tát solid bodies.**^**In Latin:*impetus*or*vis*. This fourth power was the intellectual precursor to tát the English Latinism*momentum*, also called*quantity of motion*.**^**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 and English.

## References[edit]

**^**J. D. Jackson,*Classical Electrodynamics*, 2nd edition, Wiley, Thủ đô New York, (1975). p. 137**^**Spackman, M. A. (1992). "Molecular electric moments from x-ray diffraction data".*Chemical Reviews*.**92**(8): 1769–1797. doi:10.1021/cr00016a005.**^**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**^**Baumann, Daniel (2009). "TASI Lectures on Inflation". arXiv:0907.5424 [hep-th].- ^
^{a}^{b}Mersenne, Marin (1634).*Les Méchaniques de Galilée*. Paris. pp. 7–8. - ^
^{a}^{b}^{c}^{d}^{e}Clagett, Marshall (1964–84).*Archimedes in the Middle Ages*(5 vols in 10 tomes). Madison, WI: University of Wisconsin Press, 1964; Philadelphia: American Philosophical Society, 1967–1984. **^**ῥοπή. Liddell, Henry George; Scott, Robert;*A Greek–English Lexicon*at the Perseus Project**^**Clagett, Marshall (1959).*The Science of Mechanics in the Middle Ages*. Madison, WI: University of Wisconsin Press.**^**Dijksterhuis, E. J. (1956).*Archimedes*. Copenhagen: E. Munksgaard. p. 288.**^**"moment".*Oxford English Dictionary*. 1933.**^**Galluzzi, Paolo (1979).*Momento. Studi Galileiani*. Rome: Edizioni dell' Ateneo & Bizarri.**^**Euler, Leonhard (1765).*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.]*(in Latin). Rostock and Greifswald (Germany): A. F. Röse. p. 166. 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 tát any axis is the sum of all of the products, which arise, if the individual elements of the body toàn thân are multiplied by the square of their distances from the axis.)**^**Huygens, Christiaan (1673).*Horologium oscillatorium, sive de Motu pendulorum ad horologia aptato demonstrationes geometricae*(in Latin). p. 91.**^**Huygens, Christiaan (1977–1995). "Center of Oscillation (translation)". Translated by Mahoney, Michael S. Retrieved 22 May 2022.**^**Poisson, Siméon-Denis (1811).*Traité de mécanique, tome premier*. p. 67.**^**Thompson, Silvanus Phillips (1893).*Dynamo-electric machinery: A Manual For Students Of Electrotechnics*(4th ed.). Thủ đô New York, Harvard publishing teo. p. 108.**^**Thomson, James; Larmor, Joseph (1912).*Collected Papers in Physics and Engineering*. University Press. p. civ.**^**Pearson, Karl (October 1893). "Asymmetrical Frequency Curves".*Nature*.**48**(1252): 615–616. Bibcode:1893Natur..48..615P. doi:10.1038/048615a0. S2CID 4057772.**^**Venn, J. (September 1887). "The Law of Error".*Nature*.**36**(931): 411–412. Bibcode:1887Natur..36..411V. doi:10.1038/036411c0. S2CID 4098315.**^**Walker, Helen M. (1929).*Studies in the history of statistical method, with special reference to tát certain educational problems*. Baltimore, Williams & Wilkins Co. p. 71.

## External links[edit]

- Media related to tát Moment (physics) at Wikimedia Commons
- [1] A dictionary definition of moment.

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