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In
physics Physics is the natural science that studies matter, its fundamental constituents, its motion and behavior through space and time, and the related entities of energy and force. "Physical science is that department of knowledge which ...
, a moment is a mathematical expression involving the product of a
distance Distance is a numerical or occasionally qualitative measurement of how far apart objects or points are. In physics or everyday usage, distance may refer to a physical length or an estimation based on other criteria (e.g. "two counties over"). ...
and
physical quantity A physical quantity is a physical property of a material or system that can be quantified by measurement. A physical quantity can be expressed as a ''value'', which is the algebraic multiplication of a ' Numerical value ' and a ' Unit '. For exam ...
. 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 In physics, a force is an influence that can change the motion of an object. A force can cause an object with mass to change its velocity (e.g. moving from a state of rest), i.e., to accelerate. Force can also be described intuitively as a ...
, often called
torque In physics and mechanics, torque is the rotational equivalent of linear force. It is also referred to as the moment of force (also abbreviated to moment). It represents the capability of a force to produce change in the rotational motion of th ...
, 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 Mass is an intrinsic property of a body. It was traditionally believed to be related to the quantity of matter in a physical body, until the discovery of the atom and particle physics. It was found that different atoms and different ele ...
es, and
electric charge Electric charge is the physical property of matter that causes charged matter to experience a force when placed in an electromagnetic field. Electric charge can be ''positive'' or ''negative'' (commonly carried by protons and electrons res ...
distributions.


Elaboration

In its most basic form, a moment is the product of the distance to a point, raised to a power, and a physical quantity (such as force or electrical charge) at that point: : \mu_n = r^n\,Q, where Q 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 In mathematics, an integral assigns numbers to functions in a way that describes displacement, area, volume, and other concepts that arise by combining infinitesimal data. The process of finding integrals is called integration. Along with ...
of that quantity's density over space: :\mu_n = \int r^n \rho(r)\,dr where \rho 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 r^n \rho(r) 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 r 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'' = 1; the 2nd moment to ''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 A quadrupole or quadrapole is one of a sequence of configurations of things like electric charge or current, or gravitational mass that can exist in ideal form, but it is usually just part of a multipole expansion of a more complex structure ref ...
'', especially in the context of electric charge distributions.


Examples

* The ''moment of force'', or ''
torque In physics and mechanics, torque is the rotational equivalent of linear force. It is also referred to as the moment of force (also abbreviated to moment). It represents the capability of a force to produce change in the rotational motion of th ...
'', is a first moment: \mathbf = rF, or, more generally, \mathbf \times \mathbf. * Similarly, ''
angular momentum In physics, angular momentum (rarely, moment of momentum or rotational momentum) is the rotational analog of linear momentum. It is an important physical quantity because it is a conserved quantity—the total angular momentum of a closed syst ...
'' is the 1st moment of
momentum In Newtonian mechanics, momentum (more specifically linear momentum or translational momentum) is the product of the mass and velocity of an object. It is a vector quantity, possessing a magnitude and a direction. If is an object's mass ...
: \mathbf = \mathbf \times \mathbf. Note that momentum itself is ''not'' a moment. * The ''
electric dipole moment The electric dipole moment is a measure of the separation of positive and negative electrical charges within a system, that is, a measure of the system's overall polarity. The SI unit for electric dipole moment is the coulomb- meter (C⋅m). ...
'' is also a 1st moment: \mathbf = q\,\mathbf for two opposite point charges or \int \mathbf\,\rho(\mathbf)\,d^3r for a distributed charge with charge density \rho(\mathbf). Moments of mass: * The ''total
mass Mass is an intrinsic property of a body. It was traditionally believed to be related to the quantity of matter in a physical body, until the discovery of the atom and particle physics. It was found that different atoms and different ele ...
'' is the zeroth moment of mass. * The ''
center of mass In physics, the center of mass of a distribution of mass in space (sometimes referred to as the balance point) is the unique point where the weighted relative position of the distributed mass sums to zero. This is the point to which a force may ...
'' is the 1st moment of mass normalized by total mass: \mathbf = \frac 1M \sum_i \mathbf_i m_i for a collection of point masses, or \frac 1M \int \mathbf \rho(\mathbf) \, d^3r for an object with mass distribution \rho(\mathbf). * The ''
moment of inertia The moment of inertia, otherwise known as the mass moment of inertia, angular mass, second moment of mass, or most accurately, rotational inertia, of a rigid body is a quantity that determines the torque needed for a desired angular accele ...
'' is the 2nd moment of mass: I = r^2 m for a point mass, \sum_i r_i^2 m_i for a collection of point masses, or \int r^2\rho(\mathbf) \, d^3r for an object with mass distribution \rho(\mathbf). Note that 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''
potential Potential generally refers to a currently unrealized ability. The term is used in a wide variety of fields, from physics to the social sciences to indicate things that are in a state where they are able to change in ways ranging from the simple r ...
may be expressed as a series of
spherical harmonics In mathematics and physical science, spherical harmonics are special functions defined on the surface of a sphere. They are often employed in solving partial differential equations in many scientific fields. Since the spherical harmonics form ...
: : \Phi(\mathbf) = \int \frac \, d^3r' = \sum_^\infty \sum_^\ell \left( \frac \right) q_\, \frac The coefficients q_ are known as ''multipole moments'', and take the form: : q_ = \int (r')^\ell\, \rho(\mathbf)\, Y^*_(\theta',\varphi')\, d^3r' where \mathbf' expressed in spherical coordinates \left(r', \varphi', \theta'\right) is a variable of integration. A more complete treatment may be found in pages describing
multipole expansion A multipole expansion is a mathematical series representing a function that depends on angles—usually the two angles used in the spherical coordinate system (the polar and azimuthal angles) for three-dimensional Euclidean space, \R^3. Similar ...
or
spherical multipole moments Spherical multipole moments are the coefficients in a series expansion of a potential that varies inversely with the distance R to a source, ''i.e.'', as 1/''R''. Examples of such potentials are the electric potential, the magnetic potential ...
. (Note: the convention in the above equations was taken from Jackson – the conventions used in the referenced pages may be slightly different.) When \rho represents an electric charge density, the q_ are, in a sense, projections of the moments of electric charge: q_ is the monopole moment; the q_ are projections of the dipole moment, the q_ 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 The electric potential (also called the ''electric field potential'', potential drop, the electrostatic potential) is defined as the amount of work energy needed to move a unit of electric charge from a reference point to the specific point in ...
and the
gravitational potential In classical mechanics, the gravitational potential at a location is equal to the work (energy transferred) per unit mass that would be needed to move an object to that location from a fixed reference location. It is analogous to the electric ...
. 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 calculate interaction energies and intermolecular forces. The technique can also be used to determine the properties of an unknown distribution \rho. 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, but has also been applied to the universe itself, being for example the technique employed by the WMAP and Planck experiments to analyze the
cosmic microwave background In Big Bang cosmology the cosmic microwave background (CMB, CMBR) is electromagnetic radiation that is a remnant from an early stage of the universe, also known as "relic radiation". The CMB is faint cosmic background radiation filling all spac ...
radiation.


History

In works believed to stem from
Ancient Greece Ancient Greece ( el, Ἑλλάς, Hellás) was a northeastern Mediterranean civilization, existing from the Greek Dark Ages of the 12th–9th centuries BC to the end of classical antiquity ( AD 600), that comprised a loose collection of cu ...
, the concept of a moment is alluded to by the word ῥοπή (''rhopḗ'', "inclination") and composites like ἰσόρροπα (''isorropa'', "of equal inclinations"). The context of these works is
mechanics Mechanics (from Ancient Greek: μηχανική, ''mēkhanikḗ'', "of machines") is the area of mathematics and physics concerned with the relationships between force, matter, and motion among physical objects. Forces applied to objec ...
and
geometry Geometry (; ) is, with arithmetic, one of the oldest branches of mathematics. It is concerned with properties of space such as the distance, shape, size, and relative position of figures. A mathematician who works in the field of geometry is c ...
involving the
lever A lever is a simple machine consisting of a beam or rigid rod pivoted at a fixed hinge, or '' fulcrum''. A lever is a rigid body capable of rotating on a point on itself. On the basis of the locations of fulcrum, load and effort, the lever is d ...
. In particular, in extant works attributed to
Archimedes Archimedes of Syracuse (;; ) was a Greek mathematician, physicist, engineer, astronomer, and inventor from the ancient city of Syracuse in Sicily. Although few details of his life are known, he is regarded as one of the leading scientis ...
, the moment is pointed out in phrasings like: :" Commensurable magnitudes ( ) and Bare equally balanced () if their distances o the center Γ, i.e., ΑΓ and ΓΒare inversely proportional () to their weights ()." Moreover, in extant texts such as '' The Method of Mechanical Theorems'', moments are used to infer the
center of gravity In physics, the center of mass of a distribution of mass in space (sometimes referred to as the balance point) is the unique point where the weighted relative position of the distributed mass sums to zero. This is the point to which a force ma ...
, area, and volume of geometric figures. In 1269, William of Moerbeke translates various works of
Archimedes Archimedes of Syracuse (;; ) was a Greek mathematician, physicist, engineer, astronomer, and inventor from the ancient city of Syracuse in Sicily. Although few details of his life are known, he is regarded as one of the leading scientis ...
and Eutocious into
Latin Latin (, or , ) is a classical language belonging to the Italic languages, Italic branch of the Indo-European languages. Latin was originally a dialect spoken in the lower Tiber area (then known as Latium) around present-day Rome, but through ...
. The term ῥοπή is transliterated into ''ropen''. Around 1450, Jacobus Cremonensis translates ῥοπή in similar texts into the
Latin Latin (, or , ) is a classical language belonging to the Italic languages, Italic branch of the Indo-European languages. Latin was originally a dialect spoken in the lower Tiber area (then known as Latium) around present-day Rome, but through ...
term ''momentum'' ( "movement"). The same term is kept in a 1501 translation by Giorgio Valla, and subsequently by
Francesco Maurolico Francesco Maurolico (Latin: ''Franciscus Maurolycus''; Italian: ''Francesco Maurolico''; gr, Φραγκίσκος Μαυρόλυκος, 16 September 1494 - 21/22 July 1575) was a mathematician and astronomer from Sicily. He made contributions t ...
,
Federico Commandino Federico Commandino (1509 – 5 September 1575) was an Italian humanist and mathematician. Born in Urbino, he studied at Padua and at Ferrara, where he received his doctorate in medicine. He was most famous for his central role as translat ...
, Guidobaldo del Monte,
Adriaan van Roomen Adriaan van Roomen (29 September 1561 – 4 May 1615), also known as Adrianus Romanus, was a mathematician, professor of medicine and medical astronomer from the Duchy of Brabant in the Habsburg Netherlands who was active throughout Central Europ ...
, Florence Rivault, Francesco Buonamici,
Marin Mersenne Marin Mersenne, OM (also known as Marinus Mersennus or ''le Père'' Mersenne; ; 8 September 1588 – 1 September 1648) was a French polymath whose works touched a wide variety of fields. He is perhaps best known today among mathematicians for ...
, and
Galileo Galilei Galileo di Vincenzo Bonaiuti de' Galilei (15 February 1564 – 8 January 1642) was an Italian astronomer, physicist and engineer, sometimes described as a polymath. Commonly referred to as Galileo, his name was pronounced (, ). He ...
. That said, why was the word ''momentum'' chosen for the translation? One clue, according to
Treccani The ''Enciclopedia Italiana di Scienze, Lettere e Arti'' ( Italian for "Italian Encyclopedia of Science, Letters, and Arts"), best known as ''Treccani'' for its developer Giovanni Treccani or ''Enciclopedia Italiana'', is an Italian-language ...
, 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). In 1554,
Francesco Maurolico Francesco Maurolico (Latin: ''Franciscus Maurolycus''; Italian: ''Francesco Maurolico''; gr, Φραγκίσκος Μαυρόλυκος, 16 September 1494 - 21/22 July 1575) was a mathematician and astronomer from Sicily. He made contributions t ...
clarifies the Latin term ''momentum'' in the work ''Prologi sive sermones''. Here is a Latin to English translation as given by Marshall Clagett:
" ..equal weights at unequal distances do not weigh equally, but unequal weights t these unequal distances mayweigh 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 and weight—and this they call moment. Therefore, a body acquires weight from both quantity .e., sizeand quality .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 f the weightsare equal, as
Archimedes Archimedes of Syracuse (;; ) was a Greek mathematician, physicist, engineer, astronomer, and inventor from the ancient city of Syracuse in Sicily. Although few details of his life are known, he is regarded as one of the leading scientis ...
demonstrated in '' The Book on Equal Moments''. Therefore, weights or athermoments 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 physics, the center of mass of a distribution of mass in space (sometimes referred to as the balance point) is the unique point where the weighted relative position of the distributed mass sums to zero. This is the point to which a force ma ...
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.
Aristotle Aristotle (; grc-gre, Ἀριστοτέλης ''Aristotélēs'', ; 384–322 BC) was a Greek philosopher and polymath during the Classical period in Ancient Greece. Taught by Plato, he was the founder of the Peripatetic school of ...
investigates it in ''On Mechanical Questions'', and it is completely different from hethree aforesaid owers or magnitudes ..
in 1586, Simon Stevin uses the Dutch term ''staltwicht'' ("parked weight") for momentum in '' De Beghinselen Der Weeghconst''. In 1632,
Galileo Galilei Galileo di Vincenzo Bonaiuti de' Galilei (15 February 1564 – 8 January 1642) was an Italian astronomer, physicist and engineer, sometimes described as a polymath. Commonly referred to as Galileo, his name was pronounced (, ). He ...
publishes '' Dialogue Concerning the Two Chief World Systems'' and uses the Italian ''momento'' with many meanings, including the one of his predecessors. In 1643, Thomas Salusbury translates some of Galilei's works into
English English usually refers to: * English language * English people English may also refer to: Peoples, culture, and language * ''English'', an adjective for something of, from, or related to England ** English national ...
. Salusbury translates Latin ''momentum'' and Italian ''momento'' into the English term ''moment''. In 1765, the
Latin Latin (, or , ) is a classical language belonging to the Italic languages, Italic branch of the Indo-European languages. Latin was originally a dialect spoken in the lower Tiber area (then known as Latium) around present-day Rome, but through ...
term ''momentum inertiae'' (
English English usually refers to: * English language * English people English may also refer to: Peoples, culture, and language * ''English'', an adjective for something of, from, or related to England ** English national ...
: ''
moment of inertia The moment of inertia, otherwise known as the mass moment of inertia, angular mass, second moment of mass, or most accurately, rotational inertia, of a rigid body is a quantity that determines the torque needed for a desired angular accele ...
'') is used by
Leonhard Euler Leonhard Euler ( , ; 15 April 170718 September 1783) was a Swiss mathematician, physicist, astronomer, geographer, logician and engineer who founded the studies of graph theory and topology and made pioneering and influential discoveries ...
to refer to one of
Christiaan Huygens Christiaan Huygens, Lord of Zeelhem, ( , , ; also spelled Huyghens; la, Hugenius; 14 April 1629 – 8 July 1695) was a Dutch mathematician, physicist, engineer, astronomer, and inventor, who is regarded as one of the greatest scientists o ...
's quantities in '' Horologium Oscillatorium''. 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.) Huygens 1673 work involving finding the center of oscillation had been stimulated by
Marin Mersenne Marin Mersenne, OM (also known as Marinus Mersennus or ''le Père'' Mersenne; ; 8 September 1588 – 1 September 1648) was a French polymath whose works touched a wide variety of fields. He is perhaps best known today among mathematicians for ...
, who suggested it to him in 1646. In 1811, the French term ''moment d'une force'' (
English English usually refers to: * English language * English people English may also refer to: Peoples, culture, and language * ''English'', an adjective for something of, from, or related to England ** English national ...
: ''moment of force'') with respect to a point and plane is used by Siméon Denis Poisson in ''Traité de mécanique''
An English translation
appears in 1842. In 1884, the term ''
torque In physics and mechanics, torque is the rotational equivalent of linear force. It is also referred to as the moment of force (also abbreviated to moment). It represents the capability of a force to produce change in the rotational motion of th ...
'' is suggested by James Thomson in the context of measuring rotational forces of
machine A machine is a physical system using power to apply forces and control movement to perform an action. The term is commonly applied to artificial devices, such as those employing engines or motors, but also to natural biological macromolecul ...
s (with
propeller A propeller (colloquially often called a screw if on a ship or an airscrew if on an aircraft) is a device with a rotating hub and radiating blades that are set at a pitch to form a helical spiral which, when rotated, exerts linear thrust upon ...
s and
rotor Rotor may refer to: Science and technology Engineering * Rotor (electric), the non-stationary part of an alternator or electric motor, operating with a stationary element so called the stator *Helicopter rotor, the rotary wing(s) of a rotorcraft ...
s). Today, a
dynamometer A dynamometer or "dyno" for short, is a device for simultaneously measuring the torque and rotational speed ( RPM) of an engine, motor or other rotating prime mover so that its instantaneous power may be calculated, and usually displayed by ...
is used to measure the torque of machines. In 1893,
Karl Pearson Karl Pearson (; born Carl Pearson; 27 March 1857 – 27 April 1936) was an English mathematician and biostatistician. He has been credited with establishing the discipline of mathematical statistics. He founded the world's first university st ...
uses the term ''n-th moment'' and \mu_n in the context of
curve-fitting Curve fitting is the process of constructing a curve, or mathematical function, that has the best fit to a series of data points, possibly subject to constraints. Curve fitting can involve either interpolation, where an exact fit to the data is ...
scientific measurements. Pearson wrote in response to
John Venn John Venn, Fellow of the Royal Society, FRS, Fellow of the Society of Antiquaries of London, FSA (4 August 1834 – 4 April 1923) was an English mathematician, logician and philosopher noted for introducing Venn diagrams, which are used in l ...
, who, some years earlier, observed a peculiar pattern involving
meteorological Meteorology is a branch of the atmospheric sciences (which include atmospheric chemistry and physics) with a major focus on weather forecasting. The study of meteorology dates back millennia, though significant progress in meteorology did not ...
data and asked for an explanation of its cause. In Pearson's response, this analogy is used: the mechanical "center of gravity" is the
mean There are several kinds of mean in mathematics, especially in statistics. Each mean serves to summarize a given group of data, often to better understand the overall value ( magnitude and sign) of a given data set. For a data set, the '' ar ...
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 Statistics (from German: '' Statistik'', "description of a state, a country") is the discipline that concerns the collection, organization, analysis, interpretation, and presentation of data. In applying statistics to a scientific, industr ...
function involving the sum of the th powers of deviations was noticed by several earlier, including
Laplace Pierre-Simon, marquis de Laplace (; ; 23 March 1749 – 5 March 1827) was a French scholar and polymath whose work was important to the development of engineering, mathematics, statistics, physics, astronomy, and philosophy. He summarized ...
,
Kramp Kramp is a surname. Notable people with the surname include: *Christian Kramp Christian Kramp (8 July 1760 – 13 May 1826) was a French mathematician, who worked primarily with factorials. Christian Kramp's father was his teacher at grammar ...
,
Gauss Johann Carl Friedrich Gauss (; german: Gauß ; la, Carolus Fridericus Gauss; 30 April 177723 February 1855) was a German mathematician and physicist who made significant contributions to many fields in mathematics and science. Sometimes refer ...
,
Encke Encke may refer to: *Johann Franz Encke (1791–1865), a nineteenth-century German astronomer ** Encke (crater), a lunar crater ** Encke Division, a dark gap in the rings of Saturn **Comet Encke, a short-period comet ***Encke (horse) (2009–2014) ...
, Czuber, Quetelet, and De Forest.


See also

*
Torque In physics and mechanics, torque is the rotational equivalent of linear force. It is also referred to as the moment of force (also abbreviated to moment). It represents the capability of a force to produce change in the rotational motion of th ...
(or ''moment of force''), see also the article
couple (mechanics) In mechanics, a couple is a system of forces with a resultant (a.k.a. net or sum) moment of force but no resultant force.''Dynamics, Theory and Applications'' by T.R. Kane and D.A. Levinson, 1985, pp. 90-99Free download/ref> A better term is forc ...
*
Moment (mathematics) In mathematics, the moments of a function are certain quantitative measures related to the shape of the function's graph. If the function represents mass density, then the zeroth moment is the total mass, the first moment (normalized by total m ...
*
Mechanical equilibrium In classical mechanics, a particle is in mechanical equilibrium if the net force on that particle is zero. By extension, a physical system made up of many parts is in mechanical equilibrium if the net force on each of its individual parts is zero ...
, 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 The moment of inertia, otherwise known as the mass moment of inertia, angular mass, second moment of mass, or most accurately, rotational inertia, of a rigid body is a quantity that determines the torque needed for a desired angular accele ...
\left(I = \Sigma m r^2\right), analogous to
mass Mass is an intrinsic property of a body. It was traditionally believed to be related to the quantity of matter in a physical body, until the discovery of the atom and particle physics. It was found that different atoms and different ele ...
in discussions of rotational motion. It is a measure of an object's resistance to changes in its rotation rate * Moment of momentum (\mathbf = \mathbf \times m\mathbf), the rotational analog of linear
momentum In Newtonian mechanics, momentum (more specifically linear momentum or translational momentum) is the product of the mass and velocity of an object. It is a vector quantity, possessing a magnitude and a direction. If is an object's mass ...
. *
Magnetic moment In electromagnetism, the magnetic moment is the magnetic strength and orientation of a magnet or other object that produces a magnetic field. Examples of objects that have magnetic moments include loops of electric current (such as electroma ...
\left(\mathbf = I\mathbf\right), a
dipole In physics, a dipole () is an electromagnetic phenomenon which occurs in two ways: *An electric dipole deals with the separation of the positive and negative electric charges found in any electromagnetic system. A simple example of this system ...
moment measuring the strength and direction of a magnetic source. *
Electric dipole moment The electric dipole moment is a measure of the separation of positive and negative electrical charges within a system, that is, a measure of the system's overall polarity. The SI unit for electric dipole moment is the coulomb- meter (C⋅m). ...
, 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 (\mathbf = q \mathbf) *
Bending moment In solid mechanics, a bending moment is the reaction induced in a structural element when an external force or moment is applied to the element, causing the element to bend. The most common or simplest structural element subjected to bending mo ...
, a moment that results in the bending of a structural element *
First moment of area The first moment of area is based on the mathematical construct moments in metric spaces. It is a measure of the spatial distribution of a shape in relation to an axis. The first moment of area of a shape, about a certain axis, equals the sum ove ...
, a property of an object related to its resistance to shear stress *
Second moment of area The second moment of area, or second area moment, or quadratic moment of area and also known as the area moment of inertia, is a geometrical property of an area which reflects how its points are distributed with regard to an arbitrary axis. The ...
, a property of an object related to its resistance to bending and deflection *
Polar moment of inertia The second polar moment of area, also known (incorrectly, colloquially) as "polar moment of inertia" or even "moment of inertia", is a quantity used to describe resistance to torsional deformation (deflection), in cylindrical (or non-cylindrical) ...
, a property of an object related to its resistance to torsion * Image moments, statistical properties of an image *
Seismic moment Seismic moment is a quantity used by seismologists to measure the size of an earthquake. The scalar seismic moment M_0 is defined by the equation M_0=\mu AD, where *\mu is the shear modulus of the rocks involved in the earthquake (in pascals (Pa), ...
, quantity used to measure the size of an earthquake * Plasma moments, fluid description of plasma in terms of density, velocity and pressure *
List of area moments of inertia The following is a list of second moments of area of some shapes. The second moment of area, also known as area moment of inertia, is a geometrical property of an area which reflects how its points are distributed with respect to an arbitrary axis ...
*
List of moments of inertia Moment of inertia, denoted by , measures the extent to which an object resists rotational acceleration about a particular axis, it is the rotational analogue to mass (which determines an object's resistance to ''linear'' acceleration). The momen ...
*
Multipole expansion A multipole expansion is a mathematical series representing a function that depends on angles—usually the two angles used in the spherical coordinate system (the polar and azimuthal angles) for three-dimensional Euclidean space, \R^3. Similar ...
*
Spherical multipole moments Spherical multipole moments are the coefficients in a series expansion of a potential that varies inversely with the distance R to a source, ''i.e.'', as 1/''R''. Examples of such potentials are the electric potential, the magnetic potential ...


Notes


References


External links

*{{Commonscatinline, Moment (physics)

A dictionary definition of moment. Length Physical quantities el:Ροπή sq:Momenti