In physics, moment is a tendency to produce motion, especially, a rotation about a point or axis. Moments are generally defined with respect to a fixed reference point; they deal with physically measurable quantities such as forces or electric charge distributions in relation to that reference point.
In its most simple and basic form, a moment is the product of a distance from some reference point, raised to some power, multiplied by some physical quantity such as a force, charge distribution, etc.:
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 over space:
where 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 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 cancels 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, especially in the context of electric charge distributions.
- The moment of force, or torque, is a 1st moment: , or, more generally,
- The electric dipole moment is also a 1st moment: for two opposite point charges or for a distributed charge with charge density
- The moment of inertia is a 2nd moment: for a point mass, for a collection of point masses, or for an object with mass distribution
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. The conventions used in the referenced pages may be slightly different.)
This expression applies to 1/r potententials, examples of which include the electric potential, the magnetic potential and the gravitational potential. The expression can be used, for example, to approximate the strength of a field produced by a complex distribution of charges by calculating the first few multipole moments. For sufficiently large r, a good approximation can be obtained from just the monopole and dipole moments (for electric and gravitational fields; classical magnetic fields have no monopole moment).
This technique can also be used to determine the properties of an unknown distribution . 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 radiation.
The concept of moment in physics is derived from the mathematical concept of moments.[clarification needed]. The principle of moments is derived from Archimedes' discovery of the operating principle of the lever. In the lever one applies a force, in his day most often human muscle, to an arm, a beam of some sort. Archimedes noted that the amount of force applied to the object, the moment of force, is defined as M = rF, where F is the applied force, and r is the distance from the applied force to object. However, historical evolution of the term 'moment' and its use in different branches of science, such as mathematics, physics and engineering, is unclear.
- 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 force – see also the article couple (mechanics)
- Moment of inertia – is analogous to mass in discussions of rotational motion. It is a measure of an object's resistance to changes in its rotation rate
- Moment of momentum – is the rotational analog of linear momentum.
- Magnetic moment – is 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
- Torque or moment, the tendency of a force to rotate an object about an axis
- 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
- 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
- J. D. Jackson, Classical Electrodynamics, 2nd edition, Wiley, New York, (1975). p. 137
- Baumann, D., TASI Lectures on Inflation, 2009, ArXiv e-prints, arXiv:0907.5424
- Robertson, D.G.E.; Caldwell, G.E.; Hamill, J.; Kamen, G.; and Whittlesey, S.N. (2004) Research Methods in Biomechanics. Champaign, IL:Human Kinetics Publ., p. 285.
-  A clear definition of moment
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