Moment (physics)

In physics, a moment is a physical quantity which takes into account how a physical property is located or arranged. Moments are usually defined with respect to a fixed reference point or axis; they deal with physical quantities as measured at some distance from that reference point or axis. In principle, any physical quantity can be multiplied by distance to produce a moment. The moment of momentum of a particle is the distance (between the particle and some origin) times the particle's momentum, in other words its angular momentum. Likewise the moment of force acting on an object is the torque acting on the object. The moment of a point mass is distance (between the particle and some origin) times the particle's mass. Likewise for electric charge. In all these cases, the moment of a quantity depends on the origin.

One can also define a "moment of a moment", or "second moment", and iterate. In general, the nth moment of some property is the nth power of distance times that property.

The term "moment" is also prominent in multipole expansions. One can take an electric charge distribution, and calculate monopole moments (charge by itself), dipole moments (between pairs of charges), quadrupole moments (between pairs of dipoles), octupole moments (between pairs of quadrupoles), and so on. Each moment depends on how the charges are arranged in the distribution. In this case, the 2ⁿ-fold multipole moment has a dependence on the nth power of distance from the centre of the multipole configuration to a point in space.

Elaboration

In its most simple and basic form, a moment is the product of the distance to some point, raised to some power, multiplied by some physical quantity such as the force, charge, etc. 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 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, especially in the context of electric charge distributions.

Examples

• The moment of force, or torque, is a 1st moment: $\mathbf{\tau} = rF$, or, more generally, $\mathbf{r} \times \mathbf{F}$
• The electric dipole moment is also a 1st moment: $\mathbf{p} = q\,\mathbf{d}$ for two opposite point charges or $\int \mathbf{r}\,\rho(\mathbf{r})\,d^3r$ for a distributed charge with charge density $\rho(\mathbf{r})$
• The moment of inertia is a 2nd moment: $I = r^2 m$ for a point mass, $\sum_i r_i^2 m_i$ for a collection of point masses, or $\int r^2\rho(\mathbf{r}) \, d^3r$ for an object with mass distribution $\rho(\mathbf{r})$

Multipole moments

Assuming a density function that is finite and localized to a particular region, outside that region a 1/r potential may be expressed as a series of spherical harmonics:

$\Phi(\mathbf{r}) = \int \frac{\rho(\mathbf{r'})}{|\mathbf{r}-\mathbf{r'}|}d^3r' = \sum_{l=0}^{\infty} \sum_{m=-l}^{l} \left( \frac{4\pi}{2l+1} \right) q_{lm}\, \frac{Y_{lm}(\theta, \phi)}{r^{l+1}}$

The coefficients $q_{lm}$ are known as multipole moments, and take the form:

$q_{lm} = \int (r')^{l}\, \rho(\mathbf{r'})\, Y^*_{lm}(\theta',\phi')\, d^3r'$

where $\mathbf{r}'$ expressed in spherical coordinates $(r',\phi',\theta')$ 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 $\rho$ represents an electric charge density, the $q_{lm}$ are, in a sense, projections of the moments of electric charge: $q_{00}$ is the monopole moment; the $q_{1m}$ are projections of the dipole moment, the $q_{2m}$ 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 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,^[2][3] but has also been applied to the universe itself,^[4] being for example the technique employed by the WMAP and Planck experiments to analyze the Cosmic microwave background radiation.

History

The concept of moment in physics is derived from the mathematical concept of moments.^[5] . 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.

Federico Commandino, in 1565, translated into Latin from Archimedes:

The center of gravity of each solid figure is that point within it, about which on all sides parts of equal moment stand.^[6]

This was apparently the first use of the word moment (Latin, momentorum) in the sense which we now know it: a moment about a center of rotation.^[7]

The word moment was first used in Mechanics in its now rather old-fashioned sense of 'importance' or 'consequence,' and the moment of a force about an axis meant the importance of the force with respect to its power to generate in matter rotation about the axis... But the word 'moment' has also come to be used by analogy in a purely technical sense, in such expressions as the 'moment of a mass about an axis,' or 'the moment of an area with respect to a plane,' which require definition in each case. In those instances there is not always any corresponding physical idea, and such phrases stand, both historically and scientifically, on a different footing. - A. M. Worthington, 1920^[8]

See also

• 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 force – see also the article couple (mechanics)
• Moment of inertia – $(I = \Sigma m r^2)$ 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 – $(\mathbf{L} = \mathbf{r} \times m\mathbf{v})$ is the rotational analog of linear momentum.
• Magnetic moment – $(\mathbf{\mu}=I\mathbf{A})$ 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 $(\mathbf{p}=q \mathbf{d})$
• 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

References

1. J. D. Jackson, Classical Electrodynamics, 2nd edition, Wiley, New York, (1975). p. 137
2. M.A. Spackman, 'Molecular Electric Moments from X-Ray Diffraction Data, Chem. Rev., 92 (1992), p. 1769
3. 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, http://www.springer.com/978-3-642-30807-9
4. Baumann, D., TASI Lectures on Inflation, 2009, ArXiv e-prints, arXiv:0907.5424
5. 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.
6. Commandini, Federici (1565). Liber de Centro Gravitatis Solidorum., (at Google books)
7. Crew, Henry; Smith, Keith Kuenzi (1930). Mechanics for Students of Physics and Engineering. The Macmillan Company, New York. p. 25.
8. Worthington, Arthur M. (1920). Dynamics of Rotation. Longmans, Green and Co., London. p. 7., (at Google books)

External links

• A dictionary definition of moment.