Editing Moment (physics) (section)

== 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.