Moment (physics)

Product of a distance and physical quantity

A moment is a mathematical expression involving the product of a distance and a physical quantity such as a force or electric charge. Moments are usually defined with respect to a fixed reference point and refer to physical quantities located some distance from the reference point. 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 the object. In principle, any physical quantity can be multiplied by a distance to produce a moment. Commonly used quantities include forces, masses, and electric charge distributions; a list of examples is provided later.

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:

${\displaystyle \mu _{n}=r^{n}\,Q,}$

where ${\displaystyle Q}$ 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:

${\displaystyle \mu _{n}=\int r^{n}\rho (r)\,dr}$

where ${\displaystyle \rho }$ 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 ${\displaystyle r^{n}\rho (r)}$ 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 ${\displaystyle r}$ 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 first moment: ${\displaystyle \mathbf {\tau } =rF}$, or, more generally, ${\displaystyle \mathbf {r} \times \mathbf {F} }$.
• Similarly, angular momentum is the 1st moment of momentum: ${\displaystyle \mathbf {L} =\mathbf {r} \times \mathbf {p} }$. Momentum itself is not a moment.
• The electric dipole moment is also a 1st moment: ${\displaystyle \mathbf {p} =q\,\mathbf {d} }$ for two opposite point charges or ${\textstyle \int \mathbf {r} \,\rho (\mathbf {r} )\,d^{3}r}$ for a distributed charge with charge density ${\displaystyle \rho (\mathbf {r} )}$.

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: ${\textstyle \mathbf {R} ={\frac {1}{M}}\sum _{i}\mathbf {r} _{i}m_{i}}$ for a collection of point masses, or ${\textstyle {\frac {1}{M}}\int \mathbf {r} \rho (\mathbf {r} )\,d^{3}r}$