# Moment (physics)

In physics, the moment of force (often just moment, though there are other quantities of that name such as moment of inertia) is a pseudovector quantity that represents the magnitude of force applied to a rotational system at a distance from the axis of rotation. The concept of the moment arm, this characteristic distance, is key modelling the operation of the lever, pulley, gear, and most other simple machines involving a mechanical advantage. The SI unit for moment is the newton meter (Nm).

Moment = Magnitude of Force × Force arm [the perpendicular distance to the pivot (Fd)]

## Overview

In general, the (first) moment M of a vector B is

${\displaystyle \mathbf {M_{A}} =\mathbf {r} \times \mathbf {B} \,}$

where

r is the position where quantity B is applied.
× represents the cross product of the vectors.

If r is a vector relative to point A, then the moment is the "moment M with respect to the axis that goes through the point A", or simply "moment M around A". If A is the origin, one often omits A and says simply moment.

## Parallel axis theorem

Since the moment is dependent on the given axis, the moment expression possess a common y,

${\displaystyle \mathbf {M_{B}} =\mathbf {R} \times \mathbf {B} +\sum _{i=0}{\mathbf {r_{i}} \times \mathbf {b_{i}} }\,}$

where

${\displaystyle \mathbf {B} =\sum _{i=0}{\mathbf {b_{i}} }\,}$

or alternatively,

${\displaystyle \mathbf {M_{B}} =\mathbf {R} \times \mathbf {B} +\mathbf {M_{A}} \,}$

## Principle of Moments

The Principle of Moments, also known as Varignon's Theorem states that the moment of a force is equal to the sum of the components of that force. This allows resolution of a moment into its component moments to solve more complex problems.

## Related quantities

Some notable physical quantities arise from the application of moments:

• Angular momentum (${\displaystyle L=I\omega }$ ), the rotational analog of momentum.
• Moment of inertia (${\displaystyle I=\sum mr^{2}}$), which is analogous to mass in discussions of rotational motion.
• Magnetic moment (${\displaystyle \mathbf {\mu } =I\mathbf {A} }$), a dipole moment measuring the strength and direction of a magnetic source.

## History

The principle of moments is derived from Archimedes' discovery of the operating principle of the lever[citation needed]. In the lever one applies a force (in his day most often human muscle), to an arm 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.