Moment (physics)

See also moment (mathematics) for a more abstract concept of moments that evolved from this concept of physics.

These two articles, one originally titled moment, the other titled moment (physics) need to get merged.

In physics, the moment M of a vector B is

M_A = r × B,

where r is the position where the quantity B is applied. If r is a vector relative to a point A, then the moment is the "moment M with respect to 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.

Since the moment is dependent on the given axis, the moment expression possess a common property when the observation axis is changed. If M_A is the moment around A, then the moment around the axis that goes through a point B is

M_B = M_A + R×B,

where R is the vector from point B to point A. This expression is usually referred to as the parallel axis theorem. For cases when the moment is the sum of individual "submoments", such as in rigid body dynamics where each particle of the body contribute to a moment, the axis change is the sum of a macroscopic and microscopic quantity,

M_B = R×B + ∑_ir_i×b_(i),

where B = ∑_ib_i , or in the form

M_B = R×B + M_A.

There are certain three kinds of important momentums in physics.

B = mv: Angular momentum, which is typically the cause of both rotation and translation movements of a body.
B = m ω×r: Moment of inertia, which, when describing rotational motions, plays the same role as the mass does for translation motion. But this "mass" is often dependent on time and also on the reference axis.
B = F: Torque, which is a force applied on a position of the body, but not on the center of mass. When no torque is applied, the angular momentum is conserved.

The concept of moment is important in physics and illustrates the magnification of force in rotational systems due to the distance between the application of the force and where the force is applied. The concept of the moment arm, this characteristic distance, is key to the operation of the lever, pulley, and most other simple machines that generate a mechanical advantage.

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=Fd, where F is the applied force, and d is the distance from the applied force to object.

An example:

A force of 85 N is applied to the lever arm shown above. The length of the moment arm is 0.6 m. The moment of force at the joint to the arm can be calculated by:

M=Fd

=85 x 0.6
=51 N.m clockwise

This moment can then be applied at any point along the arm. For instance, if the object being moved was located 0.2m from the pivot point, known as the fulcrum, the amount of force applied would be 51 N.m / 0.2m = 256 N, considerably more force than was put into the arm. For this reason a suitable arm and pivot point can allow human muscle to move objects that would otherwise be immobile. This is the principle behind the crowbar for instance, where the blade operates applies the entire force being applied at the far end of the bar. With a common 1m bar with a blade of perhaps 5cm long, the crowbar multiplies the force by 20x.