# Moment of Force

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## moment of force

[′mō·mənt əv ′fȯrs]*The Great Soviet Encyclopedia*(1979). It might be outdated or ideologically biased.

## Moment of Force

a quantity that characterizes the rotational effect of a force acting on a rigid body; the moment of a force is a fundamental concept of mechanics. A distinction is made between the moment of a force about a center (a point) and about an axis.

The moment of a force about a center O is a vector quantity. The magnitude of the moment is *MO = Fh*, where *F* is the force and *h* is the arm, that is, the length of the perpendicular drawn from *O* to the line of action of the force (see Figure 1); the vector MO is perpendicular to the plane that contains the center *O* and the force F and is pointed in a direction such that the rotation caused by the force appears counterclockwise (in a right-handed coordinate system). The moment of a force is expressed as a vector product, by the equation M_{O} = [rF], where r is the radius vector drawn from *O* to the point of application

of the force. The dimensions of the moment of a force are *L ^{2}MT^{2}*, and the units of measurement are newton ⋅ meters (n ⋅ m), dyne ⋅ cm (1 N ⋅ m = 10

^{7}dyne ⋅ cm), or kilogram-force ⋅ meters.

The moment of a force about an axis is a scalar quantity equal to the projection, on this axis, of the moment of the force about any point *O* of the axis or equal to the numerical value of the moment of the projection *F _{xy}* of the force F on the

*xy*-plane, which is perpendicular to the

*z*-axis, taken with respect to the point of intersection of the axis with the plane. In other words,

*MZ* = M_{0} cos γ = ± *F _{xy}h_{1}*

The plus sign in this expression is selected when the rotation caused by the force F is counterclockwise when viewed from the positive end of the z-axis (also in a right-handed system). The moments of a force about the *x-, y-*, and z-axes can also be calculated from the formulas

*M _{x}* =

*YF*−

_{z}*ZF*

_{y}*M*=

_{y}*z*

*F*−

_{x}*x*

*F*

_{z}where *F _{x}, F_{y}*, and

*F*are the projections of the force F on the axes;

_{z}*x,y*, and

*z*are the coordinates of the point

*A*of application of the force.

If a system of forces has a resultant, then the moment of the system is calculated according to Varignon’s theorem.

S. M. TARG