Moment of a Force
Moment of a Force: A force acting on a rigid body is a sliding vector: it may be moved along its line of action without changing the effect on the body. With the aid of the notion of the couple moment, we now will investigate how a force may be moved to a parallel line of action. Consider in Fig. 1 a force F whose line of action f is assumed to be moved to the line f′, which is parallel to f and passes through point 0. The perpendicular distance of the two lines is given by
h. As a first step, the forces F and −F are introduced on the line f′. These two forces are in equilibrium. One of the forces and the originally given force (action line f) represent a couple. The couple moment is given by its magnitude M(0) = hF and the sense of rotation. The system consisting of force F with action line f′
and couple moment M(0) = hF is statically equivalent to force F with action line f. The quantity M(0) = hF is called the moment of the force F about (with respect to) point 0. The superscript (0) indicates the reference point. The perpendicular distance of point 0 from the action line f is called the lever arm of force F with
respect to 0. The sense of rotation of the moment is given by the sense of rotation of force F about 0.
It should be noted that a couple moment is independent of the point of reference, whereas the magnitude and sense of rotation of the moment of a force depend on this point. Often it is advantageous to replace a force F by its Cartesian components Fx = Fx ex and Fy = Fy ey (Fig. 2). Adopting the commonly used sign convention that a moment is positive if it tends to rotate the body counterclockwise when viewed from above (), the moment of the force F about point 0 in Fig. 2 is given by M(0) = hF. Using the relations
Hence, the moment is equal to the sum of the moments of the force components about 0. Note the senses of rotation of the respective components: they determine the algebraic signs in the summation. Consider now two forces F1 and F2 and their resultant R (Fig. 3). The moments of the two forces with respect to point 0 are
Therefore, it is immaterial whether the forces are added first and then the moment is determined or if the sum of the individual moments is calculated. This property holds for an arbitrary number of forces:
The sum of the moments of single forces is equal to the moment of their resultant.