# Moment of a Force

Moment about an Axis: Consider the force passing through point A as shown in the figure.

 The view from the y axis reveals that is perpendicular to the x axis and that its line of action does not intersect the x axis. Therefore, the moment of about the x axis is found as where is the moment arm of the force with respect to the x axis. In this case, the moment axis is pointing in the positive x direction as shown. Similarly, the view from the x axis reveals that is perpendicular to the y axis as well. Hence, the moment of about the y axis is found as where is the moment arm of the force with respect to the y axis. In this case, the moment axis is pointing in the negative y direction. Also as to be expected, since is parallel to the z axis.

We can now expand this discussion to the case of calculating the moment about an arbitrary line aa. Two commonly encountered cases are described next.

 Case 1: The line of action of the force is perpendicular to aa, and that the two lines do not intersect each other. In this case, the moment about aa is found as (6)     An example of this case is shown in the figure where the line of action of the force is in z direction and line aa is in xy plane; clearly, they are perpendicular to each other. Case 2: The line of action of the force is NOT perpendicular to aa, and that the two lines do not intersect each other. In this case, the moment about aa is found in two steps using the vector approach. First, the moment about a point lying on line aa is calculated as Then, the projection of along line aa is found using the dot product (Magnitude of the component) The previous two equations can be combined into a triple scalar product as (7) If comes out negative, it simply means that its direction is opposite to that defined by This projection can also be put in vector form as (8)

Example 6

Example 7 (LiveMath)