Principle of Moments | Moment About an Axis

**Vector Representation:** The moment of a
force about an arbitrary moment center "O" can also be described by
the vector equation

(4) |

where is the position vector measured from the moment center to any point along the line of action of the force vector . The directional sense of the moment is found by first aligning the position and the force vectors tail to tail, then curling the four fingers of the right hand from to with the thumb pointing in the direction of the moment vector. The axis of the moment vector passes through the moment center and is perpendicular to the plane containing and . The magnitude of moment is measured in units of force times length (e.g., lb·in or N·m).

To calculate the moment of a force using the vector approach, we must know:

Force vector

Location of the moment center

Position vector (measured from the moment center to any point along the line of action of the force vector)

The following example describes the calculation of moment using the vector approach.

It is important to note that the position vector can be measured
from the moment center to **any** point along the line of action
of as
shown in the figure below. The resulting cross products in all cases will
be equal. That is

The reason all cross products in the above equation give the same answer is because

and

where θ is the angle between the tails of and .

Therefore, if we slide the force vector to any location along
its line of action, its moment with respect to any arbitrary point
or moment center will remain unchanged. This attribute is generally
known as the In the case of two or more forces, Eq. (4) is expanded to (5) |

where N is the number of forces and represents the position vector measured from the moment center to any point on the line of action of . The following example shows the calculation of moment due to multiple forces.