Question 2.3: Determine (1) the moment of the force F about point C; and (...

Part 1

The moment of a force about point C can be computed by either the scalar method (M_{C} = Fd), or the vector method (M_{C} =r×F). In this problem the scalar method would be inconvenient, because we have no easy means of determining d (the perpendicular distance between C and the line AB). Therefore, we use the vector method, which consists of the following three steps: (1) write F in vector form; (2) choose an r, and write it in vector form; and (3) compute M_{C} =r × F.

Step 1: Write F in vector form.Referring to the figure, we obtain

which yields

F=185.7i − 371.4j + 278.6k N

Step 2: Choose an r, and write it in vector form. The vector r is a vector from point C to any point on the line of action of F. From the figure we see that there are two convenient choices for r—the vector from point C to either point A or point B. As shown in the figure, let us choose r to be r_{CA}. (As an exercise, you may wish to solve this problem by choosing r to be the vector from point C to point B.) Now we have

r=r_{CA}= −2i m

Step 3: Calculate M_{C} = r × F. The easiest method for evaluating the cross product is to use the determinant expansion:

Expanding this determinant gives

M_{C} =557.2j + 742.8k N · m

Part 2

The magnitude of M_{C} is

The perpendicular distance d from point C to the line of action of F may be determined by

Observe that, instead of using the perpendicular distance to determine the moment, we have used the moment to determine the perpendicular distance.Caution A common mistake is choosing the wrong sense for r in Eq. (2.4) M_{o}=r × F Note that r is directed from the moment center to the line of action of F. If the sense of r is reversed, r × F will yield the correct magnitude of the moment, but the wrong sense. To avoid this pitfall, it is strongly recommended that you draw r on your sketch before attempting to write it in vector form.