Question 2.4: Determine the moment of the force F in Fig. (a) about point ...

The force F and point A lie in the xy-plane. Problems of this type may be solved using either the vector method (r × F) or the scalar method (Fd). For illustrative purposes, we use both methods.

Vector Solution

Recall that the three steps in the vector method are to write F in vector form, choose r and write it in vector form, and then evaluate the cross product r × F. Writing F in vector form, we get

There are several good choices for r in this problem, three of which are r_{AB}, r_{AC}, and r_{AD}. Choosing

r=r_{AB}= −120i + 180j

the moment about point A is

Expanding this determinant, we obtain

M_{A} =k[(800)(−120) + (800)(180)]=72000k N · mm

The magnitude of M_{A} is 72000 N · mm. Note that the direction of M_{A} is the positive z direction, which by the right-hand rule means that the moment about point A is counterclockwise.

Scalar Solution

In Fig. (b), we have resolved the force into the rectangular components F_{1} and F_{2} at point B. The moment arm of each component about point A (the perpendicular distance between A and the line of action of the force) can be determined by inspection. The moment arms are d_{1} = 180mm for F_{1} and d_{2} = 120mm for F_{2}, as shown in Fig. (b).The moment of F about A now can be obtained by the principle of moments. Nothing that the moment of F_{1} is counterclockwise, whereas the moment of F_{2} is clockwise, we obtain

Note that the sense of M_{A} is counterclockwise. Applying the right-hand rule, the vector representation of the moment is

M_{A}=72000k N·mm

Recall that a force, being a sliding vector, can be moved to any point on its line of action without changing its moment. In Fig. (c) we have moved F to point C. Now the moment arm of F_{1} about A is d_{1} = 90mm., and the moment arm of F_{2} is zero. Hence, the moment of F about A is

counterclockwise, as before.Another convenient location for F would be point D in Fig. (c). Here the moment arm of F_{1} about A is zero, whereas the moment arm of F_{2} is 120 mm, which again yields M_{A} = 72000N · mm counterclockwise.