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Chapter 2

Q. 2.3

Determine (1) the moment of the force F about point C; and (2) the perpendicular distance between C and the line of action of F.

Determine (1) the moment of the force F about point C; and (2) the perpendicular distance between C and the line of action of F.

Step-by-Step

Verified Solution

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

F = 500λ_{AB} = 500\frac{\overrightarrow{AB} }{|\overrightarrow{AB} |}=500 \left(\frac{2i−4j+3k}{5.385}\right)

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:

M_{C} = r×F= r_{CA}×F=\left|\begin{matrix}i&j&k\\-2&0&0\\185.7&−371.4& 278.6\end{matrix} \right|

Expanding this determinant gives

M_{C}=557.2j+742.8k\:N\cdot m

Part 2

The magnitude of M_{C} is

M_{C} = \sqrt{(557.2)^2 + (742.8)^2 }= 928.6\:N\cdot m

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

d=\frac{M_{C}}{F}=\frac{928.6}{500}= 1.857\:m

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                 (2.4)

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.

Determine (1) the moment of the force F about point C; and (2) the perpendicular distance between C and the line of action of F.