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Chapter 2

Q. 2.2

Three ropes are attached to the post at A in Fig. (a). The forces in the ropes are F_{1}=260\:lb, F_{2} = 75\:lb, and F_{3} = 60\:lb. Determine (1) the magnitude of the force R that is equivalent to the three forces shown, and (2) the coordinates of the point where the line of action of R intersects the yz-plane.

Three ropes are attached to the post at A in Fig. (a). The forces in the ropes are F1 = 260 lb, F2 = 75 lb, and F3 = 60 lb. Determine (1) the magnitude of the force R that is equivalent to the three forces shown, and (2) the coordinates of the point where the line of action of R intersects

Step-by-Step

Verified Solution

Part 1
The forces are concurrent at point A and thus may be added immediately. Because the forces do not lie in a coordinate plane, it is convenient to use vector notation. One method for expressing each of the forces in vector notation is to use the form F=Fλ, where λ is the unit vector in the direction of the force F. Thus

F_{1}=260λ_{AB}=260\frac{\overrightarrow{AB}}{|\overrightarrow{AB|}}=260\left(\frac{−3i−12j+4k}{13} \right) =−60i−240j+80k\:lb

 

F_{2}=75λ_{AC}=75\frac{\overrightarrow{AC}}{|\overrightarrow{AC|}}=75\left(\frac{−3i+4k}{5} \right) =−45i+60k\:lb

 

F_{3}=-60j\:lb

The resultant force is given by
R =\Sigma{F}=F_{1}+F_{2}+F_{3}=(−60i−240j+80k)+(−45i+60k)+(−60j)=−105i−300j+140k\:lb

The magnitude of R is
R =\sqrt{(−105)^2+(−300)^2+(140)^2}=347.3\:lb

Part 2
The unit vector λ in the direction of R is
λ=\frac{R}{R}=\frac{−105i−300j+140k}{347.3}=−0.3023i−0.8638j+0.4031k

Let D be the point where λ intersects the yz-plane, as shown in Fig.\:(b). The coordinates of D can be determined by proportions:
\frac{|λ_{x}|}{3}=\frac{|λ_{y}|}{12-y_{D}}=\frac{|λ_{z}|}{z_{D}}

Substituting the components of λ, this becomes
\frac{0.3023}{3}=\frac{0.8638}{12-y_{D}}=\frac{0.4031}{z_{D}}

yielding
y_{D} =3.43\:ft             z_{D}=4.0\:ft

Three ropes are attached to the post at A in Fig. (a). The forces in the ropes are F1 = 260 lb, F2 = 75 lb, and F3 = 60 lb. Determine (1) the magnitude of the force R that is equivalent to the three forces shown, and (2) the coordinates of the point where the line of action of R intersects