Question 2.12: The roof is supported by cables as shown in the photo. If th......
The roof is supported by cables as shown in the photo. If the cables exert forces F_{AB} = 100 N and F_{AC} = 120 N on the wall hook at A as shown in Fig. 2-38a, determine the resultant force acting at A. Express the result as a Cartesian vector.
The resultant force F_R is shown graphically in Fig. 2-38b. We. can express this force as a Cartesian vector by first formulating F_{AB} and F_{AC} as Cartesian vectors and then adding their components. The directions of F_{AB} and F_{AC} are specified by forming unit vectors u_{AB} and u_{AC} along the cables. These unit vectors are obtained from the associated position vectors r_{AB} and r_{AC}. With reference to Fig. 2-38a, to go from A to B, we must travel {-4k} m and, then {-4i} m. Thus,
r_{AB}=\{4i\ -\ 4k\}\ m \\ r_{AB}= \sqrt{(4\ m)^2 +(-4\ m)^2} =5.66\ m \\ F_{AB}= F_{AB}\left(\frac{r_{AB}}{r_{AB}} \right) = (100\ N)\left(\frac{4}{5.66}i\ -\ \frac{4}{5.66}k \right) \\ F_{AB}= \{70.7i\ -\ 70.7k\}\ NTo go from A to C, we must travel {-4k} m, then {2j} m, and finally {4j}. Thus,
r_{AC}=\{4i\ +\ 2j\ -\ 4k\}\ m \\ r_{AC}= \sqrt{(4\ m)^2 +(2\ m)^2+(-4\ m)^2} =6\ m \\ F_{AC}= F_{AC}\left(\frac{r_{AC}}{r_{AC}} \right) = (120\ N)\left(\frac{4}{6}i\ +\ \frac{2}{6}j\ -\ \frac{4}{6}k \right) \\ \quad \quad= \{80i + 40j – 80k\}\ N
The resultant force is therefore
F_R = F_{AB} + F_{AC} = { 70.7i – 70.7k } N + { 80i + 40j – 80k } N
= { 151i + 40j – 151k } N
