Question 3.10: Three cables are attached to a bracket as shown. Replace the......
Three cables are attached to a bracket as shown. Replace the forces exerted by the cables with an equivalent force-couple system at A .
We first determine the relative position vectors drawn from point A to the points of application of the various forces and resolve the forces into rectangular components. Observing that { F}_{B}\,=\,(700\mathrm{~N})L_{B E} where
L_{B E}=\frac{{{\overrightarrow{{B E}} }}}{B E}=\frac{75\mathrm{i}-\,150\mathrm{j}+50\mathrm{k}}{175}we have, using meters and newtons,
{r}_{R/A}={\overrightarrow{{ AB}} }=0.075\mathrm{i}+{\mathrm{0.050k}}\qquad{F}_{B}=300\mathrm{i}-{\mathrm{600j}}+200\mathrm{k} \\ \\ {r}_{C/ A}= \overrightarrow{A C} = 0.075{\mathrm{i}}-0.050\mathrm{k}\qquad{F}_{C}=707 {\mathrm{i}}\qquad\qquad-\,{ {7}}07 {\mathrm{k}} \\ \\ {r}_{D/A}=\overrightarrow{A D} = 0.100\mathrm{i}-0.100\mathrm{j}\qquad {F}_{D}=600\mathrm{i}+\mathrm{1039j}The force-couple system at A equivalent to the given forces consists of a force R~=~\Sigma {F} and a couple M_A^R~=~\Sigma {(r\times F)} . The force R is readily obtained by adding respectively the x, y , and z components of the forces:
{ R}\,=\,\Sigma{ F}\,=\,(1607~{ N})\mathrm{i}\,+\,(439~{\ N})\mathrm{j}\,-\,(507~{ N})\mathrm{k}The computation of {M}_{\mathrm{A}}^{R} will be facilitated if we express the moments of the forces in the form of determinants (Sec. 3.8):
{r}_{B/A}\times{F}_{B}= \begin{vmatrix} \mathrm i & \mathrm j & \mathrm k \\ 0.075 & 0 & 0.050 \\ 300 & -600 & 200 \end{vmatrix} =\,30\mathrm{i}\qquad\qquad\qquad-45\mathrm{k} \\ \\ {r}_{C/A}\times{F}_{C}= \begin{vmatrix} \mathrm i & \mathrm j & \mathrm k \\ 0.075 & 0 & -0.050 \\ 707 & 0 & -707 \end{vmatrix} =\qquad\qquad17.68 \mathrm j \\ \\ {r}_{D/A}\times{F}_{D}= \begin{vmatrix} \mathrm i & \mathrm j & \mathrm k \\ 0.100 & -0.100 & 0 \\ 600 & 1039 & 0 \end{vmatrix} =~~~\qquad\qquad\qquad163.9\mathrm{k}Adding the expressions obtained, we have
{M}_{\Lambda}^{R}=\Sigma(\mathrm{r}\times\mathrm{F})\,=\,(30\ \mathrm{N}\cdot\mathrm{m})\mathrm{i}\,+\,(17.68\ \mathrm {N}\cdot\mathrm{m})\mathrm{j}\,+\,(118.9\ \mathrm{N}\cdot\mathrm{m})\mathrm{k}The rectangular components of the force R and the couple {M}_{A}^{R} are shown in the adjoining sketch.
