Question 3.SP.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.
STRATEGY: 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. Then, sum the forces and moments.
MODELING and ANALYSIS: Note that \pmb{F}_B=(700 N) \pmb{\lambda}_{BE} where
\pmb{\lambda}_{B E}=\frac{\overrightarrow{B E}}{B E}=\frac{75 \pmb{i}-150 \pmb{j}+50 \pmb{k}}{175}Using meters and newtons, the position and force vectors are
\begin{aligned}\pmb{r}_{B / A} &=\overrightarrow{A B}=0.075 \pmb{i}+0.050 \pmb{k} & \pmb{F}_B=300 \pmb{i}-600 \pmb{j}+200 \pmb{k} \\\pmb{r}_{C / A} &=\overrightarrow{A C}=0.075 \pmb{i}-0.050 \pmb{k} & \pmb{F}_C=707 \pmb{i} \quad-707 \pmb{k} \\\pmb{r}_{D / A} &=\overrightarrow{A D}=0.100 \pmb{i}-0.100 \pmb{j} & \pmb{F}_D=600 \pmb{i}+1039 \pmb{j}\end{aligned}The force-couple system at A equivalent to the given forces consists of a force R = ΣF and a couple \pmb{M}_A^R=\Sigma(\pmb{r} \times \pmb{F}). Obtain the force R by adding respectively the x, y, and z components of the forces:
R = ΣF =(1607 N)i +(439 N)j −(507 N)k
The computation of \pmb{M}_A^R is facilitated by expressing the moments of the forces in the form of determinants (Sec. 3.1F). Thus,
\pmb{r}_{B / A} \times \pmb{F}_B=\left|\begin{array}{ccc}\pmb{i} & \pmb{j} & \pmb{k} \\ 0.075 & 0 & 0.050 \\ 300 & -600 & 200\end{array}\right|=30 \pmb{i} \quad-45 \pmb{k}\pmb{r}_{C / A} \times \pmb{F}_C=\left|\begin{array}{ccc}\pmb{i} & \pmb{j} & \pmb{k} \\ 0.075 & 0 & -0.050 \\ 707 & 0 & -707\end{array}\right|=17.68 \pmb{j}
\pmb{r}_{D / A} \times \pmb{F}_D=\left|\begin{array}{ccc}\pmb{i} & \pmb{j} & \pmb{k} \\ 0.100 & -0.100 & 0 \\ 600 & 1039 & 0\end{array}\right|=\quad 163.9 \pmb{k}
Adding these expressions, you have
\pmb{M}_A^R=\Sigma(\pmb{r} \times \pmb{F})=(30 N \cdot m) \pmb{i}+(17.68 N \cdot m) \pmb{j}+(118.9 N \cdot m) \pmb{k}Figure 1 shows the rectangular components of the force R and the couple \pmb{M}_A^R.
REFLECT and THINK: The determinant approach to calculating moments shows its advantages in a general three-dimensional problem such as this.
