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 Determine the force components acting on the ball- and-socket at  \(A\) , the reaction at the roller  \(B\)  and the tension  on the cord  \(C D\)  needed for cquilibrium of the quarter  circular plate. 

 

 

\( \stackrel{+}\swarrow\sum F_{x}=0 \longrightarrow \quad A_ x=0 \)

\( \sum F_{y}=0 \longrightarrow A_ y=0 \)

\(\sum M_{x}=0 \)

\( -200(3 \sin 60)-200(3)+B_ z(3)=0 \)

\(∴ B _z=373.2 N \uparrow \)

\( \sum M_ y=0 \)

\( 350(2)+200\left(3 cos 60^{\circ}\right)-A_{z}(3)=0 \)

\(∴ A_z=333.3 N \uparrow \)

\(+\uparrow\sum F_z=0\longrightarrow 333.3 A_z+B_z+F_d=350-200-200=0\longrightarrow\ ∴ F_{CD}=43.5 N\uparrow\)

 Member  \(A B\)  is supported at  \(B\)  by a cable and at  \(A\)  by  a smooth fixed square rod which fits loosely through the  square hole of the collar. Determine the tension in cable  \(B C\) if the force  \(\mathbf{F}=\{-4.5 \mathbf{k}\} \mathrm{kN}\) .

\( \overline{r}_{B C}=\overline{r}_{C}-\overline{r}_{B} \)

\( =(0 i-2.4\hat j+1.8 k) \)

\( =(3.6 i-1.2\hat j+0 k) \)

\( =(-3.6\hat i-1.2\hat j+1.8 k) m \)

\( \left|\overrightarrow{r}_{B C}\right|=4 \cdot 2 m \)

\( \vec{U}_{B C}=\frac{\vec{r}_{B C}}{\left|\vec{r}_{B C}\right|} \)      \( =\left(-\frac{3.6}{4.2}\hat i-\frac{1.2}{4.2} \hat{j}+\frac{1.8}{4.2} k\right) \)

                         \( =\left(-\frac{6}{7} \hat{i}-\frac{2}{7} \hat{\jmath}+\frac{3}{7} k\right) \)

\( \vec{F}_{D C} =\left(-\frac{6}{7} F_{B C} \hat{\imath}\right. -\frac{2}{7} {F}_{DC} \hat{j} +\frac{3}{7}F_{BC}k) \)

\( +\uparrow\sum F_ z=0\quad \longrightarrow\quad\frac{3}{7} F_{B C}-F=0\longrightarrow\frac{3}{7} F_{B C}=F \)

\( ∴ F_{B C}=10.5 \mathrm{kN} \)

 

 

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