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Draw the free-body diagram of member \(A B,\)  which is  supported by a roller at  \(A\) and a pin at  \(B\) . Explain the  significance of each force on the diagram. (See Fig. } 5-7 b .

 

 Determine the horizontal and vertical components  of reaction at the pin  \(A\)  and the tension developed in cable \(B C\) used to support the stecl frame. 

 

 

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

\( T_{BC}\left(\frac{4}{5}\right)(1)+T_{BC}\left(\frac{3}{5}\right)(3)-60(1)-30=0 \)

\(∴ T_{B C}=34.62 \mathrm{kN} \)

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

\(A_X-T_{BC}(\frac{3}{5})=0\)

\(∴ A _X=20.77 \mathrm{kN} \rightarrow \)

\( +\uparrow \sum F_ y=0 \)

\( A_y-60-T_{BC}\left(\frac{4}{5}\right)=0 \)

\(∴ A_ y=87.7 \mathrm{kN}\quad\uparrow \)

The horizontal beam is supported by springs at its ends. Each spring has a stiffness of \( k=5 \mathrm{kN} / \mathrm{m}\)  and is originally unstretched so that the beam is in the horizontal  position. Determine the angle of tilt of the beam if a load of \(800 \mathrm{N}\)  is applied at point  \(C\)  as shown. 

 

 

\( +\uparrow \sum F _y=0 \)

\( F_{A}+C_{B}=800 N \)

\( \sum M_{A}=0. \)

\( F_{B}(3)-800(1)=0 \)

\(∴ F_{B}=267 \mathrm{N} \)

\( F_{A}=800-F_{B}=800-267=533 N \)

\( F=k _s \quad \longrightarrow \begin{array}{l}{S_{A}=\frac{F_{A}}{k}=\frac{533}{5}=0.107 \mathrm{m}} \\ {S_{B}=\frac{F_{B}}{K}=\frac{267}{5}=0.053 \mathrm{m}}\end{array} \)

\( \theta=\tan ^{-1}\left(\frac{(S_ A-5 _B)}{3}\right) \)

\( =1.02^{\circ} \)

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