the gusset plate is subjected to the forces of three members as shown in (figure 1). the forces are…

the gusset plate is subjected to the forces of three members as shown in (figure 1). the forces are concurrent at point o. take f = 2 kn. determine the magnitude of the tension force in member c for equilibrium. express your answer to three significant figures and include the appropriate units. determine the angle θ for equilibrium. express your answer in degrees to three significant figures.

the gusset plate is subjected to the forces of three members as shown in (figure 1). the forces are concurrent at point o. take f = 2 kn. determine the magnitude of the tension force in member c for equilibrium. express your answer to three significant figures and include the appropriate units. determine the angle θ for equilibrium. express your answer in degrees to three significant figures.

Answer

Explanation:

Step1: Set up equilibrium equations

For equilibrium, $\sum F_x = 0$ and $\sum F_y=0$. Let the force in member $C$ be $T$. Resolve force $F = 2$ kN into components. $F_x=F\cos30^{\circ}$ and $F_y = F\sin30^{\circ}$. $\sum F_x=F\cos30^{\circ}-T\cos\theta = 0$ and $\sum F_y=9 - F\sin30^{\circ}-T\sin\theta=0$. Given $F = 2$ kN, $F_x = 2\cos30^{\circ}=\sqrt{3}$ kN and $F_y=2\sin30^{\circ}=1$ kN.

Step2: Solve for $T$ from $x -$ component equation

From $\sum F_x = 0$, we have $T\cos\theta=\sqrt{3}$. From $\sum F_y = 0$, we have $9 - 1-T\sin\theta=0$, so $T\sin\theta=8$.

Step3: Square and add the two - equations

$(T\cos\theta)^2+(T\sin\theta)^2 = (\sqrt{3})^2 + 8^2$. Using $\sin^{2}\theta+\cos^{2}\theta = 1$, we get $T^{2}=3 + 64=67$.

Step4: Find the magnitude of $T$

$T=\sqrt{67}\approx 8.19$ kN.

Step5: Find the angle $\theta$

Since $T\cos\theta=\sqrt{3}$ and $T=\sqrt{67}$, then $\cos\theta=\frac{\sqrt{3}}{\sqrt{67}}$. $\theta=\cos^{-1}(\frac{\sqrt{3}}{\sqrt{67}})\approx 77.4^{\circ}$

Answer:

Magnitude of tension force in member $C$: $8.19$ kN Angle $\theta$: $77.4^{\circ}$