Step 1: Define the Force Components

The three forces $T_1$, $F$, and $T_2$ are acting along members A, B, and C, respectively. Since they are concurrent at point $O$, we use the equilibrium equations:

$\sum F_x = 0$ $\sum F_y = 0$

Force $T_1$ (Along Member A)

• Since member A is along the positive y-axis, its components are:
  $T_{1x} = 0$ $T_{1y} = 9 \text{ kN}$

Force $F$ (Along Member B)

• Member B has a given right triangle with sides 3, 4, and hypotenuse 5.
  • $\cos\theta_B = \frac{3}{5}$, $\sin\theta_B = \frac{4}{5}$
  • Force $F$ is acting toward the origin, so its components are:
    $F_x = -F \frac{3}{5}$ $F_y = -F \frac{4}{5}$

Force $T_2$ (Along Member C)

• Given $\theta = 90^\circ$, Member C is perpendicular to Member B.
• Member C must be at an angle of $-\theta_B$ because it's in the fourth quadrant:
  • $\cos\theta_C = \frac{4}{5}$, $\sin\theta_C = -\frac{3}{5}$
  • Since $T_2$ is pointing away from the origin:
    $T_{2x} = T_2 \frac{4}{5}$ $T_{2y} = -T_2 \frac{3}{5}$

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Step 2: Solve for $F$ using Equilibrium Conditions

Sum of Forces in $x$-Direction:

$\sum F_x = 0 \Rightarrow -F \frac{3}{5} + T_2 \frac{4}{5} = 0$ $T_2 \frac{4}{5} = F \frac{3}{5}$ $T_2 = \frac{3}{4} F$

Sum of Forces in $y$-Direction:

$\sum F_y = 0 \Rightarrow 9 - F \frac{4}{5} - T_2 \frac{3}{5} = 0$ Substituting $T_2 = \frac{3}{4} F$:

$9 - F \frac{4}{5} - \left(\frac{3}{4}F\right) \frac{3}{5} = 0$

$9 - \frac{4}{5}F - \frac{9}{20}F = 0$

$9 - \left(\frac{16}{20}F + \frac{9}{20}F \right) = 0$

$9 - \frac{25}{20}F = 0$

$\frac{25}{20}F = 9$

$F = 9 \times \frac{20}{25}$

$F = 7.20 \text{ kN}$

Final Answer:

$\boxed{7.20 \text{ kN}}$

Let me know if you need further details or clarifications!

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Related Questions:

1. How do we determine the components of a force along an inclined member?
2. Why is equilibrium used to solve for unknown forces in trusses?
3. How would the problem change if $\theta$ were not $90^\circ$?
4. What would happen if the force $F$ were acting away from the origin instead?
5. How do right triangle trigonometry principles help in force resolution?

Tip:

Always check the direction of forces carefully—whether they push toward or pull away from the origin—since it affects the sign of force components.