The problem asks to determine the forces in members $AB$ and $AC$ of the truss using the method of joints. The given truss has a force of $50 \, \text{kN}$ applied at joint $C$. The geometry includes a horizontal span of $3.6 \, \text{m}$ and a vertical height of $6 \, \text{m}$.

Solution Using the Method of Joints

Step 1: Analyze Joint A

• Identify forces acting on the joint:
  1. Force in member $AB$ (let's denote it as $F_{AB}$).
  2. Force in member $AC$ (denote it as $F_{AC}$).
  3. Reaction forces at the supports (assumed to be at $A$).

Step 2: Resolve Reaction Forces at A

• Since the truss is symmetric and a vertical force of $50 \, \text{kN}$ is applied at joint $C$, the vertical reaction force at $A$ is $25 \, \text{kN}$.

Step 3: Geometry of Members

• The slope of member $AC$ is determined by its rise and run:
  • Vertical height = $6 \, \text{m}$
  • Horizontal length = $3.6 \, \text{m}$
  • The length of $AC$: $AC = \sqrt{(3.6)^2 + (6)^2} = \sqrt{12.96 + 36} = \sqrt{48.96} = 7 \, \text{m}$
• The angle $\theta$ between member $AC$ and the horizontal can be found using: $\tan \theta = \frac{6}{3.6}, \quad \theta = \tan^{-1}\left(\frac{6}{3.6}\right)$

Step 4: Apply Equilibrium Equations at Joint A

1. Horizontal Force Balance ($\sum F_x = 0$): $F_{AB} - F_{AC} \cdot \cos(\theta) = 0$
2. Vertical Force Balance ($\sum F_y = 0$): $25 \, \text{kN} - F_{AC} \cdot \sin(\theta) = 0$

From these equations, solve for $F_{AC}$ and $F_{AB}$.

Determining Tension or Compression:

• $F_{AC}$ is pulling away from the joint: Tension.
• $F_{AB}$ is pushing towards the joint: Compression.

Would you like a detailed step-by-step solution for the force calculations, or do you have any specific questions?

Here are some related questions you might find helpful:

1. How do you calculate the reaction forces in a truss system?
2. What are the conditions for equilibrium in a truss joint?
3. How do you distinguish between tension and compression in truss members?
4. How does the method of sections compare to the method of joints?
5. How can trigonometric identities simplify solving truss problems?

Tip: Always draw a free-body diagram for each joint to visualize forces clearly!