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To determine the force in bar **AF** from the truss in the image, we'll use **method of joints** or **method of sections**, which are standard approaches for analyzing trusses.

### Assumptions:
1. The truss is static and in equilibrium.
2. The external loads are known: 10 kN at F and 5 kN at G.
3. Supports at A and C offer reactions in the x- and y-directions.

### Step-by-Step Plan:
1. **Determine the reactions at the supports** using equilibrium equations.
2. **Analyze Joint A** and use equilibrium conditions to solve for the force in member **AF**.

### Step 1: Summing moments about A
To determine \(C_y\), we take moments about point A.

\[
\Sigma M_A = 0
\]
This gives the equation:
\[
- (10 \, \text{kN})(3 \, \text{m}) - (5 \, \text{kN})(6 \, \text{m}) + C_y(6 \, \text{m}) = 0
\]
Solving for \(C_y\):
\[
C_y = \frac{(10 \times 3) + (5 \times 6)}{6} = \frac{30 + 30}{6} = 10 \, \text{kN}
\]

### Step 2: Summing forces in the y-direction
\[
\Sigma F_y = 0 \quad \Rightarrow \quad A_y + C_y = 10 \, \text{kN} + 5 \, \text{kN}
\]
\[
A_y + 10 = 15 \quad \Rightarrow \quad A_y = 5 \, \text{kN}
\]

### Step 3: Summing forces in the x-direction
No horizontal loads are applied, so:
\[
\Sigma F_x = 0 \quad \Rightarrow \quad A_x = C_x = 0
\]

### Step 4: Analyzing Joint A
Now, let's analyze joint A, where member **AF** meets. We have forces in member AB, AF, and the reaction \(A_y = 5 \, \text{kN}\).

Using equilibrium conditions for joint A:

#### In the y-direction:
\[
\Sigma F_y = 0 \quad \Rightarrow \quad F_{AF} \sin \theta = 5 \, \text{kN}
\]

The angle \(\theta\) of bar AF can be determined from the truss geometry. The vertical distance is 6 m, and the horizontal distance is 3 m. So:
\[
\sin \theta = \frac{6}{\sqrt{6^2 + 3^2}} = \frac{6}{\sqrt{36 + 9}} = \frac{6}{\sqrt{45}} = \frac{6}{3\sqrt{5}} = \frac{2}{\sqrt{5}}
\]

Substituting into the force equation:
\[
F_{AF} \times \frac{2}{\sqrt{5}} = 5 \quad \Rightarrow \quad F_{AF} = 5 \times \frac{\sqrt{5}}{2}
\]

Calculating:
\[
F_{AF} = 5 \times 1.118 = 5.59 \, \text{kN}
\]

Thus, the force in member **AF** is approximately **5.59 kN** (tension).

Would you like any further clarification on the steps or details?

Here are five questions to expand on the concept:
1. How do we differentiate between tension and compression in a truss?
2. What is the significance of reaction forces in truss analysis?
3. How does the method of sections compare to the method of joints for truss analysis?
4. What role does trigonometry play in determining the force components in a truss?
5. Can the method of moments be used to solve for forces in other types of structures?

**Tip**: Always double-check the geometry of the truss to accurately calculate angles and lengths for precise force analysis.

Learn how to solve for the force in bar AF of a truss using the method of joints. This detailed solution includes equilibrium equations, moment calculations, and trigonometric steps. External forces include 10 kN and 5 kN, and the truss structure has members with known geometry.

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