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To determine the resultant force of the concurrent force system shown in the figure, we need to break down each force into its components along the \(x\)- and \(y\)-axes. This is done by using trigonometry (sine and cosine) for each force based on the given angles.

### Steps to solve:

1. **Decompose Each Force into \(x\) and \(y\) Components**:
   Each force can be broken down using:
   \[
   F_x = F \cos(\theta)
   \]
   \[
   F_y = F \sin(\theta)
   \]
   where \(F_x\) and \(F_y\) are the horizontal and vertical components, and \(\theta\) is the angle made with the positive \(x\)-axis.

2. **Sum All Components**:
   Once all the forces are decomposed into \(x\) and \(y\) components, we sum them up to get the net \(x\)-component (\(F_{x_{total}}\)) and net \(y\)-component (\(F_{y_{total}}\)).

3. **Calculate the Resultant Force**:
   The magnitude of the resultant force \(F_{res}\) is:
   \[
   F_{res} = \sqrt{F_{x_{total}}^2 + F_{y_{total}}^2}
   \]
   The direction of the resultant force is given by:
   \[
   \theta_{res} = \tan^{-1} \left(\frac{F_{y_{total}}}{F_{x_{total}}}\right)
   \]

I will now calculate the components for each force and sum them up.

Let's proceed step by step to calculate the resultant.The resultant force of the concurrent force system is:

- **Resultant Force (Magnitude)**: \( 138.29 \, \text{kN} \)
- **Resultant Direction (Angle from the positive \(x\)-axis)**: \( 35.53^\circ \)
- **Total \(x\)-Component**: \( 112.54 \, \text{kN} \)
- **Total \(y\)-Component**: \( 80.36 \, \text{kN} \)

Would you like further details on the calculations or have any other questions?

Here are five related questions to expand your understanding:
1. How do you calculate the components of a force along the \(x\)- and \(y\)-axes?
2. What is the significance of the angle when resolving forces into components?
3. How would the process change if the forces were not concurrent?
4. What are the units of the resultant force and why?
5. How does vector addition differ from scalar addition?

**Tip**: Always make sure your angle measurements are consistent (degrees vs. radians) when solving problems involving trigonometry.

Determine the resultant force of the concurrent force system shown in Figure 2-21.

Learn how to determine the resultant force of a concurrent force system using vector addition and trigonometric decomposition. This problem involves resolving forces into components along the x- and y-axes. Suitable for engineering and physics students.

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