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To determine if this system is in equilibrium, we need to analyze both the forces and the moments acting on the system.

### Step 1: Sum of Forces
For a system to be in equilibrium, the sum of forces in both the vertical and horizontal directions should be zero.

#### Vertical Forces:
- \( 10 \, \text{kN} \) acting downwards on the left.
- \( 6 \, \text{kN} \) acting downwards on the right.
- \( 4 \, \text{kN} \) acting upwards from the center.

**Sum of vertical forces:**
\[
F_{\text{vertical}} = -10 \, \text{kN} - 6 \, \text{kN} + 4 \, \text{kN} = -12 \, \text{kN}
\]
Since the sum of vertical forces is not zero, **the system is not in equilibrium** in terms of forces.

### Step 2: Sum of Moments
For rotational equilibrium, the sum of moments about any point should also be zero. We can calculate the moments about the center point (where the \( 4 \, \text{kN} \) force is applied) because it's the simplest option.

- Moment due to the \( 10 \, \text{kN} \) force at a distance of \( 5 \, \text{m} \):
\[
M_1 = 10 \, \text{kN} \times 5 \, \text{m} = 50 \, \text{kNm} \, (\text{clockwise})
\]

- Moment due to the \( 6 \, \text{kN} \) force at a distance of \( 5 \, \text{m} \):
\[
M_2 = 6 \, \text{kN} \times 5 \, \text{m} = 30 \, \text{kNm} \, (\text{counterclockwise})
\]

- There is also an additional clockwise moment of \( 80 \, \text{kNm} \) applied.

**Sum of moments:**
\[
M_{\text{total}} = 80 \, \text{kNm} + 50 \, \text{kNm} - 30 \, \text{kNm} = 100 \, \text{kNm} \, (\text{clockwise})
\]
Since the sum of moments is not zero, the system is not in rotational equilibrium.

### Conclusion:
The system is **not in equilibrium** because both the sum of forces and the sum of moments are not zero.

Do you have any questions, or would you like more details on any specific part?

Here are 5 related questions:
1. How do we determine equilibrium for forces acting in both the vertical and horizontal directions?
2. What would happen if the sum of the moments were zero but the forces were not?
3. Can you explain the concept of a moment and how it relates to rotational equilibrium?
4. What is the significance of the direction (clockwise vs. counterclockwise) in moment calculations?
5. How would this system change if an additional vertical force was added?

**Tip:** When calculating moments, always choose a point of rotation that simplifies the math, such as a point where multiple forces intersect.

Analyze whether a mechanical system with forces of 10 kN, 6 kN, and 4 kN, and a moment of 80 kNm is in equilibrium. The solution includes the sum of forces and moments to check both translational and rotational equilibrium.

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