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To determine if the given system is in equilibrium, we need to check two primary conditions:

1. **Sum of Forces in Vertical Direction = 0** (to ensure translational equilibrium).
2. **Sum of Moments about any point = 0** (to ensure rotational equilibrium).

### Step 1: Sum of Forces in the Vertical Direction

There are three vertical forces acting on the system:

- \( 10 \, \text{kN} \) acting upwards (at the left).
- \( 5 \, \text{kN} \) acting downwards (in the middle).
- \( 5 \, \text{kN} \) acting downwards (on the right).

Summing these forces:
\[
\Sigma F_y = 10 \, \text{kN} \, (\text{up}) - 5 \, \text{kN} \, (\text{down}) - 5 \, \text{kN} \, (\text{down}) = 10 - 5 - 5 = 0 \, \text{kN}
\]
So, the sum of the vertical forces is zero, satisfying the first condition for equilibrium.

### Step 2: Sum of Moments about a Point

We can choose any point to calculate the moments. Let's take moments about the point where the 5 kN downward force is applied in the middle (at a distance of 5 m from the left side).

- **Moment due to the 10 kN force**: This force is 5 m away from the point, and it causes a **counterclockwise** moment.
  \[
  M_{10 \, \text{kN}} = 10 \, \text{kN} \times 5 \, \text{m} = 50 \, \text{kN} \cdot \text{m} \, (\text{counterclockwise})
  \]

- **Moment due to the right 5 kN force**: This force is 10 m away from the point, and it causes a **clockwise** moment.
  \[
  M_{5 \, \text{kN}} = 5 \, \text{kN} \times 10 \, \text{m} = 50 \, \text{kN} \cdot \text{m} \, (\text{clockwise})
  \]

Summing the moments:
\[
\Sigma M = 50 \, \text{kN} \cdot \text{m} \, (\text{counterclockwise}) - 50 \, \text{kN} \cdot \text{m} \, (\text{clockwise}) = 0
\]
The sum of the moments is zero, satisfying the second condition for equilibrium.

### Conclusion

Since both the sum of vertical forces and the sum of moments about any point are zero, the system is in equilibrium.

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Would you like a more detailed explanation or have any questions?

Here are 5 related questions you could explore:

1. What is the difference between translational and rotational equilibrium?
2. How do we determine the moment arm in various mechanical systems?
3. What happens if only one of the equilibrium conditions is satisfied?
4. Can we check the equilibrium by taking moments around different points?
5. How does the distribution of forces affect the stability of structures?

**Tip**: Always verify both the force balance and moment balance in mechanics problems to fully check equilibrium.

Learn how to determine if a system is in equilibrium by balancing forces and moments. This statics problem involves calculating the sum of vertical forces and moments to ensure that the system satisfies equilibrium conditions. Suitable for high school students studying physics or mechanics.

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