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The problem describes a horizontal beam in equilibrium under the action of three forces:

- \( F_1 = 10 \, N \) acting upwards at the left end,
- \( F_2 = 15 \, N \) acting downwards at the center, and
- \( F_3 = 5 \, N \) acting upwards at the right end.

The question asks what the necessary condition is for the beam to remain in equilibrium.

In physics and engineering, for a body (such as a beam) to be in static equilibrium, two main conditions must be satisfied:

1. **The sum of all forces must be zero** (this includes both vertical and horizontal forces).
2. **The sum of all moments (torques) about any point must be zero**.

Looking at the answer choices:

1. "La suma de los momentos debe ser cero." (The sum of the moments must be zero.)  
   This is one of the conditions of equilibrium.
   
2. "La suma de las fuerzas horizontales debe ser cero." (The sum of the horizontal forces must be zero.)  
   This would apply if there were horizontal forces, but none are mentioned in this case, so it is not relevant here.

3. "La suma de las fuerzas verticales debe ser cero." (The sum of the vertical forces must be zero.)  
   Since only vertical forces are mentioned, this condition is indeed necessary for vertical equilibrium.

4. "La viga debe estar en movimiento." (The beam must be in motion.)  
   This contradicts the concept of equilibrium, where the object must be stationary.

**Correct answers are:**
- The sum of the moments must be zero.
- The sum of the vertical forces must be zero.

Depending on the question type (if it's single-answer or multiple-choice), the appropriate condition would typically be the first one:  
**"La suma de los momentos debe ser cero."**  
This encompasses the broader concept of equilibrium.

Would you like further details or clarification on this?  
Here are 5 related questions you might find useful:

1. How are moments (torques) calculated in equilibrium problems?
2. What is the difference between static and dynamic equilibrium?
3. How do forces at different points affect the stability of a beam?
4. What happens if only the forces, but not the moments, are balanced?
5. How is equilibrium maintained in a real-world engineering system?

**Tip**: In equilibrium problems, always check both force balance (in all directions) and moment balance to ensure the system is fully stable.

En un sistema de ingeniería, una viga horizontal está en equilibrio bajo la acción de tres fuerzas: F1 = 10 N hacia arriba en el extremo izquierdo, F2 = 15 N hacia abajo en el centro, y F3 = 5 N hacia arriba en el extremo derecho. ¿Cuál es la condición necesaria para que la viga esté en equilibrio?

Learn how to solve an equilibrium problem involving a horizontal beam subjected to multiple vertical forces. This example uses the core principles of statics, focusing on the sum of forces and moments. Suitable for engineering students.

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