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The image shows a problem where eight forces \(F_1, F_2, \dots, F_8\) are applied to a point in a plane. These forces are represented in a sketch with their respective angles, and the goal is to find the force \(F_H\) that would balance the system (keep it in equilibrium).

We have the following information:

### Forces:
- \( F_1 = 2000 \, \text{N} \)
- \( F_2 = 4000 \, \text{N} \)
- \( F_3 = 3000 \, \text{N} \)
- \( F_4 = 3500 \, \text{N} \)
- \( F_5 = 6000 \, \text{N} \)
- \( F_6 = 4500 \, \text{N} \)
- \( F_7 = 3500 \, \text{N} \)
- \( F_8 = 6000 \, \text{N} \)

### Angles (in degrees):
- \( F_1 \) is along the positive x-axis (0°)
- \( F_2 \) is at 90° (positive y-axis)
- \( F_3 \) is at 135°
- \( F_4 \) is at 45°
- \( F_5 \) is at 150°
- \( F_6 \) is at 210°
- \( F_7 \) is at 315°
- \( F_8 \) is at 30°

### Task:
We need to find \( F_H \), which is the force that keeps the system in equilibrium. This means that the sum of all the horizontal components (x-components) and vertical components (y-components) of the forces must add up to zero.

We will first resolve each force into its x and y components using trigonometry, and then sum the components along each axis.

Let me calculate that.The force \( F_H \) required to keep the system in equilibrium is approximately:

- Magnitude: \( 9915.16 \, \text{N} \)
- Angle: \( -95.39^\circ \) (or \( 95.39^\circ \) below the negative x-axis)

This means that \( F_H \) should act in a direction slightly below the negative y-axis to balance the forces.

Would you like more details on the calculations, or do you have any further questions?

Here are some related questions to deepen understanding:
1. How do you resolve a force into its x and y components using trigonometry?
2. Why does the equilibrium condition require the sum of forces to be zero?
3. What would happen if \( F_H \) had a different magnitude or direction?
4. How can the angle \( -95.39^\circ \) be interpreted with respect to the positive x-axis?
5. How can this concept of equilibrium be applied to real-world systems like bridges or buildings?

**Tip:** When solving equilibrium problems, always ensure that both the sum of horizontal and vertical forces are zero to maintain balance.

Find the force FH that keeps the system of eight forces in equilibrium, as shown in the diagram, with given magnitudes and angles.

Learn how to find the force FH that keeps a system of eight forces in equilibrium, using trigonometric methods to resolve force components and apply the equilibrium conditions in a plane. Includes detailed calculations of magnitudes and angles.

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