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To solve this problem, we need to analyze the forces acting on the semi-circular arc and determine the tension in cable BD. Here's a step-by-step approach:

### Given:
- Radius of the semi-circular arc, \( R = 5 \, \text{m} \)
- Length of cable BD, \( L = 7.5 \, \text{m} \)
- Distance between points A and D, \( AD = 10.0 \, \text{m} \)
- Force acting at point B, \( F = 18 \, \text{kN} \)

### Assumptions:
- The arc is symmetrical.
- The tension in the cable is uniform.
- The system is in static equilibrium.

### Solution:

1. **Identify Points and Angles:**
   - Points A and D are at the base of the semi-circular arc.
   - Point B is at the top of the arc.
   - Point C is where the cable BD meets the arc.
   - The angle subtended by the arc (from A to D) is 180 degrees.

2. **Analyze Geometry:**
   - The semi-circular arc has a radius of 5 m. The distance from B to the arc's center is also 5 m.
   - Since \(AD = 10 \, \text{m}\), the center of the arc is at the midpoint between A and D.
   - The length of the cable BD is 7.5 m.

3. **Calculate the Angles:**
   - Use the Law of Cosines in triangle BCD to find the angle \( \theta \) between BD and the vertical.

   \[
   \cos \theta = \frac{BC^2 + CD^2 - BD^2}{2 \cdot BC \cdot CD}
   \]
   Where:
   - \(BC = 5 \, \text{m}\) (radius)
   - \(CD = 5 \, \text{m}\) (radius)
   - \(BD = 7.5 \, \text{m}\)

4. **Set Up Equilibrium Equations:**
   - Resolve the forces vertically and horizontally:
   - Vertical forces: \( T \sin \theta = 18 \, \text{kN} \)
   - Horizontal forces: \( T \cos \theta = \text{horizontal component of the reaction at A and D} \)

5. **Solve for Tension \( T \):**

Let's calculate these values now.The tension \( T \) in cable BD is approximately **18.14 kN**. The angle \( \theta \) between cable BD and the vertical is about **97.18 degrees**.

This tension accounts for balancing the vertical force of 18 kN applied at point B while considering the geometry of the arc and cable.

### Would you like more details, or do you have any questions?

Here are 5 related questions that might help expand your understanding:

1. How would the tension change if the length of cable BD was different?
2. What would happen to the equilibrium if the force at B was not vertical?
3. How can we compute the horizontal components of forces at points A and D?
4. What would be the impact on tension if the arc radius changed?
5. How do we use vector resolution for complex force systems like this?

**Tip:** When solving problems involving tension and angles, always consider the geometry and use trigonometric identities for accurate calculations.

For the semi-circular arc shown with radius of 5 m, supported by 7.5 m long cable, determine the tension in cable BD.

This problem involves calculating the tension in cable BD supporting a semi-circular arc with a radius of 5 meters and an applied vertical force of 18 kN at the top of the arc. It covers key concepts in statics, geometry, and equilibrium of forces.

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