Determine the angle u between cables AB and AC 5 ft 60° Z 3 ft O 6 ft 4 ft FAB= 70 lb A FAC = 60 lb 2 ft
Determine the angle u between cables AB and AC 5 ft 60° Z 3 ft O 6 ft 4 ft FAB= 70 lb A FAC = 60 lb 2 ft
Chapter2: Loads On Structures
Section: Chapter Questions
Problem 1P
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Question
![### Determining the Angle Between Cables AB and AC
**Objective:** Determine the angle \( \theta \) between cables AB and AC.
#### Diagram Description
This illustration is a mechanical diagram showcasing a setup where two cables (denoted as AB and AC) are attached at different points. The diagram is three-dimensional, including axes labeled as x, y, and z. Below is a detailed breakdown of the diagram:
**Points and Measurements:**
1. **Point A:** The point where both cables AB and AC converge.
2. **Point B:**
- Located 4 ft vertically above point A on the z-axis.
3. **Point C:**
- Located 5 ft from Point O, with a horizontal distance of 6 ft from the y-axis and making an angle of 60° with Point O.
4. **Point O:**
- An anchor point possibly at ground level, 6 ft horizontally from Point A on the y-axis.
5. **Cables:**
- \( F_{AB} = 70 \text{ lb} \): The tension force in cable AB.
- \( F_{AC} = 60 \text{ lb} \): The tension force in cable AC.
**Measurement Coordinates:**
- \( OB = 4 \text{ ft} \)
- \( OC = 5 \text{ ft} \) at a 60° angle with the x-y plane.
#### Steps to Solve
To determine the angle \( \theta \) between the two cables, you should follow these steps:
1. **Identify Position Vectors:**
- Find the position vectors for points B and C with respect to A.
2. **Calculate Vector Components:**
- Calculate the components of the vectors \( \vec{AB} \) and \( \vec{AC} \).
3. **Use Dot Product:**
- Utilize the dot product formula to relate vectors:
\[
\vec{A} \cdot \vec{B} = |\vec{A}| |\vec{B}| \cos(\theta)
\]
4. **Calculate Cosine of the Angle:**
- Extract angle \( \theta \) using the inverse cosine function.
5. **Result Interpretation:**
- Compute the angle \( \theta \) to understand the spatial relation between the two cables in degrees.
By following this structured approach, the](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F7a85083f-c711-4fe4-a0ea-d798364bd78d%2F69a2d153-ad7a-46dd-93e5-0a43e33523a3%2F9tskfkp_processed.jpeg&w=3840&q=75)
Transcribed Image Text:### Determining the Angle Between Cables AB and AC
**Objective:** Determine the angle \( \theta \) between cables AB and AC.
#### Diagram Description
This illustration is a mechanical diagram showcasing a setup where two cables (denoted as AB and AC) are attached at different points. The diagram is three-dimensional, including axes labeled as x, y, and z. Below is a detailed breakdown of the diagram:
**Points and Measurements:**
1. **Point A:** The point where both cables AB and AC converge.
2. **Point B:**
- Located 4 ft vertically above point A on the z-axis.
3. **Point C:**
- Located 5 ft from Point O, with a horizontal distance of 6 ft from the y-axis and making an angle of 60° with Point O.
4. **Point O:**
- An anchor point possibly at ground level, 6 ft horizontally from Point A on the y-axis.
5. **Cables:**
- \( F_{AB} = 70 \text{ lb} \): The tension force in cable AB.
- \( F_{AC} = 60 \text{ lb} \): The tension force in cable AC.
**Measurement Coordinates:**
- \( OB = 4 \text{ ft} \)
- \( OC = 5 \text{ ft} \) at a 60° angle with the x-y plane.
#### Steps to Solve
To determine the angle \( \theta \) between the two cables, you should follow these steps:
1. **Identify Position Vectors:**
- Find the position vectors for points B and C with respect to A.
2. **Calculate Vector Components:**
- Calculate the components of the vectors \( \vec{AB} \) and \( \vec{AC} \).
3. **Use Dot Product:**
- Utilize the dot product formula to relate vectors:
\[
\vec{A} \cdot \vec{B} = |\vec{A}| |\vec{B}| \cos(\theta)
\]
4. **Calculate Cosine of the Angle:**
- Extract angle \( \theta \) using the inverse cosine function.
5. **Result Interpretation:**
- Compute the angle \( \theta \) to understand the spatial relation between the two cables in degrees.
By following this structured approach, the
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