Algebra & Trigonometry with Analytic Geometry
13th Edition
ISBN:9781133382119
Author:Swokowski
Publisher:Swokowski
Chapter8: Applications Of Trigonometry
Section8.3: Vectors
Problem 47E
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![### Understanding Force Vectors in Equilibrium
**Problem Statement:**
Consider a 200-N weight suspended by two wires as shown in the figure. Find the magnitude and components of the force vectors \( F_1 \) and \( F_2 \).
**Diagram Explanation:**
The diagram illustrates a 200-N weight suspended by two wires at different angles. The weight is hanging from the junction point of the two wires. The angles and forces are labeled as follows:
- The wire on the left exerts a force \( F_1 \) at an angle of 30° from the horizontal.
- The wire on the right exerts a force \( F_2 \) at an angle of 45° from the horizontal.
- The weight suspended is marked as 200 N acting downward due to gravity.
**Steps to Solve:**
1. **Identify Forces:**
- \( F_1 \): Tension in the left wire.
- \( F_2 \): Tension in the right wire.
- \( W = 200 \, \text{N} \): Weight acting downward.
2. **Resolve Forces into Components:**
- Horizontal components:
- \( F_{1x} = F_1 \cos(30^\circ) \)
- \( F_{2x} = F_2 \cos(45^\circ) \)
- Vertical components:
- \( F_{1y} = F_1 \sin(30^\circ) \)
- \( F_{2y} = F_2 \sin(45^\circ) \)
3. **Equilibrium Conditions:**
- Horizontal equilibrium: \( F_{1x} = F_{2x} \)
- Vertical equilibrium: \( F_{1y} + F_{2y} = W \)
Using the equilibrium conditions and the resolved components, we can set up a system of equations to solve for \( F_1 \) and \( F_2 \):
\[
F_1 \cos(30^\circ) = F_2 \cos(45^\circ)
\]
\[
F_1 \sin(30^\circ) + F_2 \sin(45^\circ) = 200 \text{ N}
\]
By solving these equations simultaneously, the magnitudes and components of the forces \( F_1 \) and \( F_2 \) can be determined](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F41a1af84-f79e-495e-9bf6-e155c46aa5b8%2F6c44203e-5637-4075-9d05-ce8622fac92c%2F9e0uyc8_processed.jpeg&w=3840&q=75)
Transcribed Image Text:### Understanding Force Vectors in Equilibrium
**Problem Statement:**
Consider a 200-N weight suspended by two wires as shown in the figure. Find the magnitude and components of the force vectors \( F_1 \) and \( F_2 \).
**Diagram Explanation:**
The diagram illustrates a 200-N weight suspended by two wires at different angles. The weight is hanging from the junction point of the two wires. The angles and forces are labeled as follows:
- The wire on the left exerts a force \( F_1 \) at an angle of 30° from the horizontal.
- The wire on the right exerts a force \( F_2 \) at an angle of 45° from the horizontal.
- The weight suspended is marked as 200 N acting downward due to gravity.
**Steps to Solve:**
1. **Identify Forces:**
- \( F_1 \): Tension in the left wire.
- \( F_2 \): Tension in the right wire.
- \( W = 200 \, \text{N} \): Weight acting downward.
2. **Resolve Forces into Components:**
- Horizontal components:
- \( F_{1x} = F_1 \cos(30^\circ) \)
- \( F_{2x} = F_2 \cos(45^\circ) \)
- Vertical components:
- \( F_{1y} = F_1 \sin(30^\circ) \)
- \( F_{2y} = F_2 \sin(45^\circ) \)
3. **Equilibrium Conditions:**
- Horizontal equilibrium: \( F_{1x} = F_{2x} \)
- Vertical equilibrium: \( F_{1y} + F_{2y} = W \)
Using the equilibrium conditions and the resolved components, we can set up a system of equations to solve for \( F_1 \) and \( F_2 \):
\[
F_1 \cos(30^\circ) = F_2 \cos(45^\circ)
\]
\[
F_1 \sin(30^\circ) + F_2 \sin(45^\circ) = 200 \text{ N}
\]
By solving these equations simultaneously, the magnitudes and components of the forces \( F_1 \) and \( F_2 \) can be determined
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