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Choose the nearest magnitude and direction of F so that the particle is in equilibrium. 8 kN] 30° 60° 4 kN 0 5 kN F -X

Choose the nearest magnitude and direction of F so that the particle is in equilibrium. 8 kN] 30° 60° 4 kN 0 5 kN F -X

Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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### Equilibrium of Forces #### Problem Statement: Choose the nearest magnitude and direction \( \theta \) of \( \mathbf{F} \) so that the particle is in equilibrium. #### Diagram Explanation: The diagram provided shows a particle at the origin with forces acting on it in a two-dimensional plane. Four forces are acting on the particle: 1. An 8 kN force directed upward to the left at an angle of 30° above the negative x-axis. 2. A 5 kN force directed horizontally to the right along the positive x-axis. 3. A 4 kN force directed downward to the left at an angle of 60° below the negative x-axis. 4. A force \( \mathbf{F} \) acting in a downward direction making an angle \( \theta \) with the negative y-axis. #### Objective: To determine the magnitude and angle \( \theta \) of force \( \mathbf{F} \) such that the particle remains in equilibrium. #### Formal Analysis: To ensure equilibrium, the sum of all forces acting on the particle should be zero both in the x and y components. This can be expressed as: 1. \( \sum F_x = 0 \) 2. \( \sum F_y = 0 \) Break down each force into its x and y components and sum them up to find the values of \( \mathbf{F} \) and \( \theta \). #### Step-by-Step Solution: 1. **Resolve the 8 kN force**: - \( F_{x1} = 8 \cos(30^\circ) \) - \( F_{y1} = 8 \sin(30^\circ) \) 2. **Resolve the 5 kN force**: - \( F_{x2} = 5 \) - \( F_{y2} = 0 \) 3. **Resolve the 4 kN force**: - \( F_{x3} = 4 \cos(60^\circ) \) - \( F_{y3} = -4 \sin(60^\circ) \) 4. **Resolve force \( \mathbf{F} \)**: - \( F_{x4} = -F \sin(\theta) \) - \( F_{y4} = -F \cos(\theta) \) Which satisfies: \[ \sum F
Transcribed Image Text:### Equilibrium of Forces #### Problem Statement: Choose the nearest magnitude and direction \( \theta \) of \( \mathbf{F} \) so that the particle is in equilibrium. #### Diagram Explanation: The diagram provided shows a particle at the origin with forces acting on it in a two-dimensional plane. Four forces are acting on the particle: 1. An 8 kN force directed upward to the left at an angle of 30° above the negative x-axis. 2. A 5 kN force directed horizontally to the right along the positive x-axis. 3. A 4 kN force directed downward to the left at an angle of 60° below the negative x-axis. 4. A force \( \mathbf{F} \) acting in a downward direction making an angle \( \theta \) with the negative y-axis. #### Objective: To determine the magnitude and angle \( \theta \) of force \( \mathbf{F} \) such that the particle remains in equilibrium. #### Formal Analysis: To ensure equilibrium, the sum of all forces acting on the particle should be zero both in the x and y components. This can be expressed as: 1. \( \sum F_x = 0 \) 2. \( \sum F_y = 0 \) Break down each force into its x and y components and sum them up to find the values of \( \mathbf{F} \) and \( \theta \). #### Step-by-Step Solution: 1. **Resolve the 8 kN force**: - \( F_{x1} = 8 \cos(30^\circ) \) - \( F_{y1} = 8 \sin(30^\circ) \) 2. **Resolve the 5 kN force**: - \( F_{x2} = 5 \) - \( F_{y2} = 0 \) 3. **Resolve the 4 kN force**: - \( F_{x3} = 4 \cos(60^\circ) \) - \( F_{y3} = -4 \sin(60^\circ) \) 4. **Resolve force \( \mathbf{F} \)**: - \( F_{x4} = -F \sin(\theta) \) - \( F_{y4} = -F \cos(\theta) \) Which satisfies: \[ \sum F
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