Problem Statement
ABCDEF is a regular hexagon. Forces of magnitudes 2N, 4√3N, 8N, 2√3N, and 4N act at point A in directions AB, AC, AD, AE, and AF respectively. Determine the resultant force completely.
Step-by-Step Solution
Key Geometric Properties
- AB and AE are perpendicular (90° angle)
- Internal angles = 120°
- Diagonals make 30°, 60°, and 90° angles
Step 1: Resolve Forces Along AB (Horizontal)
2 + 4√3·cos30° + 8·cos60° + 4·cos120°
= 2 + (4√3)(√3/2) + 8(1/2) + 4(-1/2)
= 2 + 6 + 4 – 2 = 10 N
= 2 + (4√3)(√3/2) + 8(1/2) + 4(-1/2)
= 2 + 6 + 4 – 2 = 10 N
Step 2: Resolve Forces Along AE (Vertical)
2√3 + 4√3·sin30° + 8·sin60° + 4·sin120°
= 2√3 + (4√3)(1/2) + 8(√3/2) + 4(√3/2)
= 2√3 + 2√3 + 4√3 + 2√3 = 10√3 N
= 2√3 + (4√3)(1/2) + 8(√3/2) + 4(√3/2)
= 2√3 + 2√3 + 4√3 + 2√3 = 10√3 N
Final Resultant
R = √(10² + (10√3)²) = √(100 + 300) = √400 = 20 N
Angle with AB: tan⁻¹(10√3/10) = 60°
Angle with AB: tan⁻¹(10√3/10) = 60°
Conceptual Explanation & Use Cases
Core Concepts:
- Vector Resolution: Breaking forces into perpendicular components (AB and AE)
- Hexagon Geometry: Using 30°, 60°, 90° angles from regular hexagon properties
- Superposition Principle: Algebraic sum of components in each direction
- Trigonometric Application: Using cosθ and sinθ for force components
Real-World Applications:
- Mechanical Engineering: Analyzing forces in hexagonal bolt patterns
- Civil Engineering: Calculating loads in hexagonal truss structures
- Physics Education: Demonstrating vector addition principles
- Robotics: Force analysis in hexagonal joint mechanisms
Why It Works:
The regular hexagon’s symmetry allows using standard angles for component resolution. Perpendicular axes (AB and AE) simplify calculations by making components independent. The resultant 20N force at 60° matches the hexagon’s internal angle pattern, demonstrating geometric-force relationships.
