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11-30-07 - Vector Addition
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11-30-07 - Vector Addition

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  • 1. Vector Addition Concurrent and Equilibrant Forces
  • 2. Definitions
    • Concurrent Forces – Acting at the same time and same place
    • Resultant – Sum of 2 or more vectors
    • Equilibrant Force –
      • A single, additional force that is exerted on an object
      • Same magnitude, but opposite direction of the Resultant
      • When combined with the Resultant, it produces equilibrium
      • Net force = 0
  • 3. Example Problem
    • Question: Find the Equilibrant force of these Concurrent forces analytically
      • 12 N south, 31 N west, 29 N north, 56 N west
    • 2 ways to determine resultant
      • Simplify to 2 vectors (use Parallelogram method to find resultant) OR
      • Draw all 4 Head-to-Tail (to find resultant – Start at the beginning and end at the end)
  • 4. Example Problem
    • 12 N south, 31 N west, 29 N north, 56 N west
    • Simplify to 2 vectors (use Parallelogram method to find resultant)
      • 12 N south + 29 N north = ?
      • (-12 N north) + 29 N north = 17 N north
      • 31 N west + 56 N west = 87 N west
  • 5. Example Problem – Not Drawn to Scale
    • 12 N south, 31 N west, 29 N north, 56 N west
    • Simplify to 17 N north and 87 N west
    • Use Parallelogram method to find resultant
    17 N 87 N F R
  • 6. Example Problem – Not Drawn to Scale
    • 17 N north and 87 N west
    • Solve for F R using Pythagorean Theorem
    • a 2 + b 2 = c 2
    • (17 2 ) + (87 2 ) = F R 2
    • √ ( 7858) = 89 N
    17 N 87 N F R = 89 N
  • 7. Example Problem – Not Drawn to Scale
    • Magnitude of F R = 89 N
    • To find direction ( Θ ) we will use the Tangent function
    • TOA: T angent Θ = O pposite/ A djacent
    • tan Θ = (17 N) / (87 N)
    • Θ = tan -1 (0.2)
    • Θ = 11°
    17 N 87 N (Adjacent) F R = 89 N Θ = 11 ° 17 N (Opposite)
  • 8. Example Problem – Not Drawn to Scale
    • Magnitude of F R = 89 N
    • Direction = ?
    • 180° - 11° = 169°
    • Therefore F R = 89 N @ 169°
    Θ = 11 ° 0 ° 180 ° 169 °
  • 9. Example Problem – Not Drawn to Scale
    • F R = 89 N @ 169°
    • Original Question - Find the Equilibrant force of these concurrent forces analytically
    • Equilibrant is same magnitude, opposite direction of Resultant
    0 ° 180 ° 169 ° F R F E
  • 10. Example Problem – Not Drawn to Scale
    • F R = 89 N @ 169°
    • F E = 89 N @ ??
    • Because we know that it is the exact opposite direction – we can add 180 ° to the direction of F R
    • 169 ° + 180 ° = 349 °
    0 ° 180 ° 169 ° F R F E 180 ° 349 °
  • 11. Example Problem – Not Drawn to Scale
    • F R = 89 N @ 169°
    • F E = 89 N @ 349 °
    0 ° 180 ° 169 ° F R F E 180 ° 349 °
  • 12. Solve Analytically – Using equations
    • 30 N @ 0 ° ; 40 N @ 90 °
    • 20 N @ 180 ° ; 15 N @ 270 °
    • 18 N @ 360 ° ; 22 N @ 270 °
    • 44 N @ 270 ° ; 12 N @ 360 °
    • 10 N @ 0 ° ; 20 N @ 180 ° ; 14 N @ 90 ° ; 20 N @ 270 °

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