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# 11-30-07 - Vector Addition

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### 11-30-07 - Vector Addition

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