Vector AdditionVector Addition

Concurrent and Equilibrant Concurrent and Equilibrant ForcesForces

DefinitionsDefinitions

Concurrent ForcesConcurrent Forces – Acting at the same – Acting at the same time and same placetime and same place

ResultantResultant – Sum of 2 or more vectors – Sum of 2 or more vectors Equilibrant ForceEquilibrant Force – –

A single, additional force that is exerted on an A single, additional force that is exerted on an objectobject

Same magnitude, but opposite direction of the Same magnitude, but opposite direction of the ResultantResultant

When combined with the Resultant, it When combined with the Resultant, it produces equilibriumproduces equilibrium

Net force = 0Net force = 0

Example ProblemExample Problem

Question: Find the Equilibrant force of Question: Find the Equilibrant force of these Concurrent forces these Concurrent forces analyticallyanalytically 12 N south, 31 N west, 29 N north, 56 N 12 N south, 31 N west, 29 N north, 56 N

westwest 2 ways to determine resultant2 ways to determine resultant

Simplify to 2 vectors (use Parallelogram Simplify to 2 vectors (use Parallelogram method to find resultant) method to find resultant) OROR

Draw all 4 Head-to-Tail (to find resultant – Draw all 4 Head-to-Tail (to find resultant – Start at the beginning and end at the end)Start at the beginning and end at the end)

Example ProblemExample Problem

12 N south, 31 N west, 29 N north, 12 N south, 31 N west, 29 N north, 56 N west56 N west

Simplify to 2 vectors (use Simplify to 2 vectors (use Parallelogram method to find Parallelogram method to find resultant)resultant) 12 N south + 29 N north = ?12 N south + 29 N north = ? (-12 N north) + 29 N north = 17 N (-12 N north) + 29 N north = 17 N

northnorth

31 N west + 56 N west = 87 N west31 N west + 56 N west = 87 N west

Example Problem – Example Problem – Not Drawn to ScaleNot Drawn to Scale

12 N south, 31 N west, 29 N north, 12 N south, 31 N west, 29 N north, 56 N west56 N west

Simplify to 17 N north and 87 N Simplify to 17 N north and 87 N westwest

Use Parallelogram method to find Use Parallelogram method to find resultantresultant

17 N

87 N

FR

Example Problem – Example Problem – Not Drawn to ScaleNot Drawn to Scale

17 N north and 87 N west17 N north and 87 N west Solve for FSolve for FRR using Pythagorean using Pythagorean

TheoremTheorem aa22 + b + b22 = c = c22

(17(1722) + (87) + (8722) =) = FFRR22

√√((7858) = 89 N7858) = 89 N17 N

87 N

FR = 89 N

Example Problem – Example Problem – Not Drawn to ScaleNot Drawn to Scale

Magnitude of FMagnitude of FRR = 89 N = 89 N To find direction (To find direction (ΘΘ)) we will use the we will use the

Tangent functionTangent function TOA: TOA: TTangent angent ΘΘ = = OOpposite/pposite/AAdjacentdjacent tan tan ΘΘ = (17 N) / (87 N) = (17 N) / (87 N) ΘΘ = tan = tan-1-1 (0.2) (0.2) ΘΘ = 11° = 11°

17 N

87 N (Adjacent)

FR = 89 N

ΘΘ = 11 = 11°°17 N

(Opposite)

Example Problem – Example Problem – Not Drawn to ScaleNot Drawn to Scale

Magnitude of FMagnitude of FRR = 89 N = 89 N Direction = ?Direction = ? 180° - 11° = 169°180° - 11° = 169°

Therefore FTherefore FRR = 89 N @ 169° = 89 N @ 169°

ΘΘ = 11 = 11°°0°180°

169°

Example Problem – Example Problem – Not Drawn to ScaleNot Drawn to Scale

FFRR = 89 N @ 169° = 89 N @ 169° Original Question - Original Question - Find the Find the

Equilibrant force of these Equilibrant force of these concurrent forces analyticallyconcurrent forces analytically

Equilibrant is same magnitude, Equilibrant is same magnitude, opposite direction of Resultantopposite direction of Resultant

0°180°

169°F

R

FE

Example Problem – Example Problem – Not Drawn to ScaleNot Drawn to Scale

FFRR = 89 N @ 169° = 89 N @ 169° FFEE = 89 N @ ?? = 89 N @ ?? Because we know that it is the exact Because we know that it is the exact

opposite direction – we can add 180opposite direction – we can add 180° ° to the direction of Fto the direction of FRR

169169°° + 180 + 180°° = 349 = 349°°

0°180°

169°F

R

FE

180°°

349°°

Example Problem – Example Problem – Not Drawn to ScaleNot Drawn to Scale

FFRR = 89 N @ 169° = 89 N @ 169° FFEE = 89 N @ = 89 N @ 349349°°

0°180°

169°F

R

FE

180°°

349°°

Solve Analytically – Using Solve Analytically – Using equationsequations

1.1. 30 N @ 030 N @ 0°° ; 40 N @ 90 ; 40 N @ 90°°

2.2. 20 N @ 18020 N @ 180°° ; 15 N @ 270 ; 15 N @ 270°°

3.3. 18 N @ 36018 N @ 360°° ; 22 N @ 270 ; 22 N @ 270°°

4.4. 44 N @ 27044 N @ 270°° ; 12 N @ 360 ; 12 N @ 360°°

5.5. 10 N @ 010 N @ 0°° ; 20 N @ 180 ; 20 N @ 180°° ; 14 N @ 90 ; 14 N @ 90°° ; 20 N @ ; 20 N @ 270270°°