Answer :
The resultant force has a magnitude of approximately 230.7 pounds and is directed at an angle of approximately 73.2 degrees with the positive x-axis.
Given:
Force 1 magnitude (F₁) = 69 pounds
Force 1 angle (θ₁) = 30 degrees
Force 2 magnitude (F₂) = 101 pounds
Force 2 angle (θ₂) = 45 degrees
Force 3 magnitude (F₃) = 130 pounds
Force 3 angle (θ₃) = 120 degrees
Resolve each force into its x and y components:
For force 1:
Fx₁ = F₁ × cos(θ₁) = 69 pounds × cos(30°) ≈ 59.64 pounds
Fy₁ = F₁ × sin(θ₁) = 69 pounds × sin(30°) ≈ 34.50 pounds
For force 2:
Fx₂ = F₂ × cos(θ₂) = 101 pounds × cos(45°) ≈ 71.46 pounds
Fy₂ = F₂ × sin(θ₂) = 101 pounds × sin(45°) ≈ 71.46 pounds
For force 3:
Fx₃ = F₃ × cos(θ₃) = 130 pounds × cos(120°) ≈ -65.00 pounds
Fy₃ = F₃ × sin(θ₃) = 130 pounds × sin(120°) ≈ 112.39 pounds
Calculate the sum of the x-components and y-components:
ΣFx = Fx₁ + Fx₂ + Fx₃
= 59.64 pounds + 71.46 pounds - 65.00 pounds
≈ 66.10 pounds
ΣFy = Fy₁ + Fy₂ + Fy₃
34.50 pounds + 71.46 pounds + 112.39 pounds
≈ 218.35 pounds
Calculate the magnitude and direction of the resultant force:
The magnitude of the resultant force (R) = √(ΣFx² + ΣFy²)
= √(66.10)² + (218.35)²
≈ 230.7 pounds
Direction of the resultant force (θ) = tan⁻¹(ΣFy / ΣFx)
= tan⁻¹(218.35 / 66.10)
≈ 73.2 degrees
Therefore, the resultant force has a magnitude of approximately 230.7 pounds and is directed at an angle of approximately 73.2 degrees with the positive x-axis.
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