Physics Vector Problem: Tugboats Pulling a Ship
Question:
Three tugboats are pulling on a large ship that has gone ashore. The first tugboat pulls with a force of 2500 pounds in a compass direction of 55°, the second tugboat pulls with 2000 pounds at 95°, and the third one pulls with 1000 pounds at 120°.
- a. What is the magnitude of the resultant vector?
- b. What is the compass direction of the resultant vector?
Step 1: Resolve Each Force into Components
Use the formulas:
Fx = F × cos(θ)Fy = F × sin(θ)
Tugboat 1 (2500 lb @ 55°):
F1x = 2500 × cos(55°) = 1434.0F1y = 2500 × sin(55°) = 2049.0
Tugboat 2 (2000 lb @ 95°):
F2x = 2000 × cos(95°) = -174.5F2y = 2000 × sin(95°) = 1992.4
Tugboat 3 (1000 lb @ 120°):
F3x = 1000 × cos(120°) = -500.0F3y = 1000 × sin(120°) = 866.0
Step 2: Sum the Components
Total Horizontal Component (Rx):
Rx = 1434.0 - 174.5 - 500.0 = 759.5
Total Vertical Component (Ry):
Ry = 2049.0 + 1992.4 + 866.0 = 4907.4
Step 3: Find the Resultant Vector
Magnitude of the resultant vector:
R = √(Rx² + Ry²) =
√(759.5² + 4907.4²) = √(577,822 + 24,082,718) ≈ √24,660,540 ≈ 4966.0 lb
Direction (angle θ from east):
θ = tan⁻¹(Ry / Rx) = tan⁻¹(4907.4 / 759.5) ≈ tan⁻¹(6.46) ≈ 81.2°
Final Answers:
- Magnitude: 4966.0 lb
- Compass Direction: 81.2° from due east