Meriam, Kraige & Bolton — Introduction to Statics: Problem 1_2

⚡ Mecademy AIENG정역학 · ch1  Problem Statement 1/2 Determine the magnitude of the vector sum and the angle which makes with the positive -axis. Complete both graphical and algebraic solutions. Given: with a slope gradient of 3 (rise) over 4 (run). at an angle of from the negative -axis. Problem 1/2 (a) Algebraic Solution ● Calculation Process 1. Resolve into components: The slope gradient for is 3:4, which implies a 3-4-5 right triangle. In vector form: 2. Resolve into components: is directed into the second quadrant, making with the negative -axis (or from the positive -axis). In vector form: 3. Sum the components to find : Vector sum: V=V + 1 V 2 θ x V x V = 1 10 units V = 2 12 units60 ∘ x V 1 V 1 V = 1x 10⋅ = 5 4 8 units V = 1y 10⋅ = 5 3 6 units V = 1 8i+6j V 2 V 2 60 ∘ x120 ∘ x V = 2x −12cos60= ∘ −12(0.5)=−6 units V = 2y 12sin60= ∘ 12 ≈ ( 2 3 )10.392 units V = 2 −6i+10.392j V V = x V + 1x V = 2x 8−6=2 units V = y V + 1y V = 2y 6+10.392=16.392 units V=2i+16.392j 4. Calculate magnitude : 5. Calculate angle : Since both components are positive, the vector is in the first quadrant. ● Final Conclusion: Magnitude Angle (b) Graphical Solution ● Calculation Process 1. Determine the internal angles: Angle of from the positive -axis: . Angle of from the positive -axis: . The angle between the two vectors is . 2. Apply Law of Cosines (Triangle Rule): Using the tip-to-tail method, the angle opposite the resultant is . 3. Apply Law of Sines to find the angle: Let be the angle between a