Meriam, Kraige & Bolton — Distributed Forces: Centroids and Centers of Gravity: Problem 5_187

⚡ Mecademy AIENG정역학 · ch5  Problem Statement Calculate the -, -, and -coordinates of the mass center of the bracket formed from the steel plate of uniform thickness. Problem 5/187 Since the bracket is made of a steel plate of uniform thickness and density, its mass center coincides with the centroid of its total area. We can determine the coordinates of the mass center by decomposing the complex shape into four simpler geometric areas: three rectangles and one triangle. Decomposition and Geometric Properties: Part 1: A rectangular plate in the -plane. Dimensions: , . Area . Centroid , , . Part 2: A rectangular plate in the -plane (middle section). Dimensions: , . Area . Centroid , , . Part 3: A rectangular portion of the right section in the -plane. Dimensions: , height from to ( ). Area . Centroid , , . Part 4: A triangular portion of the right section in the -plane. Vertices in -plane: , , . Area . xyz xz w = x 175 mmh=400 mm A = 1 175×400=70,000 mm 2 C : 1 =xˉ 1 = 2 175 87.5 mm =yˉ 1 0 mm =zˉ 1 = 2 400 200 mm yz w = y 150 mmh=400 mm A = 2 150×400=60,000 mm 2 C : 2 =xˉ 2 0 mm =yˉ 2 = 2 150 75 mm =zˉ 2 = 2 400 200 mm yz w = y 250 mmz=200 mm400 mmh= 200 mm A = 3 250×200=50,000 mm 2 C : 3 =xˉ 3 0 mm =yˉ 3 150+ = 2 250 275 mm =zˉ 3 200+ = 2 200 300 mm yz yz(150,0)(150,200)(400,200) A = 4 × 2 1 250×200=25,000 mm 2 Centroid , , . Total Area: (a) Determination of the -coordinate 1. Formula: 2. Substitution: 3. Calculation: 4. Result: ● Final Conclusion: The -coordinate of the mas