Meriam, Kraige & Bolton — Distributed Forces: Centroids and Centers of Gravity: Problem 5_63

⚡ Mecademy AIENG정역학 · ch5  Problem Statement Determine the -, -, and -coordinates of the mass center of the fixture formed from thin metal plate of uniform thickness. Problem 5/63 To find the mass center of the fixture, we treat it as a collection of thin plates. Since the thickness is uniform, the mass center coincides with the centroid of the total surface area. We decompose the fixture into five components: a back vertical plate (with a cutout), a top horizontal plate, a bottom horizontal plate, and two side plates forming a trough. Geometry and Coordinate System: The origin is at the bottom-left corner of the back plate. - The back plate lies in the -plane (). - The -axis is depth (forward), the -axis is width (right), and the -axis is height (up). - Component 1 (Back Plate): Outer dimensions . Cutout dimensions . The cutout is from the right edge, so it spans and . - Component 2 (Trough Top): Rectangular plate at , width (), depth . - Component 3 (Trough Bottom): Rectangular plate at , width . The front edge is slanted at from the vertical, so the depth at the bottom is . - Components 4 & 5 (Trough Sides): Two identical right trapezoids in the -plane at and . Height , top base , bottom base . (a) Calculation of the -coordinate 1. Formula: 2. Substitution: , , , , 3. Calculation: xyz O yzx=0xy z6.5× ′′ 6 ′′ 5× ′′ 3 ′′ 1 ′′ y∈ [0.5,5.5]z∈[1.5,4.5]z=4.5 ′′ 5 ′′ y∈[0.5,5.5]L = top 5 ′′ z=1.5 ′′ 5 ′′ 30 ∘ L = bot 5−3tan30= ∘ 5− ≈ 33.268 ′′ xzy=0.5 ′′ y=5.5 ′′ 3 ′′ 5 ′′ 3.268