Meriam, Kraige & Bolton — Distributed Forces: Centroids and Centers of Gravity: Problem 5_79

⚡ Mecademy AIENG정역학 · ch5  Problem Statement Calculate the weight of the aluminum casting shown. The solid is generated by revolving the trapezoidal area shown about the -axis through . Problem 5/79 (a) Calculation of the weight of the aluminum casting 1. Formula: The weight of the casting is the product of its volume and the material's weight density . According to the Second Theorem of Pappus-Guldinus, the volume of a solid generated by revolving a plane area through an angle about a non-intersecting axis is: where: is the angle of revolution in radians. is the perpendicular distance from the axis of revolution to the centroid of the area . is the cross-sectional area of the trapezoid. 2. Substitution: From the given diagram, we identify the following parameters: Rotation angle . Cross-sectional area (trapezoid with vertical parallel sides): Inner parallel side Outer parallel side Horizontal width (height of trapezoid) Inner radius from the -axis to the area: Weight density of aluminum (from typical Table D/1 values): (approximately ). W z180 ∘ W WV ρ W=ρV V Aθ V=θArˉ θ rˉ A A θ=180= ∘ π rad A a=1+ ′′ 2+ ′′ 1= ′′ 4 in. b=2 in. h=3 in. zR = i 2 in. ρ= 0.0975 lb/in. 3 168 lb/ft. 3 First, calculate the area : Next, determine the distance from the longer side to the centroid of the trapezoid: The radial distance from the -axis to the centroid is: 3. Calculation: Calculate the volume : Calculate the weight : 4. Result: ● Final Conclusion: The weight of the aluminum casting, gen