Meriam, Kraige & Bolton — Distributed Forces: Centroids and Centers of Gravity: Problem 5_18

⚡ Mecademy AIENG정역학 · ch5  Problem Statement Determine by direct integration the coordinates of the centroid of the rectangular tetrahedron. Problem 5/18 (a) x-coordinate of the centroid, ● Calculation Process 1. Define the geometry and volume: From the diagram, the tetrahedron has vertices at the origin , on the -axis at , on the -axis at (since the hypotenuse in the -plane is ), and on the -axis at . The equation of the inclined plane is . The total volume is: 2. Present Final Formula: 3. Substitute Values and Perform Operations: Inner integral: Middle integral: Let . Outer integral: 4. Final Calculation: xˉ (0,0,0)x(b,0,0)y(0,b,0) xy b 2z(0,0,h) + b x + b y = h z 1V V= dxd y d z = ∫ 0 h ∫ 0 b(1−z/h) ∫ 0 b(1−z/h−y/b) b h 6 1 2 =xˉ xd V = V 1 ∫∫∫ xdxd y d z bh 2 6 ∫ 0 h ∫ 0 b(1−z/h) ∫ 0 b(1−z/h−y/b) xdx= ∫ 0 b(1−z/h−y/b) b(1−z/h)−y 2 1 [] 2 C=b(1−z/h) ( C − ∫ 0 C 2 1 y)dy= 2 − (C−y) = 2 1 [ 3 1 3 ] 0 C C = 6 1 3 b (1− 6 1 3 z/h) 3 b (1− ∫ 0 h 6 1 3 z/h)dz= 3 − (1−z/h) = 6 b 3 [ 4 h 4 ] 0 h 24 bh 3 =xˉ = bh/6 2 bh/24 3 4 b ● Final Conclusion: The -coordinate of the centroid is . (b) y-coordinate of the centroid, ● Calculation Process 1. Present Final Formula: By symmetry with the -axis (since the intercepts on and are both ): 2. Substitute Values: Due to the identical limits and integrand form as : 3. Final Calculation: ● Final Conclusion: The -coordinate of the centroid is . (c) z-coordinate of the centroid, ● Calculation Process 1. Present Final Formula: Using horizontal s