Meriam, Kraige & Bolton — Distributed Forces: Centroids and Centers of Gravity: Problem 5_21

⚡ Mecademy AIENG정역학 · ch5  Problem Statement Determine the - and -coordinates of the mass center of the homogeneous plate of uniform thickness . Problem 5/21 (a) Total Area of the Plate 1. Formula: For a plate in the -plane, the area is given by the integral of the height over its width: 2. Substitution: The boundary curve is given as . Since the plate passes through the point , we have: The equation of the curve in terms of as a function of is: Substituting this into the area integral: 3. Calculation: xy t xyA y(x) A= y dx ∫ 0 b x=ky 2 (b,h) b=kh⟹ 2 k= h 2 b yx x= y ⟹ h 2 b 2 y= h b x A= h dx ∫ 0 b b x A= x dx b h ∫ 0 b 1/2 A= x = b h [ 3 2 3/2 ] 0 b b b h ( 3 2 3/2 ) A= bh 3 2 4. Result: The area of the parabolic plate is . ● Final Conclusion: The area is , which is needed to calculate the coordinates of the mass center. (b) x-coordinate of the Mass Center 1. Formula: For a homogeneous plate of uniform thickness, the mass center's - coordinate coincides with the area centroid: 2. Substitution: 3. Calculation: 4. Result: ● Final Conclusion: The -coordinate of the mass center is . (c) y-coordinate of the Mass Center 1. Formula: The -coordinate of the area centroid is: 2. Substitution: A= bh 3 2 A= bh 3 2 x =xˉ xdA= A 1 ∫ x⋅ A 1 ∫ 0 b ydx =xˉ x h dx bh 3 2 1 ∫ 0 b ( b x ) =xˉ ⋅ 2bh 3 x dx b h ∫ 0 b 3/2 =xˉ x 2b 3/2 3 [ 5 2 5/2 ] 0 b =xˉ b = 2b 3/2 3 ( 5 2 5/2 ) b 5 3 =xˉ b= 5 3 0.6b x=xˉ b 5 3 y =yˉ dA= A 1 ∫y ~ ( y dx)= A 1 ∫ 0 b 2 y y dx 2A 1 ∫ 0 b 2 =yˉ h dx 2( bh) 3 2 1 ∫