Meriam, Kraige & Bolton — Distributed Forces: Centroids and Centers of Gravity: Problem 5_11

⚡ Mecademy AIENG정역학 · ch5  Problem Statement If the shaded area is revolved 360° about the y-axis, determine the y-coordinate of the centroid of the resulting volume. Problem 5/11 (a) Determination of the y-coordinate of the centroid for the volume of revolution 1. Formula: The y-coordinate of the centroid for a volume of revolution about the - axis is given by: Using the method of cylindrical shells, a differential volume element is defined as: The centroid of this shell element is located at . Thus, the moment of the volume about the -plane is: 2. Substitution: From the diagram, the boundaries are: Top boundary: Bottom boundary is a parabolic curve from to . Assuming the vertex is at , its equation is: At . So, . Now, substitute these into the volume and moment integrals: y =yˉ V ydV∫ dV dV=2πx(y − top y )dx bottom =y ~ 2 y+y topbottom xz d V = ∫y ~ 2πx(y −y )dx = ∫ 0 b/2 ( 2 y +y topbottom )[ topbottom ]π x( y∫ 0 b/2 y = top h (0,3h/4)(b/2,0) (0,3h/4) y = bottom − 4 3h kx 2 x=b/2,y=0⇒0= − 4 3h k ⇒ ( 2 b ) 2 k= b 2 3h y = bottom 1− 4 3h ( b 2 4 x 2 ) V= 2 π x h− 1− dx ∫ 0 b/2 [ 4 3h ( b 2 4x 2 )] 3. Calculation: Total Volume : Volume Moment : Final Calculation for : 4. Result: ● Final Conclusion: The y-coordinate of the centroid of the resulting volume is , or . ✨ Final Answer Summary (a) y d V = ∫π x h− − x dx ∫ 0 b/2 [ 2 ( 4 3h b 2 3h 2 ) 2 ] V V= 2 π x+ x dx= ∫ 0 b/2 ( 4 h b 2 3h 3 ) 2 π + [ 8 hx 2 4b 2 3hx 4 ] 0 b/2 V= 2 π + = [ 8 h(b/4) 2 4b 2 3h(b/16) 4 ] 2 π + = ( 3