Explanation Given information: The height of the parabola is h The half-width of the parabola is a . Calculation: Show the panel with rectangular strip as in Figure 1 Express the area ( A ) of the parabola as below: A = π 4 a h Write the expression for the centroid ( y ¯ ) of the parabola about xx axis as below: y ¯ = 2 5 h Write the expression for moment of inertia ( I A A ' ) ' of the right portion about A A ' as below: ( I A A ' ) R = 1 3 ( h − y ) 3 Differentiate the above equation with respect to x . d I A A ' = 1 3 ( h − y ) 3 d x (1). Consider the panel as semi parabolic, Write the expression for the distance ( y ) as below: y = h b 2 x 2 Here, b is the width of the semi parabola. Substitute a for b . y = h a 2 x 2 (2) Substitute equation (2) in equation (1). d ( I A A ' ) R = 1 3 ( h − h a 2 x 2 ) 3 d x d ( I A A ' ) R = 1 3 h 3 a 6 ( a 2 − x 2 ) 3 d x d ( I A A ' ) R = 1 3 h 3 a 6 ( a 6 − 3 a 4 x 2 + 3 a 2 x 4 − x 6 ) d x Integrate d ( I A A ' ) R from 0 to a. ( I A A ' ) R = ∫ 0 2 1 3 h 3 a 6 ( a 6 − 3 a 4 x 2 + 3 a 2 x 4 − x 6 ) d x = 1 3 h 3 a 6 ∫ 0 2 ( a 6 − 3 a 4 x 2 +

Vector Mechanics for Engineers: Statics, 11th Edition

11th Edition

ISBN: 9780077687304

Author: Ferdinand P. Beer, E. Russell Johnston Jr., David Mazurek

Publisher: McGraw-Hill Education

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Center of Gravity and Centroid

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Chapter 9.2, Problem 9.60P

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Find the depth (yP) of the point at which the resultant of the hydrostatic forces acting on the panel.

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Chapter 9.2, Problem 9.59P

Chapter 9.2, Problem 9.61P

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Chapter 9 Solutions

Vector Mechanics for Engineers: Statics, 11th Edition

Ch. 9.1 - 9.1 through 9.4 Determine by direct integration...Ch. 9.1 - 9.1 through 9.4 Determine by direct integration...Ch. 9.1 - 9.1 through 9.4 Determine by direct integration...Ch. 9.1 - 9.1 through 9.4 Determine by direct integration...Ch. 9.1 - 9.5 through 9.8 Determine by direct integration...Ch. 9.1 - 9.5 through 9.8 Determine by direct integration...Ch. 9.1 - Prob. 9.7P Ch. 9.1 - Prob. 9.8P Ch. 9.1 - 9.9 through 9.11 Determine by direct integration...Ch. 9.1 - Prob. 9.10P

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Find the depth ( y P ) of the point at which the resultant of the hydrostatic forces acting on the panel.

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Find the depth (yP) of the point at which the resultant of the hydrostatic forces acting on the panel.

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