Explanation Draw the free-body diagram of the shaded area as in Figure (1). Write the equation to find y as follows: y = k x 2 (1) Apply the boundary condition. When x = 2 a ; y = h . Substitute h for y and 2 a for x in Equation (1). h = k ( 2 a ) 2 k = h 4 a 2 Substitute h 4 a 2 for k in Equation (1). y = h 4 a 2 x 2 Consider a small strip as in Figure 1. Find the area of the small strip with respect to y -axis as follows: d A = y d x Substitute h 4 a 2 x 2 for y . d A = h 4 a 2 x 2 d x Integrate the equation and apply the limits. ∫ d A = ∫ a 2 a h 4 a 2 x 2 d x A = h 4 a 2 [ x 3 3 ] a 2 a = h 4 a 2 [ ( 2 a ) 3 3 − a 3 3 ] = h 4 a 2 [ 7 a 3 3 ] = 7 a h 12 Find the change in moment of inertia with respect to y -axis using the relation. d I y = x 2 d A Substitute h 4 a 2 x 2 d x for dA . d I y = x 2 h 4 a 2 x 2 d x = h 4 a 2 x 4 d x Integrate the equation and substitute the limits in y -axis

Vector Mechanics for Engineers: Statics and Dynamics

11th Edition

ISBN: 9780073398242

Author: Ferdinand P. Beer, E. Russell Johnston Jr., David Mazurek, Phillip J. Cornwell, Brian Self

Publisher: McGraw-Hill Education

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Chapter 9.1, Problem 9.18P

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The moment of inertia and radius of gyration of the shaded area with respect to the y axis.

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Chapter 9.1, Problem 9.17P

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Vector Mechanics for Engineers: Statics and Dynamics

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 - Prob. 9.3P Ch. 9.1 - Prob. 9.4P 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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