Explanation Draw the free-body diagram of the shaded area as in Figure (1). Refer to Figure (1). Consider the upper curve. Write the equation to find y 1 as follows: y 1 = k 1 x 2 (1) Apply the boundary condition. When x = a ; y 1 = b . Substitute b for y 1 and a for x in Equation (1). b = k 1 a 2 k 1 = b a 2 Substitute b a 2 for k 1 in Equation (1). y 1 = b a 2 x 2 Consider the lower curve. Write the equation to find y 2 as follows: y 2 = k 2 x 1 2 (2) Apply the boundary condition. When x = a ; y 2 = b . Substitute b for y 2 and a for x in Equation (2). b = k 2 a 1 2 k 2 = b a 1 2 Substitute b a 1 2 for k 2 in Equation (2). y 2 = b a 1 2 x 1 2 Consider a small strip as in Figure 1. Find the change in area using the relation as follows: d A = ( y 2 − y 1 ) d x Substitute b a 1 2 x 1 2 for y 2 and b a 2 x 2 for y 1 . d A = ( b a 1 2 x 1 2 − b a 2 x 2 ) d x Integrate the equation at substitute the limits with respect to x -axis. ∫ d A = ∫ 0 a ( b a 1 2 x 1 2 − b a 2 x 2 ) d x A = [ b 3 2 a 1 2 x 3 2 − b 3 a 2 x 3 ] 0 a = [ 2 b 3 a 1 2 a 3 2 − b 3 a 2 a 3 ] = [ 2 b a 3 − b a 3 ] = a b 3 Find the change in moment of inertia with respect to x -axis using the relation. d I x = 1 3 y 2 3 d x − 1 3 y 1 3 d x Substitute b a 1 2 x 1 2 for y 2 and b a 2 x 2 for y 1

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

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

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