Explanation Calculation: Sketch the cross section of the shaded diagram as shown in Figure 1. Refer to Figure 1. Find the area ( A 1 ) of section ABCD as shown in below: A 1 = b 1 h 1 (1) Here, b 1 is width of section ABCD and h 1 is height of section ABCD . Substitute 108 mm for b 1 and 60 mm for h 1 in Equation (1). A 1 = 108 × 60 = 6 , 480 mm 2 Find the area ( A 2 ) of section abc as shown in below: A 2 = − ( 1 2 b 2 h 2 ) (2) Substitute 72 mm for b 1 and 36 mm for h 1 in Equation (2). A 2 = − 1 2 × 72 × 36 = − 1 , 296 mm 2 Find the total area ( A ) using the relation as follows: A = A 1 + A 2 (3) Here, A 1 is area of section ABCD and A 2 is area of section abc . Substitute 6 , 480 mm 2 for A 1 , and − 1 , 296 mm 2 for A 2 in Equation (3). A = 6 , 480 − 1 , 296 = 5 , 184 mm 2 Refer to Figure 1. Find the centroid for section ABCD about x axis ( x 1 ) as shown below: x 1 = 108 2 = 54 mm . Find the centroid for section abc about x axis ( x 2 ) as shown below: x 2 = 22 + 72 3 = 22 + 24 = 46 mm Find the centroid ( x ¯ ) using the relation as follows: x ¯ = A 1 x 1 + A 2 x 2 A 1 + A 2 (4) Substitute 6 , 480 mm 2 for A 1 , − 1 , 296 mm 2 for A 2 , 54 mm for x 1 , 36 mm for x 2 in Equation (4). x ¯ = 6 , 480 × 54 + ( − 1 , 296 × 46 ) 6 , 480 + 1 , 296 = 349 , 920 − 59 , 616 7 , 776 = 56 mm Find the moment of inertia for section 1 about x axis ( I x 1 ) as shown below: ( I x 1 ) = 1 12 b 1 h 1 3 (5) Substitute 108 mm for b 1 and 60 mm for h 1 in Equation (5). ( I x 1 ) = 1 12 108 × 60 3 = 1.944 × 10 6 mm 4 Find the moment of inertia for section 2 about x axis ( I x 2 ) as shown below: ( I x 2 ) = 2 [ 1 36 b 2 h 2 3 + ( 1 2 × b 2 h 2 3 ) t ] (6) Substitute 72 mm for b 2 , 18 mm for h 2 , 6 mm for t in Equation (6). ( I x 2 ) = 2 [ 1 36 ( 72 × 18 3 ) + ( 1 2 × 72 × 18 ) 6 2 ] = 2 [ 11 , 664 + 23 , 328 ] = 69 , 984 mm 4 = 0

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, Problem 9.188RP

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Find the moment of inertia about x and y axes with respect to centroid axes.

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Chapter 9, Problem 9.187RP

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Find the moment of inertia about x and y axes with respect to centroid axes.

Solution Summary: The author calculates the moment of inertia about x and y axes with respect to centroid

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Find the moment of inertia about x and y axes with respect to centroid axes.

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