(a) Explanation Calculation: Sketch the cross section as shown in Figure 1. Refer to Figure 1. Find the area ( A 1 ) of half circle as shown below. A 1 = π 2 ( a b ) 2 (1) Substitute 54 mm for a and 36 mm for b in Equation (1). A 1 = π 2 ( 54 ) ( 36 ) = 3 , 053.6 mm 2 Find the area ( A 2 ) of semicircle as shown below. A 2 = − π 2 ( r ) 2 (2) Substitute 18 in . for r in Equation (2). A 2 = − π 2 ( 18 ) 2 = − 508.9 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 3 , 053.6 mm 2 for A 1 , and − 508.9 mm 2 for A 2 in Equation (3). A = 3 , 053.6 − 508.9 = 2 , 544.7 mm 2 Refer to Figure 1. Find the centroid for section 1 about y axis ( y 1 ) as shown below: y 1 = 4 r 3 π (4) Substitute 18 in . for r in Equation (4). y 1 = 4 ( 36 ) 3 π = 144 3 π = 48 π = 15.2789 mm Find the centroid for section 2 about x axis ( y 2 ) as shown below: y 2 = 4 ( 18 ) 3 π = 72 3 π = 24 π = 7 (b) To determine Find the centroid of the area.

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 9.189RP

(a)

To determine

Find the polar moment of inertia of the area with respect to point O.

(b)

To determine

Find the centroid of the area.

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Chapter 9, Problem 9.188RP

Chapter 9, Problem 9.190RP

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Chapter 9 Solutions

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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Find the polar moment of inertia of the area with respect to point O .

Solution Summary: The author calculates the polar moment of inertia of the area with respect to point O.