(a) Explanation Calculation: Sketch the cross section as shown in Figure 1. Refer to Figure 1. Find the area of section 1 as shown below: A 1 = π 4 r 4 (1) Here, r is radius of circle. Substitute 4 in . for r in Equation (1). A 1 = π 4 ( 4 ) 4 = 4 π Find the area of section 2 as shown below: A 2 = 1 2 b h (2) Here, b is width of triangle and h is height of triangle. Substitute 4 in . for h and 4 in . for b in Equation (2). A 2 = 1 2 × 4 × 4 = 8 in 2 Find the area of section 3 as shown below: A 3 = 1 2 b h (3) Substitute 4 in . for h and 4 in . for b in Equation (3). A 3 = 1 2 × 4 × 4 = 8 in 2 Find the total Area ( A ) as shown below: A = 4 π + 8 + 8 = 28.566 in 2 Find the centroid ( x 1 ) section 1 as shown below: x 1 = 16 3 π Find the centroid ( x 2 ) section 2 as shown below: x 2 = − 4 3 in . Find the centroid ( x 3 ) section 3 as shown below: x 3 = 4 3 in . Find the centroid ( x ¯ ) using the relation as follows: x ¯ = A 1 x 1 + A 2 x 2 + A 3 x 3 A 1 + A 2 + A 3 (4) Substitute 4 π for A 1 , 8 in 2 for A 2 , 8 in 2 for A 3 , 16 3 π in . for x 1 , and − 4 3 in . for x 2 , 4 3 in . for x 3 in Equation (4). x ¯ = 4 π × ( 16 3 π ) + 8 × ( − 4 3 ) + ( 8 × ( 4 3 ) ) 4 π + 8 + 8 = 21 (b) To determine Find the centroid of 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 9.2, Problem 9.46P

(a)

To determine

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

(b)

To determine

Find the centroid of area.

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Chapter 9.2, Problem 9.45P

Chapter 9.2, Problem 9.47P

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It is known that for a given area Iy = 48 x 106 mm4 and Ixy = -20 x 106 mm4, where the x and y axes are rectangular centroidal axes. If the axis corresponding to the maximum product of inertia is obtained by rotating the x axis 67.5° counterclockwise about C , use Mohr’s circle to determine (a) the moment of inertia Ix of the area, (b) the principal centroidal moments of inertia.

For the entire section shown, the moments of inertia with respect to the centroidal x and y axes at point C are Ix = 0.162(106) mm4 and Iy = 0.454(106) mm4, respectively. a. Determine the product of inertia with respect to the centroid at C, in mm4. b. Use a Mohr's Circle to determine the orientation (in degrees) of the principal axes of the section about the centroid C. c. Use the same Mohr's Circle to determine the values of the principal moments of inertia about the centroid C, in mm4.

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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.