Explanation Given: The height of the rectangular area, a = 6 in . The width of the rectangular area, b = 3 in . Show the Mohr’s circle as in Figure (1). From Figure (1), Express the moment of inertia for the rectangular area with respect to the x axis. I x = 1 3 b a 3 (I) Express the moment of inertia for the rectangular area with respect to the y axis. I y = 1 3 a b 3 (II) Express the product of inertia for the rectangular area with respect to the x and y axes. I x y = a 2 b 2 a b (III) Express the directions ( θ ) of the principal axes with its origin located at point O . tan ( 2 θ ) = − 2 I x y I x − I y (IV) Calculate the center of the circle. c = I x + I y 2 (V) Calculate the radius of the circle. R = ( I x − I y ) 2 + ( I x y ) 2 (VI) Calculate the maximum principal moments of inertia for the rectangular area. I max = c + R (VII) Calculate the minimum principal moments of inertia for the rectangular area. I min = c − R (VIII) Conclusion: Substitute 6 in . for a and 3 in . for b in Equation (I). I x = 1 3 ( 3 ) ( 6 ) 3 = 216 in . 4 Substitute 6 in . for a and 3 in . for b in Equation (II). I y = 1 3 ( 6 ) ( 3 ) 3 = 54 in . 4 Substitute 6 in . for a and 3 in . for b in Equation (III). I x y = 6 2 × 3 2 × ( 6 ) ( 3 ) = 3 ( 1.5 ) ( 6 ) ( 3 ) = 81 in . 4 Substitute 81 in . 4 for I x y , 54 in

13th Edition

ISBN: 9780132915540

Author: Russell C. Hibbeler

Publisher: Prentice Hall

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Question

Chapter 10.7, Problem 77P

To determine

The directions of the principal axes and the principal moments of inertia for the rectangular area.

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Chapter 10.7, Problem 76P

Chapter 10.7, Problem 78P

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Chapter 10 Solutions

Engineering Mechanics: Statics

Ch. 10.3 - Determine the moment of inertia of the shaded area...Ch. 10.3 - Determine the moment of inertia of the shaded area...Ch. 10.3 - Determine the moment of inertia of the shaded area...Ch. 10.3 - Determine the moment of inertia of the shaded area...Ch. 10.3 - Prob. 1P Ch. 10.3 - Prob. 2P Ch. 10.3 - Prob. 3P Ch. 10.3 - Prob. 4P Ch. 10.3 - Prob. 5P Ch. 10.3 - Prob. 6P

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