Explanation Given: The width of the rectangular area is 6 in . The height of the rectangular area is 3 in . Show the free body diagram of the rectangular area as in Figure (1). From the Figure 1, Compute the moment of inertia of the rectangular area about the x axes. I x = 1 3 b a 3 (I) Compute the moment of inertia of the rectangular area about the y axes. I y = 1 3 a b 3 (II) Compute the moment of inertia of the rectangular area about the x and y axes. I x y = a 2 b 2 a b (III) Calculate the maximum and minimum principal of moments. I max / min = I x + I y 2 ± [ ( I x − I y ) / 2 ] 2 + ( I x y ) 2 (IV) Compute the orientation of the principal axes. tan 2 θ p 2 = − 2 I x y I x − I y (V) Compute the orientation of the principal axes. θ p 1 = θ p 2 − 90 ° (VI) Conclusion: Substitute 6 in . for b and 3 in . for a in Equation (I). I x = 1 3 ( 6 ) ( 3 ) 3 = 54 in . 4 Substitute 6 in . for b and 3 in . for a in Equation (II). I y = 1 3 ( 3 ) ( 6 ) 3 = 216 in . 4 Substitute 6 in . for b and 3 in . for a in Equation (III). I x y = 3 2 × 6 2 ( 3 ) ( 6 ) = 81 in . 4 Substitute 54 in

Engineering Mechanics: Statics Plus Mastering Engineering with Pearson eText -- Access Card Package (14th Edition) (Hibbeler, The Engineering Mechanics: Statics & Dynamics Series, 14th Edition)

14th Edition

ISBN: 9780134160689

Author: Russell C. Hibbeler

Publisher: PEARSON

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Chapter 10.7, Problem 76P

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The orientation (θp1,θp2) of the principal axes having an origin at point O, and the principal moments of inertia (Imax,Imin) for the rectangular area about these axes.

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Chapter 10.7, Problem 75P

Chapter 10.7, Problem 77P

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Engineering Mechanics: Statics Plus Mastering Engineering with Pearson eText -- Access Card Package (14th Edition) (Hibbeler, The Engineering Mechanics: Statics & Dynamics Series, 14th Edition)

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The orientation ( θ p 1 , θ p 2 ) of the principal axes having an origin at point O , and the principal moments of inertia ( I max , I min ) for the rectangular area about these axes.

Solution Summary: The author shows the free body diagram of the rectangular area as in Figure 1.

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The orientation (θp1,θp2) of the principal axes having an origin at point O, and the principal moments of inertia (Imax,Imin) for the rectangular area about these axes.

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