Explanation Given: The width of the shaded area is 140 mm The height of the shaded area is 30 mm The angle is 60 ° . Show the shaded area as in Figure 1. From the Figure 1, As the shaded area is symmetrical about the x and y axes, write the product of inertia for the shaded area with respect to the x and y axes as 0 . I x y = 0 Compute the moment of inertia for the shaded area with respect to the x axes. I x = 1 12 2 a c 3 + 1 12 b ( 2 a ) 3 (I) Compute the moment of inertia for the shaded area with respect to the y axes. I y = 1 12 2 a b 3 + 2 a b [ a + b 2 ] 2 + 1 12 c ( 2 a ) 3 (II) Compute the moment of inertia for the shaded area with respect to the u axes. I u = [ I x + I y 2 ] + [ I x − I y 2 ] cos 2 θ − I x y sin 2 θ (III) Compute the moment of inertia for the shaded area with respect to the v axes. I v = [ I x + I y 2 ] − [ I x − I y 2 ] cos 2 θ + I x y sin 2 θ (IV) Compute the moment of inertia for the shaded area with respect to the u and v axes. I u v = [ I x − I y 2 ] sin 2 θ + I x y cos 2 θ (V) Conclusion: Substitute 10 mm for a , 120 mm for b , and 120 mm for c in Equation (I). I x = 1 12 2 ( 10 ) ( 120 ) 3 + 1 12 ( 120 ) [ 2 ( 10 ) ] 3 = 288 ( 10 4 ) + 8 ( 10 4 ) = 2.96 ( 10 6 ) mm 4 Substitute 10 mm for a , 120 mm for b , and 120 mm for c in Equation (II). I y = 1 12 2 ( 10 ) ( 120 ) 3 + 2 ( 10 ) ( 120 ) [ 10 + 120 2 ] 2 + 1 12 ( 120 ) [ 2 ( 10 ) ] 3 = 288 ( 10 4 ) + 1 , 176 ( 10 4 ) + 8 ( 10 4 ) = 14.72 ( 10 6 ) mm 4 Substitute 2.96 ( 10 6 ) mm 4 for I x , 14.72 ( 10 6 ) mm 4 for I y , 0 for I x y , and 60 ° for θ in Equation (III). I u = { [ 2.96 ( 10 6 ) + 14.72 ( 10 6 ) 2 ] + [ 2.96 ( 10 6 ) − 14.72 ( 10 6 ) 2 ] cos 2 ( 60 ° ) − 0 sin 2 ( 60 ° ) } = [ 2

Engineering Mechanics: Statics & Dynamics (14th Edition)

14th Edition

ISBN: 9780133915426

Author: Russell C. Hibbeler

Publisher: PEARSON

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Moment of Inertia

Inertia can be termed as a resistance offered by a body to change its state. A rigid body initially at rest will apply resistance when acted by an external force that tends to change its state of rest or velocity, or a body initially at motion will provi…

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Chapter 10.7, Problem 80P

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The moments of inertia Iu,Iv and the product of inertia Iuv for the shaded area.

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Chapter 10.7, Problem 79P

Chapter 10.7, Problem 81P

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The moments of inertia I u , I v and the product of inertia I u v for the shaded area.

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