Explanation Calculation: Sketch the cross section as shown in Figure 1. Refer to problem 9.83. The moment of inertia I ¯ x about x axis is 0.390 in 4 . The moment of inertia I ¯ y about y axis is 1.09 in 4 . Refer to problem 9.74. The product of inertia I ¯ x y about xy axis is − 0.37983 in 4 . Find the orientation of the principal axes at the origin using the equation as follows: Refer to Equation 9.25 in section “Principal axes and principal moment of inertia” in the textbook. tan 2 θ m = − 2 I x y I x − I y (1) Here, ( I x , I y , I x y ) is magnitude. Substitute − 0.37983 in 4 for I x y , 1.09 in 4 for I y , and 0.390 in 4 for I x in Equation (1). tan 2 θ m = − 2 ( − 0.37983 ) ( 0.390 ) − ( 1.09 ) = − − 0.75966 − 0.7 = − 1.08523 tan 2 θ m = − 1.08523 2 θ m = tan − 1 ( 0.62804 ) 2 θ m = − 47.341 ° θ m = − 23.7 ° 2 θ m = − 47.341 + 180 ° = 132.659 ° θ m = 66.3 ° Thus, the orientation of the principal axes at the origin is − 23.7 ° and 6 6 .3 ° _

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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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 9.3, Problem 9.89P

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Find the orientation of the principal axes at the origin and the corresponding value of moment of inertia.

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Chapter 9.3, Problem 9.88P

Chapter 9.3, Problem 9.90P

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Vector Mechanics for Engineers: Statics and Dynamics

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Find the orientation of the principal axes at the origin and the corresponding value of moment of inertia.

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