Explanation Sketch the cross section as shown in Figure 1. Refer to problem 9.72. The product of inertia of the area with respect to x and y axes by using direct parallel axis theorem is 2.40 × 10 6 mm 4 . Sketch the rectangular section as shown in Figure 1. Refer to Figure 1. There are two triangle and rectangle as shown above Figure. Find the moment of inertia ( I ¯ x ) using the relation as follows: I ¯ x = 2 [ b d 3 3 ] + 2 [ b d 3 12 ] (1) Here, b is the width of section and d is the depth of section. Substitute 60 mm for b and 40 mm for d in Equation (1). I ¯ x = 2 [ 60 × 40 3 3 ] + 2 [ 60 × 40 3 12 ] = 2 , 560 , 000 + 640 , 000 = 3.20 × 10 6 mm 4 Find the moment of inertia ( I ¯ y ) using the relation as follows: I ¯ y = 2 [ b 3 d 3 ] + 2 [ b 3 d 12 ] (2) Substitute 60 mm for b and 40 mm for d in Equation (2). I ¯ y = 2 [ 60 3 × 40 3 ] + 2 [ 60 3 × 40 12 ] = 5 , 760 , 000 + 1 , 440 , 000 = 7.20 × 10 6 mm 4 Consider the point X with co-ordinates ( I x , I x y ) and point Y with co-ordinates ( I y , − I x y ) . The diameter of the Mohr circle is defined by XY distance. It is expressed as follows: X ( 3.20 × 10 6 , 2.40 × 10 6 ) and Y ( 7.20 × 10 6 , − 2.40 × 10 6 ) . Consider that the average moment of inertia ( I a v e ) is represents the distance from origin O to the center of the circle C . Find the average moment of inertia ( I a v e ) using the relation as shown below: I a v e = 1 2 ( I ¯ x + I ¯ y ) (3) Here, I ¯ x is moment of inertia about x axis and I ¯ y is moment of inertia about y axis. Substitute 3.20 × 10 6 for I ¯ x and 7.20 × 10 6 for I ¯ y in Equation (3). I ave = 1 2 ( 3.20 + 7.20 ) × 10 6 mm 4 = 1 2 ( 10.4 ) × 10 6 mm 4 = 5.20 × 10 6 mm 4 Find the radius ( R ) using the relation as shown below: R = ( I ¯ x − I ¯ y 2 ) 2 + I ¯ x y 2 (4) Here, R is radius and I ¯ x y is product of inertia. Substitute 3.20 × 10 6 for I ¯ x , 2

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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 9.4, Problem 9.98P

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

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Chapter 9.4, Problem 9.97P

Chapter 9.4, Problem 9.99P

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

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