Explanation Calculation: Sketch the semicircle as shown in Figure 1. Find the moment of inertia ( I x ) 1 and ( I x ) 2 using the relation as follows: ( I x ) 1 = ( I x ) 2 = π 8 ( r ) 4 (1) Here, r is the radius of the semi-circular section. Substitute 6 in . for r in Equation (1). ( I x ) 1 = ( I x ) 2 = π 8 ( 6 ) 4 = 162 π in 4 Find the total moment of inertia ( I ¯ x ) about x axis using the relation as follows: ( I ¯ x ) = 2 ( I x ) 1 , 2 (2) Substitute 162 π in 4 for ( I x ) 1 , 2 in Equation (2). ( I ¯ x ) = 2 ( 162 π ) = 324 π in 4 Find the moment of inertia ( I y ) 1 and ( I y ) 2 using the relation as follows: ( I y ) 1 = ( I y ) 2 = π 8 ( r ) 4 + [ π 2 ( r ) 2 ] 3 2 (3) Here, r is the radius of the semi-circular section. Substitute 6 in . for r in Equation (3). ( I y ) 1 = ( I y ) 2 = π 8 ( 6 ) 4 + [ π 2 ( 6 ) 2 ] 3 2 = 162 π + 162 π = 324 π Find the total moment of inertia ( I ¯ y ) about y axis using the relation as follows: ( I ¯ y ) = 2 ( I y ) 1 , 2 (4) Substitute 324 π in 4 for ( I y ) 1 , 2 in Equation (4). ( I ¯ y ) = 2 ( 324 π ) = 648 π in 4 The product of inertia ( I ¯ x y ) of the area with respect to x and y axes by using direct parallel axis theorem is 864 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 (5) Here, ( I x , I y , I x y ) is magnitude

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Author: BEER

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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 9.87P

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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 9.86P

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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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