Explanation Sketch the cross section of the semicircle as shown in Figure 1. Refer to problem 9.81. 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 semicircular 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 Refer to Figure 1. 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 semicircular 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 Refer to problem 9.73. 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 . 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 ( 324 π , 864 ) and Y ( 648 π , − 864 ) . Consider that the average moment of inertia ( I ave ) is represents the distance from origin O to the center of the circle C . Find the average moment of inertia ( I ave ) using the relation as shown below: I ave = 1 2 ( I ¯ x + I ¯ y ) (5) Here, I ¯ x is moment of inertia about x axis and I ¯ y is moment of inertia about y axis. Substitute 324 π for I ¯ x and 648 π in 4 for I ¯ y in Equation (5). I ave = 1 2 ( 324 π + 648 π ) = 1 2 ( 3 , 053.628 ) = 1 , 526.81 in 4 Find the radius ( R ) using the relation as shown below: R = ( I ¯ x − I ¯ y 2 ) 2 + I ¯ x y 2 (6) Here, R is radius and I ¯ x y is product of inertia

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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 9.93P

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Find the moment and product of inertia of the area with respect to x and y axes about through 60° by using Mohr circle method.

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Chapter 9.4, Problem 9.92P

Chapter 9.4, Problem 9.94P

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Find the moment and product of inertia of the area with respect to x and y axes about through 60 ° by using Mohr circle method.

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Find the moment and product of inertia of the area with respect to x and y axes about through 60° by using Mohr circle method.

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