Explanation Calculation: Sketch the horizontal strip along the trapezoidal portion as shown in Figure 1. Refer to Figure 1. x = a − a 2 ( v a ) = a − 1 2 v Determine the area of the strip element d A as shown in below: d A = x d v Find the shaded area ( A ) using the relation: 1 2 A = ∫ 0 a d A = ∫ 0 a x d v (1) Substitute a − 1 2 v for x in Equation (1). 1 2 A = ∫ 0 a ( a − 1 2 v ) d v 1 2 A = ( a v − 1 4 v 2 ) 0 a 1 2 A = ( a ( a ) − 1 4 ( a ) 2 ) 1 2 A = a 2 − a 2 4 A = 3 a 2 4 × 2 A = 3 a 2 2 Find the moment of inertia about x axis using the relation as follows: 1 2 I x = ∫ v 2 d A = ∫ v 2 ( x d v ) (2) Substitute a − 1 2 v for x in Equation (2). 1 2 I x = ∫ 0 a v 2 ( a − 1 2 v ) d v = ∫ 0 a ( v 2 a − 1 2 v 3 ) d v = [ v 3 3 a − 1 8 v 4 ] 0 a = [ a 3 3 a − a 4 8 ] I x = [ a 4 3 − a 4 8 ] = a 4 [ 1 3 − 1 8 ] = 5 24 a 4 = 5 12 a 4 Find the moment of inertia about y axis using the relation as follows: 1 2 I y = ∫ 1 3 x 3 d v (3) Substitute a − 1 2 v for x in Equation (3)

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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 9.1, Problem 9.22P

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Find the polar moment of inertia and polar radius of gyration of the trapezoid shown with respect to point P.

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Chapter 9.1, Problem 9.21P

Chapter 9.1, Problem 9.23P

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Find the polar moment of inertia and polar radius of gyration of the trapezoid shown with respect to point P .

Solution Summary: The author calculates the polar moment of inertia and the radius of gyration of the trapezoid shown with respect to point P.

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Find the polar moment of inertia and polar radius of gyration of the trapezoid shown with respect to point P.

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