Explanation Given information: The curve equation is y = k x 2 3 . Calculation: Sketch the vertical strip shaded portion as shown in Figure 1. Write the curve equation as follows: y = k x 2 3 (1) Refer to problem 9.15 P . Write the curve equation as follows: y = k x 2 3 (1) Substitute a for x and b for y in Equation (1). b = k a 2 3 k = b a 2 3 Substitute b a 2 3 for k in Equation (1). y = b a 2 3 x 2 3 Find the area ( A ) as shown below: A = ∫ 0 a y d x (2) Substitute b a 2 3 x 2 3 for y in Equation (2). A = ∫ 0 a ( b a 2 3 x 2 3 ) d x = b a 2 3 a 5 3 5 3 = 3 5 a b Thus, the shaded area A is 3 5 a b . Thus, the curve Equation is y = b a 2 3 x 2 3 . Determine the area of the strip element d A as shown in below: d A = y d x Determine the moment of inertia ( d I y ) with respect to y axis of a rectangular strip: d I y = x 2 d A (2) Substitute y d x for d A in Equation (2). d I y = x 2 ( y ) d x (3) Substitute b a 2 3 x 2 3 for y in Equation (3). d I y = x 2 ( b a 2 3 x 2 3 ) d x Integrate equation (3) with respect to x

Vector Mechanics for Engineers: Statics and Dynamics

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ISBN: 9781259977251

Author: BEER

Publisher: MCG

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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 9.17P

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Find the moment of inertia and radius of gyration of the shaded area with respect to y axis by using direct integration.

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Chapter 9.1, Problem 9.16P

Chapter 9.1, Problem 9.18P

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Find the moment of inertia and radius of gyration of the shaded area with respect to y axis by using direct integration.

Solution Summary: The author calculates the moment of inertia and radius of gyration of the shaded area with respect to y axis by using direct integration.

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Find the moment of inertia and radius of gyration of the shaded area with respect to y axis by using direct integration.

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