Explanation Given information: The curve Equation is y = 2 b − c x 2 Calculation: Sketch the shaded portion with vertical strip as shown in Figure 1. Write the curve ( y 1 ) Equation as follows: y 1 = k x 2 (1) Write the curve ( y 2 ) Equation as follows: y 2 = b − c x 2 (2) Refer to Figure 1. For x = a 1 , y 1 = y 2 = b . Find the constant ( k 1 ) using the relation: Substitute b for ( y 1 ) and a for x in Equation (1). b = k a 2 k = b a 2 Substitute b for ( y 2 ) and a for x in Equation (2). y = 2 b − c x 2 c = b a 2 Substitute b a 2 for k in Equation (1). y 1 = ( b a 2 ) x 2 = b a 2 x 2 Substitute b a 2 for c in Equation (2). y 2 = b ( 2 − x 2 a 2 ) Determine the area of the strip element d A as shown in below: d A = ( y 2 − y 1 ) d x (3) Substitute b ( 2 − x 2 a 2 ) for y 2 and b a 2 x 2 for y 1 in Equation (3). d A = ( b ( 2 − x 2 a 2 ) − b a 2 x 2 ) d x = 2 b a 2 ( a 2 − x 2 ) d x Find the shaded area ( A ) using the relation: A = ∫ d A (4) Substitute 2 b a 2 ( a 2 − x 2 ) d x for d A in Equation (4). A = ∫ 2 b a 2 ( a 2 − x 2 ) d x = 2 b a 2 [ a 2 x − 1 3 x 3 ] 0 a = 2 b a 2 [ a 2 ( a ) − 1 3 ( a ) 3 ] = 2 b a 2 [ a 3 − 1 3 ( a ) 3 ] = 2 b a 2 a 3 2 3 A = 4 3 a b Determine the moment of inertia ( d I x ) with respect to x axis: Refer Equation 9.2 in section 9.1B determining the moment of inertia of an area by integration: d I x = ( 1 3 y 2 3 − 1 3 y 1 3 ) d x (5) Substitute b ( 2 − x 2 a 2 ) for y 2 and b a 2 x 2 for y 1 in Equation (5)

Vector Mechanics for Engineers: Statics and Dynamics

12th Edition

ISBN: 9781259977251

Author: BEER

Publisher: MCG

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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, Problem 9.187RP

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Find the moment of inertia and radius of gyration of the shaded area shown with respect to x axis.

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Chapter 9, Problem 9.186RP

Chapter 9, Problem 9.188RP

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Vector Mechanics for Engineers: Statics and Dynamics

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Find the moment of inertia and radius of gyration of the shaded area shown with respect to x axis.

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Find the moment of inertia and radius of gyration of the shaded area shown with respect to x axis.

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