Explanation Given information: The curve equation is y = ( a 2 3 − x 2 3 ) 3 2 Calculation: Sketch the thin steel wire as shown in Figure 1. Write the Equation of curve ( y ) as shown below: y = ( a 2 3 − x 2 3 ) 3 2 (1) Differentiate Equation (1) with respect to x . y = ( a 2 3 − x 2 3 ) 3 2 d y d x = − x 1 3 ( a 2 3 − x 2 3 ) 1 2 Square on both sides, ( d y d x ) 2 = ( − x 1 3 ) 2 ( a 2 3 − x 2 3 ) 1 2 ( 2 ) ( d y d x ) 2 = x − 2 3 ( a 2 3 − x 2 3 ) (2) Adding 1 on both sides in Equation (2). 1 + ( d y d x ) 2 = 1 + x − 2 3 ( a 2 3 − x 2 3 ) = ( a x ) 2 3 Find the differential element of mass ( d m ) as shown below: Consider d L = 1 + ( d y d x ) 2 d x d m = m ′ d L = m ′ 1 + ( d y d x ) 2 d x = m ′ ( a 3 ) 1 3 d x (3) Integrate Equation (3) and apply the limits. m = ∫ d m = ∫ 0 a m ′ a 1 3 x 1 3 d x = 3 2 m ′ a 1 3 [ x 2 3 ] 0 a = 3 2 m ′ a 1 3 [ a 2 3 ] = 3 2 m ′ a ( 3 3 ) m = 3 2 m ′ a Find the mass moment of inertia of the wire with respect to x axis by using direct integration

Vector Mechanics for Engineers: Statics and Dynamics

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ISBN: 9781259977251

Author: BEER

Publisher: MCG

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Chapter 9.5, Problem 9.126P

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Find the mass moment of inertia of the wire with respect to x, z, and y axis by using direct integration.

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Chapter 9.5, Problem 9.125P

Chapter 9.5, Problem 9.127P

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Find the mass moment of inertia of the wire with respect to x , z , and y axis by using direct integration.

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Find the mass moment of inertia of the wire with respect to x, z, and y axis by using direct integration.

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