Explanation Given information: The weight per unit length of 0.033 lb / ft . The unit vector λ is ( − 3 i + 6j + 2k ) 7 . Calculation: Sketch the homogeneous wire as shown in figure 1. Find the mass of each component 1 as shown below: m 1 = W g = 1 g ( W / L ) A L L (1) Here, ( W / L ) is weight per unit length and L is the length of the wire. Substitute 0.033 lb / ft for ( W / L ) , 32.2 ft / s 2 for g , and ( 2 π × 16 in . ) for L in Equation (1). m 1 = 1 32.2 × 0.033 × ( 2 π × 16 in . ) × 1 ft 12 in . = 8.5857 × 10 − 3 lb ⋅ s 2 / ft Find the mass of each component as shown below: The mass of components 2,3,4,5 are equal. m 2 = m 3 = m 4 = m 5 = W g = 1 g ( W / L ) A L L (2) Here, ( W / L ) is weight per unit length and L is the length of the wire. Substitute 0.033 lb / ft for ( W / L ) , 32.2 ft / s 2 for g , and ( 8 in . ) for L in Equation (2). m 2 = m 3 = m 4 = m 5 = 1 32.2 × ( 0.033 ) ( 8 ) × 1 12 = 0.6832 × 10 − 3 lb ⋅ ft / s 2 The centroid products of inertia of each component is zero due to symmetry. I ¯ x ′ y ′ = 0 I ¯ y ′ z ′ = 0 I ¯ z ′ x ′ = 0 The centroidal distances of sections 1, 4 and 5 about the x axis is equal to zero. The centroidal distances of section 1 about the y axis is equal to zero. The centroidal distances of sections 1, 2 and 3 about the z axis is equal to zero. x ¯ 1 = x ¯ 4 = x ¯ 5 = 0 y ¯ 1 = 0 z ¯ 1 = z ¯ 2 = z ¯ 3 = 0 Find the moment of inertia ( I x y ) about x y as shown below: ( I x y ) = ∑ ( I ¯ x ′ y ′ + m x ¯ y ¯ ) = ∑ I ¯ x ′ y ′ + m 2 x ¯ 2 y ¯ 2 + m 3 x ¯ 3 y ¯ 3 (3) Here, m 2 is mass of section 2, m 3 is mass of section 3, x ¯ 2 y ¯ 2 is centroid section about x , y axis. Substitute 0 for ∑ I ¯ x ′ y ′ , 0.6832 × 10 − 3 lb ⋅ ft / s 2 for m 2 , 0.6832 × 10 − 3 lb ⋅ ft / s 2 for m 3 , 16 in . for x ¯ 2 , 4 in . for y 2 , 12 in . x ¯ 3 , 8 in . y ¯ 3 in Equation (3). ( I x y ) = ( 0.6832 × 10 − 3 ( 16 in . ( 1 ft 12 in . ) 4 in . ( 1 ft 12 in . ) ) + 0.6832 × 10 − 3 ( 12 in . ( 1 ft 12 in

Vector Mechanics for Engineers: Statics and Dynamics

12th Edition

ISBN: 9781259638091

Author: Ferdinand P. Beer, E. Russell Johnston Jr., David Mazurek, Phillip J. Cornwell, Brian Self

Publisher: McGraw-Hill Education

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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 9.6, Problem 9.172P

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The mass moment of inertia of the component with respect to the axis through origin.

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Chapter 9.6, Problem 9.171P

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The mass moment of inertia of the component with respect to the axis through origin.

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