Explanation Given information: The density of steel is 490 lb / ft 3 . Calculation: Sketch the machine element as shown in Figure 1. Find the mass of component 1 as shown below: m 1 = ρ V (1) Here, ρ is density of steel and V is the volume of rectangular component 1. Substitute 490 lb / ft 3 for ρ and ( 3.6 × 1.2 × 6 ) in 3 for V in Equation (1). m 1 = 490 × ( 3.6 × 1.2 × 6 ) = 490 × 25.92 = 15 × 10 − 3 ρ Find the mass of component 2 as shown below: m 2 = ρ ( π r 2 2 h ) (2) Here, ρ is density of steel and V is the volume of component 2. Substitute 1.8 in for r and 1.6 in for h in Equation (2). m 2 = ρ ( π 2 ( 1.8 in . × 1 ft 12 in . ) 2 × ( 1.6 in . × 1 ft 12 in . ) ) = ρ × π 2 × 0.0225 × 0.1333 = 4.7124 × 10 − 3 ρ Find the mass of component 3 as shown below: m 3 = ρ ( π r 2 h ) (3) Here, ρ is density of steel and V is the volume of component 3. Substitute 1.5 in for r and 1.2 in for h in Equation (3). m 3 = ρ ( π ( 1.5 in . × 1 ft 12 in . ) 2 × ( 1.2 in . × 1 ft 12 in . ) ) = ρ × π ( 0.015625 ) ( 0.1 ) = 4.9087 × 10 − 3 ρ Refer to Figure 9.28, “Mass moment of inertia of common geometric shapes” in the text book. Find the moment of inertia about x axis ( I x 1 ) using the relation as shown below: I y 1 = [ 1 12 ( m 1 ) ( b 2 + h 2 ) + m 1 ( d 2 ) ] (4) Substitute 15 × 10 − 3 ρ for m 1 , 3.6 in . for b , 6 in . for h , and 3 in . for d , in Equation (4). I y 1 = [ 1 12 × 15 × 10 − 3 ρ [ ( 3.6 in . ( 1 ft 12 in . ) ) 2 + ( 6 in . ( 1 ft 12 in . ) ) 2 ] + 15 × 10 − 3 ρ ( 3 in . ( 1 ft 12 in . ) ) ] = 1 12 × 15 × 10 − 3 ρ ( 0.34 ) + 9.375 × 10 − 4 = 1.3625 × 10 − 3 ρ Find the moment of inertia about y axis for section 2 as shown below: I y 2 = I ¯ y 2 ′ + m d 2 = m 12 ( 3 r 2 + L 2 ) + m d 2 (5) Substitute 1.8 in . for r , 1.6 in . for L , 5.2 in . for d , and 4.7124 × 10 − 3 ρ for m 2 in Equation (5). I y 2 = [ 4

Vector Mechanics for Engineers: Statics and Dynamics

11th Edition

ISBN: 9780073398242

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

Publisher: McGraw-Hill Education

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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 9.5, Problem 9.144P

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Find the mass moment of inertia with respect to y axis.

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Chapter 9.5, Problem 9.143P

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Find the mass moment of inertia with respect to y axis.

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