Explanation Given information: The specific weight γ b of bronze is 0.310 lb / in 3 . The specific weight γ a of aluminum is 0.100 lb / in 3 . The specific weight γ n of neoprene is 0.0452 lb / in 3 . Calculation: Sketch the cross section of idler roller as shown in figure 1. Express the mass ( m ) of cylindrical ring as shown below: m = ρ V (1) Here, ρ is density and V is volume of cylindrical ring. Substitute ρ t for ρ and π 4 ( d 2 2 − d 1 2 ) in Equation (1). m = ( ρ t ) π 4 ( d 2 2 − d 1 2 ) = π 4 ( γ g ) t ( d 2 2 − d 1 2 ) = [ π 4 ( 1 g ) γ b t b ( d 2 2 − d 1 2 ) b + γ a t a ( d 2 2 − d 1 2 ) a + γ n t n ( d 2 2 − d 1 2 ) n ] (2) Here, t b is thickness of bronze, t a is thickness of aluminum, t n is thickness of neoprene, ( d 2 2 − d 1 2 ) b is diameter of bronze, ( d 2 2 − d 1 2 ) a is diameter of aluminum, and ( d 2 2 − d 1 2 ) n is diameter of neoprene. Refer to Figure 9.28, “Mass moment of inertia of common geometric shapes” in the textbook. Express the moment of inertia with respect to axis A A ′ as shown below: I A A ′ = 1 2 m 2 ( d 2 2 ) 2 − 1 2 m 1 ( d 1 2 ) 2 = 1 8 [ ( ρ t × π 4 d 2 2 ) d 2 2 − ( ρ t × π 4 d 1 2 ) d 1 2 ] = 1 8 [ ( ρ t × π 4 d 2 4 ) − ( ρ t × π 4 d 1 4 ) ] = 1 8 ( π 4 ρ t ) ( d 2 4 − d 1 4 ) = 1 8 ( π 4 ρ t ) ( d 2 2 − d 1 2 ) ( d 2 2 + d 1 2 ) I A A ′ = 1 8 m ( d 1 2 + d 2 2 ) Consider the roller as three concentric rings and working from the bronze outward. Substitute 32.2 ft / s 2 for g , 0.310 lb / in 3 for γ b , 0.100 lb / in 3 for γ a , 0.0452 lb / in 3 for γ n , ( 13 16 in . ) for t b , ( 11 16 in . ) for t a , ( 11 16 in . ) for t n , ( 3 8 ) in . for ( d 2 2 ) b , ( 1 4 ) in . for ( d 1 2 ) b , ( 1 2 ) in . for ( d 1 2 ) a , ( 3 8 ) in . for ( d 2 2 ) a , ( 1 1 8 ) in . for ( d 1 2 ) n ,and ( 1 2 ) in . for ( d 2 2 ) n , in Equation (2). m = π 4 × 1 32.2 { 0.310 ( lb / in 3 ) ( 13 16 in . ) [ ( 3 8 ) 2 − ( 1 4 ) 2 ] + ( 0.100 ( lb / in 3 ) ) ( 11 16 in

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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Chapter 9.5, Problem 9.127P

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Find the mass moment of inertia and radius of gyration of the idle roller with respect to axis AA′

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Chapter 9.5, Problem 9.126P

Chapter 9.5, Problem 9.128P

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Vector Mechanics for Engineers: Statics and Dynamics

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Find the mass moment of inertia and radius of gyration of the idle roller with respect to axis A A ′

Solution Summary: The author explains the mass moment of inertia and radius of gyration of the idle roller with respect to axis AAprime .

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Find the mass moment of inertia and radius of gyration of the idle roller with respect to axis AA′

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