(a) Explanation Given information: The length ( L ) of the section is 15 in . The width ( b ) of the section is 4.2 in . The specific weight of aluminum is 0.100 lb/in . 3 . Calculation: Consider the outer square as section 1 and the inner square as section 2. Show the section 1 and section 2 in Figure 1. Write the expression for the volume ( V 1 ) of the section 1 as below: V 1 = b 2 L Calculate the mass ( m 1 ) of section 1 using the relation: m 1 = ρ V 1 Substitute b 2 L for V 1 . m 1 = ρ b 2 L Write the expression for the volume ( V 2 ) of the section 2 as below: V 2 = a 2 L Calculate the mass ( m 2 ) of section 2 using the relation: m 2 = ρ V 2 Substitute a 2 L for V 2 . m 2 = ρ a 2 L Refer Figure 9.28 “Mass moments of inertia of common geometrical shapes” in Textbook for mass moment of inertia of section 1 and section 2. Calculate the mass moment of inertia ( I A A ' ) 1 with respect to A A ' axis for section 1as below: ( I A A ' ) 1 = 1 12 m 1 ( b 2 + b 2 ) + m 1 ( a 2 ) 2 Calculate the mass moment of inertia ( I A A ' ) 2 with respect to A A ' axis for section 2 as below: ( I A A ' ) 2 = 1 12 m 2 ( a 2 + a 2 ) + m 2 ( a 2 ) 2 Calculate the mass moment of inertia ( I A A ' ) with respect to A A ' axis for square hole section as below: I A A ' = ( I A A ' ) 1 − ( I A A ' ) 2 Substitute 1 12 m 1 ( b 2 + b 2 ) + m 1 ( a 2 ) 2 for ( I A A ' ) 1 and 1 12 m 2 ( a 2 + a 2 ) + m 2 ( a 2 ) 2 for ( I A A ' ) 2 . I A A ' = [ 1 12 m 1 ( b 2 + b 2 ) + m 1 ( a 2 ) 2 ] − [ 1 12 m 1 ( a 2 + a 2 ) + m 2 ( a 2 ) 2 ] Substitute ρ b 2 L for m 1 and ρ a 2 L for m 2 (b) To determine The corresponding values of the mass moment of inertia ( I A A ' ) max and the radius of gyration ( k A A ' ) with respect to the axis A A ' .

Connect 1 Semester Access Card for Vector Mechanics for Engineers: Statics and Dynamics

11th Edition

ISBN: 9781259639272

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 9.131P

(a)

To determine

Find the value of a.

(b)

To determine

The corresponding values of the mass moment of inertia (IAA')max and the radius of gyration (kAA') with respect to the axis AA'.

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Chapter 9.5, Problem 9.130P

Chapter 9.5, Problem 9.132P

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Connect 1 Semester Access Card for Vector Mechanics for Engineers: Statics and Dynamics

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