(a) Explanation Calculation: Sketch the annular area as shown in Figure 1. Refer to problem 9.26P. J O = π 2 [ R 2 4 − R 1 4 ] Express the radius of gyration using the relation as follows: k O 2 = J O A (1) Substitute π 2 [ R 2 4 − R 1 4 ] for J O and π [ R 2 2 − R 1 2 ] for A in Equation (1). k O 2 = π 2 [ R 2 4 − R 1 4 ] π [ R 2 2 − R 1 2 ] k O 2 = 1 2 [ R 2 2 + R 1 2 ] (2) Express the thickness of circular section as follows: t = R 2 − R 1 Here, R 1 is inner radius and R 2 is outer radius. Find the mean radius ( R m ) using the relation as follows: ( R m ) = 1 2 ( R 1 + R 2 ) (3) Find the radius ( R 1 ) as shown below: From Equation (3). R 1 = R m − 1 2 t Find the radius ( R 2 ) as shown below: From Equation (3) (b) To determine Find the percentage error introduced by using in place of k O following values of t / R m : 1 , 1 2 , 1 10

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 9.1, Problem 9.26P

(a)

To determine

Prove the polar radius of gyration of the annual area approximately equal to the mean radius kO=Rm_.

(b)

To determine

Find the percentage error introduced by using in place of kO following values of t/Rm:1,12,110

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Chapter 9.1, Problem 9.25P

Chapter 9.1, Problem 9.27P

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Vector Mechanics for Engineers: Statics and Dynamics

Ch. 9.1 - 9.1 through 9.4 Determine by direct integration...Ch. 9.1 - 9.1 through 9.4 Determine by direct integration...Ch. 9.1 - Prob. 9.3P Ch. 9.1 - Prob. 9.4P Ch. 9.1 - 9.5 through 9.8 Determine by direct integration...Ch. 9.1 - 9.5 through 9.8 Determine by direct integration...Ch. 9.1 - Prob. 9.7P Ch. 9.1 - Prob. 9.8P Ch. 9.1 - 9.9 through 9.11 Determine by direct integration...Ch. 9.1 - Prob. 9.10P

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Prove the polar radius of gyration of the annual area approximately equal to the mean radius k O = R m _ .

Solution Summary: The author demonstrates the polar radius of gyration of the annual area approximately equal to the mean radius.