Complete COSMOS: Online Solutions Manual Organization System
Chapter 9, Solution 131. from Consider shell to be formed by removing hemisphere of radius r hemisphere of radius r + t. For hemisphere:
I = 2 2 mr 5 Area = 1 4 r 2 = 2 r 2 2
(
)
m = V =
1 4 3 2 r = r 3 2 3 3
I = 2 2 4 3 2 r 5 r r = 53 15
Thus For hemispherical shell:
I =
4 4 5 ( r + t ) - r 5 = r 5 + 5r 4t + 10r 3t 2 + ... - r 5 15 15 4 4t r 3
Neglect terms with powers of t > 1,
I =
Mass of shell:
I = 2 r 2t
m = V = tA = t 2 r 2 = 2 r 2t
(
)
(
)2r 3
2
=
2 2 mr 3
I =
2 2 mr 3
Radius of gyration:
k2 = I 2 = r2 m 3
k = 0.816r
Vector Mechanics for Engineers: Statics and Dynamics, 8/e, Ferdinand P. Beer, E. Russell Johnston, Jr., Elliot R. Eisenberg, William E. Clausen, David Mazurek, Phillip J. Cornwell 2007 The McGraw-Hill Companies.