University:
Michigan State University
Course:
ERG 160 | Success in Science, Technology, Engineering, and Mathematics
Academic year:

2019
Views:

213

Pages:

2
Author:

Campbell Wagner

COSMOS: Complete Online Solutions Manual Organization System Chapter 9, Solution 136. First note m1 = ρV1 = ρ b 2 L And m2 = ρV2 = ρ a 2 L (a) Using Figure 9.28 and the parallel-axis theorem, have I AA′ = ( I AA′ )1 − ( I AA′ )2 2 1 a  2 2 =  m1 b + b + m1     2   12 ( ) 2 1 a  −  m2 a 2 + a 2 + m2     2   12 ( ) 1  1  5  = ρ b2 L  b2 + a 2  − ρ a 2 L  a 2  4  6  12  ( = ) ρL ( 2b 12 ( 4 + 3b 2a 2 − 5a 4 ) ) dI AA′ ρL = 6b 2a − 20a3 = 0 da 12 ( Then a=0 or ) 3 10 a=b and d 2 I AA′ ρL 1 = 6b 2 − 60a 2 = ρ L b 2 − 10a 2 2 12 2 da Also ( ) ( ) and for a=b 3 d 2 I AA′ , <0 10 da 2 a=b 3 10 Now, for a = 0, ∴ d 2 I AA′ >0 da 2 ( I AA′ )max occurs when a = 84 or 3 = 46.009 mm 10 a = 46.0 mm continued COSMOS: Complete Online Solutions Manual Organization System (b) From part (a) ( I AA′ )mass 2 4  3  3   = 2b + 3b  b − 5 b  10   10   12        ρ L  = 2 4 49 49 4 ρ Lb 4 = 2800 kg/m 3 ( 0.3 m )( 0.084 m ) 240 240 ( ) = 8.5385 × 10−3 kg ⋅ m 2 or ( I AA′ )mass 2 k AA ′ = and = 8.54 × 10−3 kg ⋅ m 2 ( I AA′ )mass m where ( 2 m = m1 − m2 = ρ L b − a 2 ) 2     3 7 2 = ρ L b −  b ρ Lb 2   =    10 10     Then 2 k AA ′ 49 ρ Lb4 7 2 7 240 = = b = ( 84 mm )2 = 2058 mm 2 7 24 24 2 ρ Lb 10 k AA′ = 45.3652 mm or k AA′ = 45.4 mm