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

2024
Views:

184

Pages:

2
Author:

Apolenajz

COSMOS: Complete Online Solutions Manual Organization System Chapter 9, Solution 169. m1 = m2 = First note that λ OA = And that 1W 2 g 1 (i + j + k ) 3 Using Figure 9.28 and the parallel-axis theorem have I x = ( I x )1 + ( I x )2  1 1W =  12  2 g  2 1W a + 2 g  a   2 2   1W  1  1 W  2 2 +   a +a + 2 g 12  2 g  ( = )  a 2  a 2     +      2     2  1 W  1 1 1 1  1W 2 +  a2 +  +  a2  = a  2 g  12 4  6 2  2 g ( )1 + ( I y )2 Iy = Iy  1  1 W  1W 2 2 =   a +a + 12 2 g 2 g    (  1  1 W +  12  2 g = )  a 2  a 2     +      2     2  2  2 1 W  2  a    a a + +  ( )     2 g   2     1 W  1 1  2  1 5  W 2 a +  a2  =  +  a +  2 g  6 2  g  12 4   continued COSMOS: Complete Online Solutions Manual Organization System I z = ( I z )1 + ( I z )2  1 1W =  12  2 g  2 1W a + 2 g   1  1 W +  12  2 g = a   2  2 1W a + 2 g  2    2  a 2   ( a ) +      2     1 W  1 1 5  5W 2  1 +  a2 +  +  a2  = a  2 g  12 4   12 4   6 g Now observe that the centroidal products of inertia, I x′y′ , I y′z ′ , and I z′x′ , of both components are zero because of symmetry. Also, y1 = 0 ( 0 ) I xy = Σ I x′y′ + mx y = m2 x2 y2 = Then ( I yz = Σ I y′z′ +0my z ( 0 ) m2 y2z2 1W a 1W 2 a ( a )   = 2 g 2   4 g 1W 2 g  a  a      2  2  1W 2 a 8 g ) I zx = Σ I z ′x′ + mz x = m1z1x1 + m z x = 1 3  a  a  1  a  a2    +  (a) = g  2  2  g 2 g Substituting into Equation (9.46) I OA = I xλx2 + I y λ y2 + I z λz2 − 2I xy λxλ y − 2I yz λ y λz − 2I zxλz λx Noting that λx2 = λ y2 = λz2 = λxλ y = λ yλz = λz λx = 1 3 Have I OA = 1 1 W 2 W 2 5 W 2 a + a + a  3 2 g g 6 g  1 W 2 1 W 2 3 W 2  a + a + a  − 2 8 g 8 g 4 g  = 1 14  3  W 2  − 2   a 3 6  4  g or I OA = 5 W 2 a 18 g