COSMOS Chapter 9 Solution 170

COSMOS: Complete Online Solutions Manual Organization System Chapter 9, Solution 170. m = ρV = ρ tA Have m1 = ρ ta 2 Then m2 = 1 ρ ta 2 2 Compute moments and moments of inertia with respect to point A Now I x′′ = ( I x′′ )1 + ( I x′′ )2  1  = ρ ta  a 2  + 12   2  =  1   a  2  2 2   2  2  1 2    + ( a )   + ρ ta   a  +  a    18   3     2    2    19 ρ ta 4 12 ( )1 + ( I y′′ )2 I y′′ = I y′′   1   a 2  2  = ρ ta 2   a 2  +   + ( a )      12   2   a 2  2a 2   1 2 1  2 2 + ρ ta   a + a  +   +     2 18  3     3  = 5 ρ ta 4 3 I z ′′ = ( I z ′′ )1 + ( I z′′ )2 1  1 1  = ρ ta 2  a 2 + a 2  + ρ ta 2  a 2  3 2   6  ( = ) 3 ρ ta 4 4 Now note symmetry implies ( I x′y′ )1 = ( I y′z′ )1 = ( I z′x′ )1 = 0 ( I x′y′ )2 = ( I y′z′ )2 = 0 continued COSMOS: Complete Online Solutions Manual Organization System I uv = I u′v′ + mu v Now 0  a  a  1 I x′′y′′ = m1x1′′ y1′′ + m2 x2′′ y2′′ = ρ ta 2    = ρ ta 4  2  2  4 Therefore 0 1 a I y′′z′′ = m1 y ″1 z ″1 + m2 y2′′ z′′2 = ρ ta 2   ( −a ) = − ρ ta 4 2 2 ( )2 + m2 z ″2 x ″2  I z′′x′′ = m1z ″1 x ″1 +  I z ′x′  1 ( I z′x′ )  = − a4 2  area  72 From Sample Problem 9.6 Then ( I z′′x′′ )2 = ρ t ( I z′x′ )2  area Then a I z′′x′′ = ρ ta 2 ( − a )   2 =− 1 ρ ta 4 72 1  1  2  1   +  − ρ ta 4 + ρ ta 2  − a  a  2  72  3  3   5 = − ρ ta 4 8 By observation λ AB = 1 (i + j − k ) 3 Now, Equation 9.46 I AB = I x′′λx2′′ + I y′′λ y2′′ + I z′′λz2′′ − 2I x′′y′′λx′′y′′ − 2I y′′z′′λ y′′λz′′ − 2I z′′x′′λz′′λx′′  19  1 2 5  1 2 3  1  2 = ρ ta    +   + −  3 3  4 3  12  3  4  1   1  1  − 2   −  4   3  3   1   1  1  2 −   −  3  2   3   5   1  1   − 2 −  −   3  3    8  or I AB = 5 ρ ta 4
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