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Analytical Mech Homework Solutions 133

Analytical Mech Homework Solutions 133 - (I I(I I 2 2 I11 I...

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( ) ( ) ( ) 1 2 3 1 2 2 1 1 3 2 1 2 1 3 3 2 0 I I I I I I I I I ω ω ω + + = ±± ω ω 1 1 1 0 K ω ω + = ±± , ( )( ) ( )( ) 3 2 2 1 3 2 3 1 2 2 1 2 3 1 3 1 2 I I I I I I I I K I I I I ω ω = − + (a) For 3 ω large and 2 0 ω = , so 1 0 K > 1 1 1 0 K ω ω + = ±± is the harmonic oscillator equation. 1 ω oscillates, but remains small. Motion is stable for initial rotation about the 3 axis if the 3 axis is the principal axis having the largest or smallest moment of inertia. (b) For 3 0 ω = and 2 ω large, 1 0 K < so 1 1 1 0 K ω ω + = ±± is the differential equation for exponential growth of 1 ω with time: 1 1 k t k t Ae Be ω = + 1 = . Motion is unstable for the initial rotation mostly about the principal axis having the median moment of inertia. 9.20 0 since either xy i i i i I m x y = i x or is zero for all six particles. Similarly, all the other products of inertia are zero. Therefore the coordinate axes are principle axes.
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