Analytical Mech Homework Solutions 187

Analytical Mech Homework Solutions 187 - 24 y3 y1 x3 x1 = 0...

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24 3131 0 yyxx −−−= We introduce three generalized coordinates Q , q 1 and q 2 that should be close to what we guess would be normal modes … 13113 2 13 ;; Qx x q xx q y y =+ =− Solve for x i , y i in terms of Q , q 1 and q 2 using the above 3 equations of constraint … () 11 2 1 3 3 ; 22 m 1 x Qq x Q x x x M = = 12 2 2 3 m yq Q y qyQ q M = 2 Thus, the kinetic energy, in terms of these generalized coordinates, is … 1 1 24 4 m Tm Q m q mq MM µ  =+ ++   ± ±± 2 where 2 M m is the mass of the H 2 S molecule The potential energy is … 2 2 2 2 1 1 1 28 8 4 m Vk Qk q kq kq µµ ++ 1 2 q M Note, that in the Lagrangian , the only cross term is one involving qq . therefore, Q is a normal mode with eigenfrequency given by the ratio … LTV 2 1 QQ Q km KM mM ω == + Constructing the residual 2x2 K and M matrices involving only q 1 and q 2 terms, which we will call Kq and Mq gives … 0 42 M and 0 M
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