M402.set.2.solutions2010

M402.set.2.solutions2010 - Math 402 Problem Set II Set II...

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Math 402 Problem Set II Set II page 1 1. The molecule consists of three atoms with masses, m1=M, m2=m3=m, where m <<< M . The rotation matrix, R(y,q,F), determines the instantaneous orientation of the molecule (with respect to the laboratory x, y and z directions) at time t. We actually carry out this rotation in the frame (the center of mass frame) which travels along with the molecule. For simplicity (and since M>>m) we let the the center of mass be on atom 1. The laboratory frame we define to be a non-rotating x,y,z frame whose origin remains fixed on the location of atom 1 at time t = 0. a . The Euler angles are: ε 0.0  Ψ t v d ( ) ε t  θ t v d ( ) 3 v d t  Φ t v d ( ) v d t  The Euler angle rotation matrix gives the primed coordinates (rotating frame components) in terms of the fixed system, x,y,z coordinates. The three rotations (about z, x' and z'' axes) are : R t v d ( ) cos ε t ( ) sin ε t ( ) 0 sin ε t ( ) cos ε t ( ) 0 0 0 1 1 0 0 0 cos 3 v d t sin 3 v d t 0 sin 3 v d t cos 3 v d t cos v d t sin v d t 0 sin v d t cos v d t 0 0 0 1  R t v d ( ) cos t v d cos 3 t v d sin t v d sin t v d sin 3 t v d sin t v d cos t v d cos 3 t v d cos t v d sin 3 t v d 0 sin 3 t v d cos 3 t v d  The inverse of the rotation matrices is given by the transpose. Rinv t v d ( ) cos t v d sin t v d 0 cos 3 t v d sin t v d cos t v d cos 3 t v d sin 3 t v d sin t v d sin 3 t v d cos t v d sin 3 t v d cos 3 t v d 
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page 2 b . We let r2cm and r3cm be position vectors of atoms 2 and 3 with respect to the center of mass of the molecule.
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