M402.set.2.description.2010

# M402.set.2.description.2010 - Math 402 Problem Set II 1 A...

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Math 402 Problem Set II 1 . A molecule consisting of three atoms of mass m 1 = M, m 2 =m 3 = m where m << M is oriented as shown at time t = 0. r 1 (t=0) = 0 r 2 (t=0) = -dsin α x^ + dcos α z^ r 3 (t=0) = dsin α x^ + dcos α z^ You can use Mathcad for this problem. See B. Hale for help. a) Assume that M>>m so that the center of mass of the molecule is given by the position of particle 1, and that the velocity of particle 1 and the Euler angles ψ (t), θ (t), and φ (t) of the molecule are given by: d r 1 (t)/dt = 3v x^ - v y^ + 2v z^ where v is a constant (no t dependence) φ (t) = [v/d]t θ (t) = [3v/d]t ψ (t) = ε t ( For question 1 ε =0 ) Write out the Euler angle rotation matrix, R ( ψ (t), θ (t), φ (t) ) as a function of v, d and t. Note that the molecule rotates as it moves and the primed coordinate system ( x ^‘ , y ^ ‘ and z^ ‘) is fixed on the molecule. The orientation of the primed coordinate system is prescribed by R . The unprimed coordinate system x ^ , y ^ and z^ is fixed. b) Find the position vectors, r 1 (t), r 2 (t), and r 3 (t) in the fixed coordinate frame (that is in terms of x
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