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Unformatted text preview: Ph-507. Homework 7 (due: Friday, April 1).PROBLEM 7-1 (4 pts.)A particle of massmis moving in two dimensions under the infuence of asymmetric harmonic potential:U(x, y) =k1x2+k2y22Here(x, y)are the coordinates of the particle in arotating coordinate system, whose angular velocity is. Find thefrequencies of the normal modes of the system. Determine the range of, for which the particle is stable at the origin,ifk1<< k2.PROBLEM 7-2 (4 pts.)Find the normal modes and their frequencies for a triangular molecule made of three identical atoms of massm.All the bonds are identical, having spring constantk.PROBLEM 7-3 (3 pts.)Find the normal frequencies of aFve-atomic linear molecule. Assume all the four bonds to have the same rigidityconstantkand all the atoms to have the same massm. Consider only longitudinal modes (i.e. those along themolecular axis).PROBLEM 7-4 (4 pts.)Lagrangian of three coupled oscillators is given by:3Xn=1mx2n2kx2n2+k(x1x2+x2x3)Findx2(t)for the following initial conditions (att= 0):(x1, x2, x3) = (x,,0) ;(x1,x2,x3) = (0,, v).1Solution to Problem 7-1:The equations of motion of the system are:mddt2xy= 2mddtxy+Cxyk1xk2yLooking for the normal modes:k1m'2+22i'm2i'mk2m'2+2x'y'= 0det='2+2k1m'2+2k2m42'2= 0'2=k1+k22m+2sk1k22m2+2...
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