This preview has intentionally blurred sections. Sign up to view the full version.
View Full Document
Unformatted text preview: Ph507. Homework 7 (due: Friday, April 1).PROBLEM 71 (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 isΩ. Find thefrequencies of the normal modes of the system. Determine the range ofΩ, for which the particle is stable at the origin,ifk1<< k2.PROBLEM 72 (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 73 (3 pts.)Find the normal frequencies of aFveatomic 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 74 (4 pts.)Lagrangian of three coupled oscillators is given by:3Xn=1∙m˙x2n2−kx2n2¸+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 71:The equations of motion of the system are:mddt2μxy¶= 2mΩ×ddtμxy¶+Cμxy¶−μk1xk2y¶Looking for the normal modes:μk1−m¡'2+Ω2¢2i'mΩ−2i'mΩk2−m¡'2+Ω2¢¶μx'y'¶= 0det=μ'2+Ω2−k1m¶μ'2+Ω2−k2m¶−4Ω2'2= 0'2=k1+k22m+Ω2±sμk1−k22m¶2+Ω2...
View
Full Document
 Spring '04
 ANON
 mechanics, Mass, Work, Boundary value problem, Normal mode, Mode shape, Longitudinal mode

Click to edit the document details