Three beads of masses , , (in that order) slide without friction on a line

PHYSICS · Multi-part · Fermi University Physics · Mechanics & Dynamical Systems · Oscillations, Waves & Acoustics · Coupled oscillators, normal modes and Fourier analysis

Action and Symmetry, Ch Small oscillations: stability, degeneracy, symmetry and spectra

Problem

Three beads of masses MM, mm, MM (in that order) slide without friction on a line. Bead 1 is joined to a fixed wall on its left by a spring of stiffness κ\kappa, bead 3 to a fixed wall on its right by a spring of stiffness κ\kappa, and beads 1 and 2, and beads 2 and 3, are each joined by a spring of stiffness κ′\kappa'. Let x1x_1, x2x_2, x3x_3 be the displacements of the three beads from equilibrium, all measured in the same direction along the line. Take M=0.050 kgM = 0.050\ \mathrm{kg}, m=0.020 kgm = 0.020\ \mathrm{kg}, κ=100 N m−1\kappa = 100\ \mathrm{N\,m^{-1}} and κ′=150 N m−1\kappa' = 150\ \mathrm{N\,m^{-1}}. ((a)) The reflection S\mathsf S through the middle bead sends x1→−x3x_1 \to -x_3, x3→−x1x_3 \to -x_1 and x2→−x2x_2 \to -x_2. Show that it leaves both TT and VV unchanged. ((b)) Build the symmetry-adapted coordinates u=x1−x3u = x_1 - x_3, which S\mathsf S leaves alone, and s=x1+x3s = x_1 + x_3 and x2x_2, which it reverses. Show that uu decouples completely from ss and x2x_2, and find the frequency of the mode it describes, in rad s−1\mathrm{rad\,s^{-1}}. ((c)) Write the remaining 2×22\times2 eigenvalue problem for ss and x2x_2 and find its two frequencies, in rad s−1\mathrm{rad\,s^{-1}}.

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