Six identical beads

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

Six identical beads, each of mass m=0.020 kgm = 0.020\ \mathrm{kg}, are threaded on a frictionless circular wire and joined to their neighbours by six identical springs of stiffness κ=50 N m−1\kappa = 50\ \mathrm{N\,m^{-1}}, forming a closed ring. Each bead moves only along the wire. ((a)) State, without computing, how many of the six modes are zero modes and how many distinct non-zero frequencies there are. ((b)) Using the monatomic-chain dispersion relation ω(qw)=2κ/m ∣sin⁡(qwdc/2)∣\omega(q_{\rm w}) = 2\sqrt{\kappa/m}\,|\sin(q_{\rm w}d_{\rm c}/2)| with qwdc=2πs/Nq_{\rm w}d_{\rm c} = 2\pi s/N, N=6N=6, s=0,…,5s=0,\ldots,5, compute all six frequencies in rad s−1\mathrm{rad\,s^{-1}}.

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