Four equal masses sit in 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

Four equal masses mm sit in a line, each joined to its neighbours and to two fixed walls, one at each end, by identical springs of stiffness κ\kappa, so that mass jj (j=1,…,4j=1,\ldots,4) obeys mx¨j=κ(xj+1−2xj+xj−1),m\ddot x_j = \kappa(x_{j+1}-2x_j+x_{j-1}), with x0=x5=0x_0=x_5=0 fixed at the walls. Take m=0.010 kgm=0.010\ \mathrm{kg} and κ=40 N m−1\kappa=40\ \mathrm{N\,m^{-1}}. ((a)) Show that a trial solution xj=Asin⁡(jθ)exp⁡(−iωt)x_j = A\sin(j\theta)\exp(-\mathrm i\omega t) automatically satisfies the condition at the left wall, and find the condition on θ\theta that satisfies the right wall. ((b)) Derive the dispersion relation ω(θ)=2κ/m sin⁡(θ/2)\omega(\theta) = 2\sqrt{\kappa/m}\,\sin(\theta/2) and list the allowed values of θ\theta. ((c)) Compute the four frequencies in rad s−1\mathrm{rad\,s^{-1}}, and say whether any of them is a zero mode.

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