Three equal masses slide on a circular hoop

PHYSICS · Multi-part · Fermi University Physics · Mathematical Methods for Physics · Oscillations, Waves & Acoustics · Coupled oscillators, normal modes and Fourier analysis

Modes, Packets and Media, Ch Normal modes of many coupled oscillators: eigenvalues, symmetry and response

Problem

Three equal masses

m = 0.40\,\mathrm{kg}

slide on a circular hoop, equally spaced around it. Each mass is joined to its two neighbours by an identical spring of stiffness

\kappa = 90\,\mathrm{N\,m^{-1}}

laid along the hoop, so the three masses and three springs form a closed ring exactly as in the ring-mode analysis. Let

\mathbf R

be the operation that rotates the three masses one position around the ring (a

120^\circ

rotation of the labels). ((a))

\mathbf R

commutes with the dynamical matrix because every mass and every spring is identical. Using

\mathbf R^3 = \mathbf 1

, state the three eigenvalues of

\mathbf R

. ((b)) Without writing down the dynamical matrix, explain why two of the three normal-mode frequencies of this system must be equal. ((c)) Using the ring-mode result

\omega_p = 2\omega_0|\sin(\pi p/N)|

with

\omega_0 = \sqrt{\kappa/m}

and

N=3

, calculate the three mode frequencies and confirm the degenerate pair predicted in part (b).

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