Three identical gliders

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

Mathematical Methods for Physics, Ch Eigenvalue problems, diagonalisation and normal modes

Problem

Three identical gliders, each of mass

m = 0.50\,\mathrm{kg}

, sit in a line on a level frictionless air track. Glider 1 is joined to glider 2, and glider 2 to glider 3, by identical ideal springs of stiffness

k = 2.0\,\mathrm{N\,m^{-1}}

. Nothing joins the chain to a wall. Let

x_1

x_2

x_3

be the displacements of the gliders from their equilibrium positions, all measured positive in the same direction. ((a)) Write the three equations of motion and assemble them as

\ddot{\mathbf{x}} = -\mathsf{A}\mathbf{x}

, giving the entries of

\mathsf{A}

\mathrm{s^{-2}}

. ((b)) Show that

(1,1,1)

is an eigenvector of

\mathsf{A}

and find its eigenvalue. Describe this motion, give the solution of the equation its normal coordinate obeys, and say whether the equilibrium is stable in the chapter's sense. ((c)) Verify that

(1,0,-1)

is an eigenvector, find its eigenvalue and describe the motion. ((d)) Without solving the cubic characteristic equation, use the trace of

\mathsf{A}

to find the third eigenvalue, then use what the symmetry of

\mathsf{A}

guarantees about eigenvectors to find the third eigenvector. Verify it by substitution and give the angular frequency of this mode.

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