A bead free to slide in a horizontal plane is held by three springs

PHYSICS · Multi-part · Fermi University Physics · Mechanics & Dynamical Systems · Oscillations and normal modes · Oscillations and normal modes

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

Problem

A bead free to slide in a horizontal plane is held by three springs, one of them compressed so that it pushes on the bead, and near the origin its potential energy is

V(x,y) = \tfrac12 k_1 x^2 + \tfrac12 k_2 y^2 + \alpha xy,

with

k_1 = 50\ \mathrm{N\,m^{-1}}

k_2 = 20\ \mathrm{N\,m^{-1}}

and

\alpha = -40\ \mathrm{N\,m^{-1}}

. Air resistance and friction are ignored. ((a)) Find the equilibrium position. ((b)) Find the eigenvalues of the Hessian of

V

at that equilibrium, in

\mathrm{N\,m^{-1}}

. ((c)) State whether the equilibrium is stable, and if not, name a direction in which it fails.

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