Two identical carts

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

Two identical carts, each of mass

m = 2.0\,\mathrm{kg}

, run on a frictionless horizontal track. Each cart is joined to a fixed wall by a spring of stiffness

k = 8.0\,\mathrm{N\,m^{-1}}

, and the two carts are joined to each other by a third spring of stiffness

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

. Let

x_1

and

x_2

be the displacements of the two carts from their equilibrium positions. The equations of motion assemble into

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

with

\mathsf{A} = \begin{pmatrix}5.0 & -1.0\\ -1.0 & 5.0\end{pmatrix}\,\mathrm{s^{-2}}.

((a)) Write down and solve the characteristic equation for

\mathsf{A}

, checking both roots against the trace and the determinant. ((b)) Find the eigenvector belonging to each eigenvalue and describe in words the motion each represents. ((c)) State, with a reason, whether the equilibrium

x_1 = x_2 = 0

is stable.

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