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

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

k' = 5.0\,\mathrm{N\,m^{-1}}

. All three springs are ideal (massless, obeying Hooke's law) and the oscillations are small enough that no spring goes slack. Let

x_1

and

x_2

be the displacements of the two carts from their equilibrium positions, both measured positive in the same direction along the track. ((a)) Using Newton's second law for each cart and Hooke's law for each spring, write the two coupled equations of motion and assemble them as

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

, giving the numerical entries of

\mathsf{A}

\mathrm{s^{-2}}

. ((b)) By symmetry, propose the two eigenvectors of

\mathsf{A}

and verify each by direct substitution, quoting the corresponding eigenvalue. ((c)) Describe, in words, the motion of the two carts in each of the two normal modes, and explain physically why the middle spring plays no part in one of them.

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