Two carts, each of mass , sit on a frictionless horizontal track. Cart 1 is joined to…

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 carts, each of mass

m = 0.20\,\mathrm{kg}

, sit on a frictionless horizontal track. Cart 1 is joined to a wall by a spring of stiffness

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

, cart 2 is joined to an identical wall on the other side by an identical spring of stiffness

k

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

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

. Let

x_1

and

x_2

be the displacements of the two carts from their equilibrium positions. ((a)) Write the two coupled equations of motion from Newton's second law and put them in the form

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

. Explain why

\mathsf A

must be symmetric. ((b)) Find the eigenvalues and normalised eigenvectors of

\mathsf A

, and give the orthogonal matrix

\mathsf P

that diagonalises it, stating what its columns represent. ((c)) At

t=0

cart 1 is pulled aside by

x_0 = 1.0\,\mathrm{cm}

, cart 2 is left at its equilibrium position, and both are released from rest. Express this initial state in normal coordinates

\eta_1, \eta_2

. ((d)) Hence find

x_1(t)

and

x_2(t)

for

t>0

, and find the earliest time at which the envelope of

x_1

falls to zero, so that cart 2 is swinging with the full amplitude

x_0

. Explain in one sentence how the spring hands the energy from cart 1 to cart 2, and what the two normal modes add to that picture.

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