Two equal masses

PHYSICS · Multi-part · Fermi University Physics · Mathematical Methods for Physics · Linear algebra and vector spaces · Linear algebra and vector spaces

Conditions of Validity, Ch The spectral theorem, degeneracy and the matrices that will not diagonalise

Problem

Two equal masses, each joined to a wall by a spring and to each other by a spring four times as stiff, obey x¨=−Ax\ddot{\mathbf x}=-\mathsf A\mathbf x with A=(5−4−45) s−2.\mathsf A=\begin{pmatrix}5&-4\\-4&5\end{pmatrix}\,\mathrm{s^{-2}}. For a function ff, write f(A)=∑nf(λn)Pnf(\mathsf A)=\sum_nf(\lambda_n)P_n as in the spectral resolution. ((a)) Find the eigenvalues of A\mathsf A and the projections PnP_n onto its eigenspaces, and use them to compute A−1/2\mathsf A^{-1/2}, taking positive square roots. Check that (A−1/2)2A=I(\mathsf A^{-1/2})^2\mathsf A=\mathsf I. ((b)) Show that x(t)=cos⁡(tA) x(0)+A−1/2sin⁡(tA) x˙(0)\mathbf x(t)=\cos(t\sqrt{\mathsf A})\,\mathbf x(0)+\mathsf A^{-1/2}\sin(t\sqrt{\mathsf A})\,\dot{\mathbf x}(0) solves x¨=−Ax\ddot{\mathbf x}=-\mathsf A\mathbf x and has the right position and velocity at t=0t=0. ((c)) The masses start at x(0)=(1,0) cm\mathbf x(0)=(1,0)\,\mathrm{cm} with velocity x˙(0)=(0,1) cm s−1\dot{\mathbf x}(0)=(0,1)\,\mathrm{cm\,s^{-1}}. Find x1(t)x_1(t) and evaluate x1(1 s)x_1(1\,\mathrm s), with the arguments of the trigonometric functions in radians.

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