Three equal masses on a smooth line are joined by two springs

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 four subspaces and the Fredholm alternative

Problem

Three equal masses on a smooth line are joined by two springs, each of stiffness k=50 N m−1k=50\,\mathrm{N\,m^{-1}}, with no walls, so static equilibrium under forces F\mathbf{F} is Kx=F\mathsf{K}\mathbf{x}=\mathbf{F} with K=k(1−10−12−10−11).\mathsf{K}=k\begin{pmatrix}1&-1&0\\-1&2&-1\\0&-1&1\end{pmatrix}. ((a)) For Fa=(2, 1, −3) N\mathbf{F}_a=(2,\,1,\,-3)\,\mathrm{N}, use the Fredholm alternative to decide whether a static solution exists. If it does, find the solution with x2=0x_2=0 and write the whole solution set. ((b)) Do the same for Fb=(2, −1, 0) N\mathbf{F}_b=(2,\,-1,\,0)\,\mathrm{N}.

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