$13\mathrm{.}6****$ In discussing the oscillation of a cart on the end of a spring, we almost always ignore the mass of the spring. Set up the Hamiltonian $H$ for a cart of mass $m$ on a spring $($force constant $k)$ whose mass $M$ is not negligible, using the extension $x$ of the spring as the generalized coordinate. Solve Hamilton's equations and show that the mass oscillates with angular frequency $=\sqrt[\phantom{2}]{\frac{k}{m+\frac{W}{3}}}.$ That is$,$ the effect of the spring's mass is to add $\frac{M}{3}$ to $m.($Assume that the spring's mass is distributed unitormily and that it stretches uniformly.$)$