$1.[10$ Marks$]$ A spring of force constant $\backslash ($ K $\backslash )$ is tied to the axle of a solid cylinder on a by a string that passes around two pulleys to a block on a tabletop. The solid cylinder and the pulleys the pulleys all have mass $\backslash ($ M $\backslash )$ and radius $\backslash ($ R $\backslash ).$ Each pulley is basically a thin hoop. Initially the spring is unstretched. Then a force $\backslash ($ F $\backslash )$ pulls the cylinder from rest a distance $\backslash ($ L $\backslash )$ along the tabletop. The cylinder rolls without slipping. Find an expression for the velocity of the cylinder after it is pulled distance $\backslash ($ L $\backslash ).$ The coefficients of friction between the tabletop and cylinder are $\backslash (\backslash$mu$\_\{\backslash$mathrm$\{$S$\}\}\backslash )$ and $\backslash (\backslash$mu$\_\{\backslash$mathrm$\{$K$\}\}\backslash ).$