Derive the characteristic equation in terms of eigenvalues $.$l that you will use to calculate the natural frequencies of vibration for an axial bar clamped at one end $($x$=0)$ and a mass $$m$$ and a spring attached at the other end of the bar $($properties E$,$ A$,$ $).$ Calculate the first natural frequency of the bar with the given boundary conditions if a mass m $=50$ kg is attached to the bar at the free end $($x $=$ l$)$ and a linear spring with spring constant K$=100$ N$/$m is attached between the mass and a fixed support. Assume that l $=12\mathrm{.}5$ m$,$ E $=200$ GPa, $=7850$ kg$/$m$3,$ A$=0\mathrm{.}001$ m$2.$