$2\mathrm{.}55-2\mathrm{.}58$ Determine the natural frequency of vibration, in bending, of the system shown in Figs. $2\mathrm{.}90($a$)($d$)$ by modeling the system as a single$-$degree$-$of$-$freedom system. Assume that the mass is $\backslash ($ m$=50\backslash$mathrm$\{$~kg$\}\backslash ),$ spring stiffness is $\backslash ($ k$=10,000\backslash )$ and the beam has a square cross section of $\backslash (5\backslash$mathrm$\{$~cm$\}\backslash$times $5\backslash$mathrm$\{$~cm$\}\backslash ),$ and is made of steel with a Young's modulus of $207$ GPa $.$

$($b$)$

$($c$)$

$($d$)$