Determine the characteristic equation for longitudinal vibration of a bar of length L$,$ elastic modulus E$,$ and mass density p$,$ that is fixed at the end x $=0$ and has a mass m at the other end x $=$ L$.$

$2.$ A ship's propeller is a $20$ m steel $($E$=210*10^9*$ N $.$m$^-2,$ p $=7800$ kg$.$m$^3)$ shaft of diameter $10-$cm$.$ The shaft is modeled as a bar fixed at the end x $=0$ and $500$ kg propeller attached to the other end x $=$ L$.$ What is the lowest natural frequency of the longitudinal vibration of the propeller shaft system?

Hint: The first solution of the equation alpha $*$ tan$($alpha$)=2\mathrm{.}45$is$*$alpha $=1\mathrm{.}137$