Problem $1$: Scaling effect on a cantilever beam $(15$ points$)$

Consider a cantilever beam made of a given material $($density: $\backslash (\backslash$rho $\backslash ),$ Young modulus: Y $),$ having dimensions L $,$ w and t $($length$,$ width and thickness, respectively$).$ Assuming the cantilever have $\backslash ($ L $\backslash )$ along $\backslash ($ x $\backslash )-$axis, $\backslash ($ w $\backslash )$ along the $\backslash ($ y $\backslash )-$axis and $\backslash ($ t $\backslash )$ along the $\backslash ($ z $\backslash )-$axis. A suitable excitation can drive the cantilever to vibrate at its own resonance frequency, $\backslash (\backslash$omega$\_\{0\}\backslash ).$ The vibrating cantilever can be used as a mass sensor by measuring its resonance frequency shift $($with respect to the unloaded value $\backslash (\backslash$omega$\_\{0\}\backslash ))$ due to a change in the mass, $\backslash ($ m $\backslash )(\backslash ($ m $\backslash$ll $\backslash )$ the weight of the beam$).$ Use scaling analysis to show the improvement in the mass sensitivity of the beam $($e$.$g$.$ resonance frequency shift to mass change $\backslash ((\backslash$mathrm$\{$d$\}\backslash$omega $/\backslash$mathrm$\{$dm$\})\backslash )),$ when the linear dimensions of the beam is scaled by a factor of $10.$