$3.(30)$ The following data is obtained for a cantilever beam vibration test. The experiment is set up the same way as in our Vibrating Beam Lab, where a proximity probe is used to measure deflection of the beam.

$\backslash$begin$\{$tabular$\}\{|$r$|$l$|$l$|$c$|$c$|\}$

$\backslash$hline $\backslash$multicolumn$\{3\}\{|$c$|\}\{$ beam dimensions $\}$ & End mass $($g$)$ & $\backslash$begin$\{$tabular$\}\{$l$\}$

Frequency $\backslash \backslash$

$\backslash ((\backslash$mathrm$\{$Hz$\})\backslash )$

$\backslash$end$\{$tabular$\}\backslash \backslash$

$\backslash$hline Length $\backslash ((\backslash$mathrm$\{$cm$\})\backslash )$ & width $\backslash ((\backslash$mathrm$\{$cm$\})\backslash )$ & thickness $\backslash ((\backslash$mathrm$\{$mm$\})\backslash )$ & $150$ & $30\backslash \backslash$

$\backslash$hline $25\mathrm{.}4$ & $2\mathrm{.}54$ & $9$ & $250$ & $25\backslash \backslash$

$\backslash$hline

$\backslash$end$\{$tabular$\}$

$($a$)$ What is the Young's modulus of the beam material, in GPa?

$($b$)$ What is the density of the beam material, in $\backslash (\backslash$mathrm$\{$kg$\}/\backslash$mathrm$\{$m$\}^\{3\}\backslash )?$

$($c$)(8)$ If we want to collect $20$ data points in each cycle and wish to record the vibration motion for $5$ seconds, please specify the sample rate and total number of points for the two loading conditions given above.

Hint: A linear "regression" can be performed with only two points.