$4.(14$ pts total$)$ Use the engineering materials that are listed in the table below $(\backslash (\backslash$rho $\backslash )$ : density, $\backslash ($ E $\backslash )$ : Young's modulus, $\backslash ($ G $\backslash )$ : shear modulus, $\backslash (\backslash$sigma$\_\{\backslash$mathrm$\{$y$\}\}\backslash )$ : yield strength, shear strength$=0\mathrm{.}6\backslash (\backslash$sigma$\_\{\backslash$mathrm$\{$y$\}\}\backslash )).$

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

$\backslash$hline Material & $\backslash$begin$\{$tabular$\}\{$c$\}$

$\backslash (\backslash$rho $\backslash )\backslash \backslash$

Please solve and explain the questions! THank you!

$\backslash (\backslash$left$(\backslash$mathrm$\{$Mg$\}/\backslash$mathrm$\{$m$\}^\{3\}\backslash$right$)\backslash )$

$\backslash$end$\{$tabular$\}$ & $\backslash$begin$\{$tabular$\}\{$c$\}$

$\backslash (\backslash$sigma$\_\{$y$\}\backslash )\backslash \backslash$

$\backslash (($M P a$)\backslash )$

$\backslash$end$\{$tabular$\}$ & $\backslash$begin$\{$tabular$\}\{$c$\}$

$\backslash ($ E $\backslash )\backslash \backslash$

$\backslash (($G P a$)\backslash )$

$\backslash$end$\{$tabular$\}$ & $\backslash$begin$\{$tabular$\}\{$c$\}$

$\backslash ($ G $\backslash )\backslash \backslash$

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

$\backslash$end$\{$tabular$\}$ & $\backslash$begin$\{$tabular$\}\{$c$\}$

Relative $\backslash \backslash$

cost$,\backslash (\backslash$bar$\{$c$\}\backslash )\backslash \backslash$

$\backslash ((\backslash$$ $/\backslash$$$)\backslash )$

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

$\backslash$hline $\backslash$begin$\{$tabular$\}\{$l$\}$

Carbon fiber$-$reinforced $\backslash \backslash$

composite

$\backslash$end$\{$tabular$\}$ & $1\mathrm{.}5$ & $775$ & $110$ & $9\mathrm{.}2$ & $80\backslash \backslash$

$\backslash$hline $\backslash$begin$\{$tabular$\}\{$l$\}$

Glass fiber$-$reinforced $\backslash \backslash$

composite

$\backslash$end$\{$tabular$\}$ & $2\mathrm{.}0$ & $151$ & $22$ & $8\mathrm{.}6$ & $40\backslash \backslash$

$\backslash$hline $\backslash$begin$\{$tabular$\}\{$l$\}$

Stainless steel $304($hot $\backslash \backslash$

finished and annealed$)$

$\backslash$end$\{$tabular$\}$ & $8\mathrm{.}0$ & $205$ & $193$ & $74$ & $4\mathrm{.}0\backslash \backslash$

$\backslash$hline $\backslash$begin$\{$tabular$\}\{$l$\}$

Aluminum alloy $(7075-\backslash \backslash$

T$6)$

$\backslash$end$\{$tabular$\}$ & $2\mathrm{.}8$ & $505$ & $71$ & $26\mathrm{.}7$ & $13\mathrm{.}4\backslash \backslash$

$\backslash$hline $\backslash$begin$\{$tabular$\}\{$l$\}$

Magnesium alloy $\backslash \backslash$

$($AZ$31$B extruded$)$

$\backslash$end$\{$tabular$\}$ & $1\mathrm{.}8$ & $220$ & $45$ & $16\mathrm{.}7$ & $12\mathrm{.}6\backslash \backslash$

$\backslash$hline $\backslash$begin$\{$tabular$\}\{$l$\}$

Titanium alloy $($Ti$-6$Al$-\backslash \backslash$

$4$V$)$

$\backslash$end$\{$tabular$\}$ & $4\mathrm{.}4$ & $830$ & $114$ & $42\mathrm{.}5$ & $110\backslash \backslash$

$\backslash$hline

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

$($a$)(10$ pts$)$ Derive a stiffness$-$to$-$mass performance index on a solid cylindrical shaft that is subjected to a torsional stress. Rank the above materials in order of best performance.

$($b$)(4$ pts$)$ Rank these materials on the basis of both performance analysis in part $($a$)$ and material cost $($in order of best selection$).$