$6.(10$ pts$)$ 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 $\backslash (=0\mathrm{.}6\backslash$sigma$\_\{$y$\}\backslash )).$ Derive a strength$-$to$-$mass performance index on a beam subjected to a force, F $,$ as shown in the right figure. Assuming bending strength in this analysis is equivalent to yield strength, Rank these materials on the basis of both performance analysis and material cost.