$4.(20$ pts$)$ Derive a stiffness$-$to$-$mass performance index for a solid cylindrical shaft that is subjected to a torsional stress. Use the engineering materials listed in the table and rank these materials based upon their performance for this application. How does this ranking change if including the material cost.

$5.(20$ pts$)$ A beam is subjected to a force F $,$ as indicated in the figure.

$($a$)$ Derive the strength$-$to$-$mass performance index expression.

$($b$)$ Derive the stiffness$-$to$-$mass performance index expression.