Values : A$=50,$ B$=1,$ D$=40,$ h$=254,$ w$=102,$ L$=5,$ k$=28,$ M$=25,$ S$1=170,$ S$2=165,$ T$=800$

Justify your selection of a suitable column if its cross$-$sectional depth and breadth are h mm and w mm respectively and length L m$,$ when fixed at both ends, by determining:

i$)$ The least second moment of area $($moment of inertia$)$

ii$)$ The least radius of gyration

iii$)$ The slenderness ratio

iv$)$ Your justifiable choice of mass per unit length of the column

v$)$ Whether the column is likely to fail under buckling or compression first $($if it supports a vertical compressive load$)$