2. The best materials for a light column that must not buckle or crush (30 points) The best choice of material for a light strong column depends on its aspect ratio, which is defined as the ratio of its height H to its diameter D. This is because short, thick columns fault by crushing and tall, slender columns fail by buckling instead. Height H and loading F is specified by the designer. Derive two performance equations for a column with a circular cross section, where lightness is important. a. Translate this problem into function, constraints, objective and free variables. b. Derive a materials index for minimized cost using the constraint that the column must not crush. (M1) c. Derive a materials index for minimized cost using the constraint that the column must not buckle. (M2) 2. The best materials for a light column that must not buckle or crush (30 points) The best choice of material for a light strong column depends on its aspect ratio, which is defined as the ratio of its height H to its diameter D. This is because short, thick columns fault by crushing and tall, slender columns fail by buckling instead. Height H and loading F is specified by the designer. Derive two performance equations for a column with a circular cross section, where lightness is important. a. Translate this problem into function, constraints, objective and free variables. b. Derive a materials index for minimized cost using the constraint that the column must not crush. (M1) c. Derive a materials index for minimized cost using the constraint that the column must not buckle. (M2)