Material indices for elastic beams with differing constraints (Figure E.4). Start each of the four parts of this problem by listing the function, the objective, and the constraints. You will need the equations for the deflection of a cantilever beam with a square cross-section t times t, given in Appendix A, Section A.3. The two that matter arc that for the deflection delta of a beam of length L under an end load F: delta =FL3/3EI and that for the deflection of a beam under a distributed load f per unit length: delta =1/8 fL4/EI where I =t4/12. Fora self-loaded beam f = rho Ag where rho is the density of the material of the beam, A its cross-sectional area and g the acceleration due to gravity. Show that the best material for a cantilever beam of given length L and given (i. e., fixed) square cross-section (t times t) that will deflect least under a given end load F is that with the largest value of the index M = E, where E is Young's modulus (neglect self-weight) (Figure E.4(a)). Show that the best material choice for a cantilever beam of given length L and with a given section (t times t) that will deflect least under its own self-weight is that with the largest value of M = E/ rho , where p is the density (Figure E.4(b)). Show that the material index for the lightest cantilever beam of length L and square section (not given, that is, the area is a free variable) that will not deflect by more than delta under its own weight is M = E/ rho 2 (Figure E.4(c)). Show that the lightest cantilever beam of length L and square section (area free) that will not deflect by more than 6 under an end load F is that made of the material with the largest value of M = E1/2/ rho (neglect self weight) (Figure E.4(d)).