You will meet the equations describing bending beams in the module but do not need to know their derivation to be able to answer this question. The deflection, d, is described by the following equation; 8 = FL 3EI Where E is the Young's modulus of the material and / is the second moment of area, which for a square section beam is: I = 64 12 Answer the following questions ensuring that you show all of your workings so that your calculations can be followed and understood. Work throughout to three significant figures. a. Including all your workings, show that the objective equation for the mass of the cantilever beam is m = = 6 Lp. (2 marks) b. Show that the constraint equation is given by d 4FL E64 (3 marks) c. Use the objective and constraint equations from parts a. and b. of this question to define an equation for the mass, m, of the beam by eliminating all free variables. (5 marks) d. Separate the variables of the combined equation into functional F, geometric G and materials M groupings. In your answer, show the combined equation clearly identifying the F, G and M groupings. (2 marks) e. From your answer to part d. of this question, define an appropriate merit index for the material selection process. Define the axes you would adopt on an Ashby chart to be able to use your merit index, and state, with a reason, whether the merit index should be maximised or minimised.