AZ Consider the two disks shown above, made with the same material. One is a solid disk of radius R, and the other is a donut-like ring of inner and outer radii ni and ro. The two disks are of equal thickness t, both are rotating around its axis of symmetry Z with angular speed w. Shown in class, the distribution of radial and tangential stresses for the rotating donut-like ring are given by 3+v 3+v 1+3v 3+v and the distribution of radial and tangential stresses for the rotating solid disk are given by - 3+ V po [R = r] , = 8 = 3+v -pa R2 8 3+v Furthermore, the maximum radial stress is given for the rotating donut-like ring by = 3+V po[r, -r] atr = rr. and for the rotating solid disk by 3+v 8 paR at r=0 Assume that the inner and outer radii of the donut-like ring are related by ro = an, with a>1. a) For a given R, derive a relationship between R and (a, n) if the solid disk and the donut- like ring are to have the same mass (6 points) b) c) d) e) Shown in class that the moment of inertia of the solid disk with respect to the symmetry axis about which it turns is given by I=IMPRE Express I of the donut-like ring with the same mass as the solid disk in terms of the solid disk moment of inertia (6 points) Derive an expression for the ratio of kinetic energy of the donut-like ring to that of the solid disk, with the same mass and rotating at the same angular speed w (6 points) Derive an expression for the ratio of the maximum radial stress experienced by the donut- like ring to that by the solid disk, with the same mass and rotating at the same angular speed w (6 points) Based on your answers to c) and d), what is your conclusion regarding increasing kinetic energy per mass of rotating disks through changes from solid disks to donut-like rings? What is (are) the limitation(s) to such an approach, e.g., what prevents you from taking a to be very close to 1? (6 points)