$4\mathrm{.}2\mathrm{.}10.$ A case$-$hardened shaft is $25$ mm diameter with a case depth of $1\mathrm{.}5$ mm $.$ Assuming the case remains perfectly elastic up to its failing stress in shear of $300\frac{N}{m}{m}^2$ and that the inner core becomes perfectly plastic at a shearing stress of $180\frac{N}{m}m2,$ calculate $($a$)$ the torque to cause elastic failure in torsion in the case and $($b$)$ the angle of twist per meter length at failure. $G=84,000\frac{N}{m}{m}^2$ for all the material while elastic.