The torsion rod with variable cross$-$section shown in Fig. $2\mathrm{.}1$ is clamped at A and carries point torques at $kN*m)$ and $D($unknown value $T)$ with the senses indicated in the figure. It is known that the diameter of $AB$ is $25mm,$ the diameter of $BC$ is $50mm$ and the diameter of $CD$ is $20mm.$ Furthermore, $6L_{AB}=6L_{BC}=5$LCD$=1200mm.$ If the shear modulus of the material is $80$GPa, find:

a$)$ The value of $T$ that would make the angle of twist at point $C$ with respect to A equal to zero.

b$)$ The maximum value of $T$ that can applied with the sense shown such that failure does not occur for an allowable shear stress of $1\mathrm{.}1$ GPa. Can the condition from part a be achieved without failure?

Fig. $2\mathrm{.}1$: Torsion rod with variable cross$-$section.