$(15$ points$)$ Torsional test data for a solid shaft shown in Figure $5$:

shaft diameter $d=50mm,$

length of the shaft $L=400mm,$

angle of twist of the shaft $2\mathrm{.}5.$

The maximum shear stress was measured for a torsion torque of $T=2,376Nm.$

Poisson ratio for this material is $0\mathrm{.}380.$

a$)(3$ points$)$ Calculate polar moment of inertia of the shaft.

b$)(3$ points$)$ Calculate maximum shear stress ${}_{\text{m}\text{a}\text{x}}$ on the shaft.

c$)(2$ points$)$ Convert angle of twist to the units of radians. $180$ degrees $=($radians$)$

d$)(3$ points$)$ Calculate shear modulus of elasticity $G.$

e$)(4$ points$)$ Calculate Young's modulus of Elasticity E$,$ using shear modulus and Poisson's ratio.