A solid steel bar of circular cross$-$section has a diameter $d=50mm,$ length $L=1\mathrm{.}5m$ and shear

modulus of elasticity $G=80$GPa. The bar is subjected to torques $T$ acting at the ends as shown

in the figure below.

If the torques have a magnitude $T=350Nm,$

$($i$)$ What is the maximum shear stress ${}_{\text{m}\text{a}\text{x}}$ in the bar?

$($ii$)$ What is the angle of twist $$ between the ends?

If the allowable shear stress is $40$MPa and the allowable angle of twist is $2$ degrees,

$($iii$)$ What is the maximum permissible torque?