A shaft $\text{AC}$ rotating at a constant speed carries a thin pulley of radius $r= 0.4\:m$ at the end $C$ which drives a belt. A motor is coupled at the end $A$ of the shaft such that it applies a torque $M_{z}$ about the shaft axis without causing any bending moment. The shaft is mounted on narrow frictionless bearings at $A$ and $B$ where $\text{AB = BC =L = 0.5 m}$. The taut and slack side tensions of the belt are $T_{1} = 300\: N$ and $T_{2} = 100\: N$, respectively. The allowable shear stress for the shaft material is $80$ $\text{MPa}$. The self-weights of the pulley and the shaft are negligible. Use the value of $\pi$ available in the on-screen virtual calculator. Neglecting shock and fatigue loading and assuming maximum shear stress theory, the minimum required shaft diameter is _____________ $\text{mm}$ (round off to $2$ decimal places).
