Question $1(30$ Marks$)$

A steel propeller shaft of hollow circular cross$-$section has an outer diameter of $100$ mm and is required to transmit $1500$ kW while rotating at $750$ revolutions per minute $($RPM$).$ The modulus of rigidity $($shear modulus$)$ of the material is $\backslash ($ G$=82\backslash$mathrm$\{$GPa$\}\backslash ).$ Two design criteria have been specified:

$($i$).$ The maximum shear stress in the shaft is not to exceed $75$ MPa $.$

$($ii$).$ The allowable angle of twist is $0\mathrm{.}5$ degrees per metre.

By considering each design criterion separately, calculate the required inner diameter of the shaft in each case, and hence specify the most appropriate wall thickness to satisfy both criteria.