i$.($a$)$ An internal combustion engine of $50$ horsepower $($hp$)$ transmits power to the car wheels of an automobile at $280$ rev$/$min $($rpm$).$

Neglecting any transmission losses, determine the minimum permissible diameter of the solid circular section steel shaft, if the maximum shear stress in the shaft is limited to $50$ MPa $.$

What will be the resulting angle of twist of the shaft, due to the applied torque, over a length of $3$ m $,$ given that the rigidity modulus, $G=70$gPa $?$ Note that $1hp=745\mathrm{.}7W.$

$($b$)$ If the shaft above was replaced with a hollow shaft with wall thickness $0\mathrm{.}002$ m determine the new maximum shear stress $($using max diameter$)$ with the same applied torque. Finally calculate the resulting angle of twist assuming the same material for the shaft and using the Torque that you have just calculated in $($a$).$