The nominal Power on a feather key is $12$W$,$ and the nominal velocity of the shaft is $2900$RPM$.$ The key's material is steelfield strength is $300$ MPa $,$ the allowed stirlace pres lomt is $50$ MPa $),$ and the shaft is manufactured from sags structural steel. The required satety factor is $3.$ It is known, that the approximating width dimension of the key is $30\%$ of the shatr's diameter. and the cross section of the key is a spware in the examined dimension's range.

a$.$ Design the shaft's minimal diameterl $($integer number$)$

b$.$ Design the key's cross$-$kection, based on the permitted surface pressure, if we know, that the key groove's depth is half of the key's total height!

c$.$ Check the key on shear stress, whether the safety factor is sufficient or not?