It is required to design a square key for fixing gear on a shaft of $0\mathrm{.}025$ m in diameter. $16$ kW power at $530$ rpm is transmitted from the shaft to the gear. The key is made of steel $50$C$4($Sy $=460$ MPa$),$ and the factor of safety is $2\mathrm{.}5.$ Use the distortion$-$energy theory and assume the yield strength to be equal to the yield strength in tension. For a shaft of $0\mathrm{.}025$ mm diameter, W $=$ H $=6\mathrm{.}0$ mm for the square key. Determine:

a$.$ Design shear stress in MPa using distortion energy theory.

b$.$ Transmitting torque in N$-$m$.$

c$.$ Minimum key length to the compression in mm

d$.$ Minimum key length due to shear in mm

e$.$ Which key length should be taken as design selection?