A solid circular cross section drive shaft with a diameter $d=4\mathrm{.}0cm$ is used to transfer energy from a motor to the wheels of a vehicle. A tensile stress of $100$ MPa is placed on the shaft. In addition, the torque, T $,$ from the motor produces a shear stress at the shaft surface according to the formula:

$=\frac{16T}{d^4}$

a$.$ If the drive shaft is constructed of ductile $1040$ steel with a yield strength of $300$ MPa $,$ what is the maximum torque that can be applied to avoid yielding?

b$.$ If the maximum applied torque is $50\%$ of the value calculated in Part a$,$ what is the maximum allowable crack size to avoid fracture? Note that the fracture toughness, $K_{Ic},$ for $1040$ steel is $54\mathrm{.}0$MPa$\sqrt[\phantom{2}]{m}.$