A shaft $($gray$)$ and gear $($white$)$ are subjected to a torque of $3000N-m$ and a press fit creates an

inner pressure $p_i$ on the gear. There is a pre$-$existing crack on the top of the gear that runs

through the gear thickness. The gear is gray cast iron $(E=200$GPa,$K_{Ic}=20$MPa$\sqrt[\phantom{2}]{m},Sy=650$

Mpa$).[$value $=4$

a$.$ Determine the internal pressure, $p_i,$ that would cause crack propagation $($assume $p_o=0).$

b$.$ For the pressure you calculated in part $($a$),$ what is the factor of safety in terms of yielding

at point $A$ on the inner radius of the gear. Use distortion energy failure criterion. Sketch

the stress element at point $A.$

c$.$ Based on your answers, will the cylinder fail first from crack propagation or yielding?