Problem $3(30$ points$).$ A gear and shaft with nominal diameter of $35$ mm are to be assembled with a medium drive fit. The gear has a hub, with an outside diameter of $60$ mm $,$ and an overall length of $50$ mm $.$ The shaft is made from AISI $1020$ CD steel, and the gear is made from steel that has been through hardened to provide $S_{ut}=700$MPa and $S_y=600$MPa.

$($a$)$ Specify dimensions with tolerances for the shaft and gear bore to achieve the desired fit.

$($b$)$ Determine the minimum and maximum pressures that could be experienced at the interface with the specified tolerances.

$($c$)$ Determine the worst$-$case static factors of safety guarding against, ielding at assembly for the shaft and the gear based on the distortion energy failure theory.

$($d$)$ Determine the maximum torque that the joint should be expected to transmit without slipping, i$.$e$.,$ when the interference pressure is at a minimum for the specified tolerances.