$[20$ points$]$ You are designing a pirate ship ride for a carnival and you need to figure out how wide the steel beam that supports the ride during operation has to be to not fail. The beam is $8\mathrm{.}0$ m long, $2\mathrm{.}0$ cm thick, and made of A$106$ steel that has a yield stress of $250$ MPa $.$ When the yield stress is reached, the beam will be permanently damaged. To avoid this, you want to design the beam width so that the maximum stress that occurs is $5$ x less than the yield stress, and no more. This would be called a safety factor of $5.$

Figure $4$: The scenario in question $6.$

$($a$)$ What width should you suggest for the steel beam if the pirate ship weights $2,500$ kg and has a maximum angular velocity of $71$ degrees$/$s$?$

$($b$)$ At what angle theta, relative to the vertical dashed line in the figure, is the maximum angular velocity reached? At this point, how much longer is the beam relative to its initial length? You might have to look up a material property to get this answer.

$($c$)$ During operation, the pirate ship is released from rest at a $45$ degree angle. What is the stress in the beam right after release?