$(25$ pts $)$ The design specifications for a ball bearing on a rotating shaft are: $(1)$ the radial load $600$ lb$,(2)$ bearing service hours $2000$ h at the speed $167$ rpm$,(3)$ the load application factor $1\mathrm{.}2,(4)$ reliability $0\mathrm{.}9,$ and $(5)$ the minimum diameter of the shaft $0\mathrm{.}675".$

$($a$)$ Run the calculation to show whether the ball bearing No: R$12$ shown in the table satisfies the design requirement.

$($b$)$ If Bearing No: R$12$ is not ok$,$ which one in the table can be used for the design?