Prob $2.(30$ points$)$

The gear transmits $2000$ lb $-$in of torque. Determine the size of key necessary to give a safety factor of $2$ against both shear and bearing failure for the design shown in Figure. Assume a shaft diameter of $1\mathrm{.}5$ in $.$ The key is steel of $\backslash (\backslash$mathrm$\{$S$\}\_\{\backslash$mathrm$\{$ut$\}\}=88\backslash )$ ksi and $\backslash (\backslash$mathrm$\{$S$\}\_\{\backslash$mathrm$\{$y$\}\}=52\backslash$mathrm$\{$ksi$\}\backslash ).$