Design Case $1(5\backslash \%).$ As a mechanical design engineer, you are asked to design a steel $(\backslash ($ E$=\backslash )207$ GPa$)$ shaft supported by two identical roller bearings at its two ends $($see Fig$\mathrm{.}1).$ Assume that the shaft is to have a uniform diameter $\backslash ($ d $\backslash )$ except in the locality of the bearing mounting, so it can be approximated as a uniform beam with simple supports. You have also been informed from the bearing manufacturer's catalogue that the allowable slope at the bearings is $\backslash (0\mathrm{.}001\backslash$mathrm$\{$~mm$\}/\backslash$mathrm$\{$mm$\}\backslash )$ without bearing life penalty. For a design factor of $1\mathrm{.}28,$ and considering the dimensions and loading illustrated in Fig. $2$:

$($a$)$ What uniform$-$diameter shaft $\backslash ($ d $\backslash )$ will support the load without penalty? $(2\mathrm{.}5\backslash \%)$

$($b$)$ What is the maximum deflection, $\backslash (\backslash$delta $\backslash ),$ of the shaft? $(2\mathrm{.}5\backslash \%)$