PART $2$: Mechanical Design Calculations $(15\%)$

Design Case $1(5\%).$ As a mechanical design engineer, you are asked to design a steel $(E=207$ 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 $d$ 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 $0\mathrm{.}001m\frac{m}{m}m$ 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 $d$ will support the load without penalty? $(2\mathrm{.}5\%)$

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