$1)[55$ marks$]$ Well$-$polished rotating shaft $($w$)$ as shown in the following Figure is supported by two ball bearings and it carries $675$ lb fly$-$wheel. The shaft is made of $1141$ steel quenched and tempered at $\backslash (1000\{\}^\{\backslash$circ$\}\backslash$mathrm$\{$F$\}\backslash ).$ Other dimensions are also given in the following Figure. Assume the shaft is massless. Consider only the fly$-$wheel force due to its mass.

Hint: Use Table A$-21$ for material properties data.

a$.$ Find the critical bearing location on the shaft where calculate the $\backslash (\backslash$sigma$\_\{\backslash$text $\{$max $\}\}=\backslash )$ Maximum stress. $(15$ pts$)$

b$.$ Below, to be consistent for all students, assume $\backslash (\backslash$sigma$\_\{\backslash$max $\}=\backslash$mathbf$\{107.5\}\backslash$mathbf$\{$K s i$\}\backslash )$ is given! Using this max. stress, what is Fatigue strength coefficient? Use both the table and exact calculation and compare them? $(\backslash (3+7\backslash )$ pts$)$

c$.$ Find the number of $\backslash ($ N $\backslash ),$ revolutions$($cycles$)$ you expect it to last under given loading. $(10$ pts$)$

d$.$ Find the maximum weight that allows for infinite life using uncorrected endurance limit and factor of safety is $1.(10$ pts$)$

e$.$ Assume $675$ lb weight, find the $\backslash ($ d $\backslash ),$ the shaft diameter for which you would predict a lifespan of $90000$ revolutions. $(10$ pts$)$