Problem No$.1$: $(49$ points$)$

Please not whether to Grade $(\backslash ($ Y $\backslash )$ or N$)$ or Bonus

A one piece, solid steel gear $+$ shaft component is being loaded by a gear tooth force, $\backslash (\backslash$mathrm$\{$F$\}\_\{\backslash$mathrm$\{$T$\}\}\backslash ),$ as shown with the dimensions in the diagram below. This gear shaft is being supported at each end $($ A and B with a ball bearing. A moment load at point B reacts the gear tooth force $\backslash (\backslash$mathrm$\{$F$\}\_\{\backslash$mathrm$\{$T$\}\}\backslash ).$ Ignore the thickness of the gear blank $($disc$).$ The material is ductile, high strength steel with a Yield Strength of $100$ ksi.

Find: The factor of safety at gear $\backslash ($ C $\backslash ).$

Schematic Diagram $-$ Not to Scale