$6.$ A torque $\backslash ($ T$=100\backslash$mathrm$\{$~N$\}\backslash$cdot $\backslash$mathrm$\{$~m$\}\backslash )$ is applied to the shaft $\backslash ($ E F G $\backslash ).$ Gear F transmits torque to shaft $\backslash ($ A B C D $\backslash )$ through gear $\backslash ($ C $\backslash ),$ which drives the chain sprocket at $\backslash ($ B $\backslash ),$ transmitting a force $\backslash ($ P $\backslash ).$ The sprocket $\backslash ($ B $\backslash ),$ gear $\backslash ($ C $\backslash ),$ and gear $\backslash ($ F $\backslash )$ have pitch diameters of $\backslash ($ a$=150,$ b$=250\backslash ),$ and $\backslash ($ c$=\backslash )125$ mm $,$ respectively. The contact force between the gears is transmitted through the pressure angle of $\backslash (20^\{\backslash$circ$\}\backslash ).$ The bearings at A $,$ D$,$ E$,$ and G are simple supports. $(15\text{'}$ in total$)$

$(1)$ Determine the tangential and radial forces that the gear $\backslash ($ F $\backslash )$ exerts on the gear $\backslash ($ C $\backslash ).$

$(2)$ Determine the torque on gear $\backslash ($ C $\backslash )$ and sprocket $\backslash ($ B $\backslash )$ and determine the force $\backslash ($ P $\backslash ).$

$(3)$ Determine the bending moment at the point $\backslash ($ B $\backslash )$ and $\backslash ($ C $\backslash ).$ Schematic Figure for Problem $\backslash$#$6$