VII. $[$total: $15\backslash \%]$ The spur gears in Figure $4$ have a module of $6$ mm and a $\backslash (25^\{\backslash$circ$\}\backslash )$ pressure angle. The $18-$teeth gear $2$ rotates at $\backslash (2100\backslash$mathrm$\{$rev$\}/\backslash$mathrm$\{$min$\}\backslash )$ clockwise and transmits $400$ kW through a gear train to gear $4$ on shaft $\backslash ($ c $\backslash ).$

$($a$)$ Find the angular velocity and the rotation direction of gear $4.[2\backslash \%]$

$($b$)$ Determine the torque that gear $3$ transmits to gear $4.[2\backslash \%]$

$($c$)$ Find the force that gear $2$ and gear $4$ export against shaft $\backslash ($ b $\backslash ).[5\backslash \%]$

$($d$)$ If the center distance between shafts $\backslash ($ a $\backslash )$ and $\backslash ($ b $\backslash )$ was incorrectly made $3$ mm larger, find the new values of the pressure angle and the pitch$-$circle diameters for the gear set of gears $2$ and $3.[6\backslash \%]$

Figure $4$