Problem $4$:

This is a problem related to spur gears and helical gears.

$(1)$ Pinion: Material SM$45$C $($HB$250),$ rotational speed $900$ rpm

Gear: Material GC$150,$ rotational speed $300$ rpm

The center distance is approximately $500$mm$,$ the pressure angle is $20,$ and the face width is $3$m $($module is also $3$m$).$ Calculate the module of the standard spur gear that can transmit $5$kW of power. Calculate the module as a whole number. $($You may ignore factors such as shock loads and dynamic coefficients.$)$

$(2)$ Using the module you calculated in $(1),$ design a helical gear made of the same material and with the same operational speed. The helix angle $\backslash$beta is $20.$ Calculate how the transmissible power changes when the helix angle is applied.