A $480$ V $,$ four$-$pole, $75hp,60Hz,Y-co$ected, three$-$phase induction motor develops its full$-$load induced torque at $3\mathrm{.}5$ percent slip when operating at $60$ Hz and $480$ V $.$ The perphase circuit model impedances of the motor are $R_1=0\mathrm{.}058,x_1=0\mathrm{.}32,x_2=0\mathrm{.}40$ and $x_M=20.$ The mechanical, core, and stray losses may be neglected.

$(1)$ Find the value of the rotor resistance $R_2.$

$(2)$ Find the ${}_{\text{m}\text{a}\text{x}},S_{\text{m}\text{a}\text{x}},$ and the rotor speed at maximum torque for this motor.

$(3)$ Find the stating torque of this motor.