A shunt DC motor at rated terminal voltage has no-load speed 200 rad/s, and speed 180 rad/s at torque 500 N·m. It drives a load with $T_L=2.78\omega_r$. Neglect rotational losses. Find steady speed in rad/s.
With constant field and terminal voltage, the motor torque-speed relation is linear. It falls from 200 rad/s at zero torque to 180 rad/s at 500 N·m, so ω=200−(20/500)T=200−0.04T.
At equilibrium T=TL=2.78ω. Therefore ω=200−0.1112ω, or 1.1112ω=200.
The speed is **179.986 rad/s**, within 177–183.
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