Which type of motor is used in lathe?

The type of motor used depends on the size of the lathe.

Small "hobby" lathes use small fractional (1/8, 1/4...) horsepower ac motors which run on household single-phase power.

The lathes which are used to produce crankshafts for huge marine diesel engines - or the rotor shafts of the massive alternators used to generate megawatts of electrical power - must use giant-sized motors, producing hundreds of horsepower.

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DC SHUNT MOTORS.
Three-phase electric motor.