Why does a rotor move at a constant angular speed? If a there is a net torque on a rotor of an electric motor produced by the electromagnetic force, then why does it rotate at a constant angular speed?
This is in reference to problem 10.35 of University physics with modern physics, 13th ed. by H. D. Young and R. A. Freedman:

(a) Compute the torque developed by an industrial motor whose output
is $150 kW$ at an angular speed of $4000 rev/min$.
(b) A drum with
negligible mass, $0.400 m$ in diameter, is attached to the motor shaft,
and the power output of the motor is used to raise a weight hanging
from a rope wrapped around the drum. How heavy a weight can the motor
lift at constant speed?
(c) At what constant speed will the weight
rise?

I'm particularly confused about the torque asked in part (a). In my understanding, it is the electromagnetic force due to the current-conducting wire attached to the rotor and the magnetic field from loop of wires in the stator which causes the rotor to rotate. The tangential electromagnetic force on the rotor produces a net torque. Then, why is it possible for the rotor to rotate at a constant angular speed?
 A: There are, of course, TWO torque sources on a motor doing work; the electromagnetic
torque that does the work, and the back-torque on that drum, 
from the load being raised, which has work done on it.
This means that we do not have any indication of a net torque, and constant
speed is to be expected:
if we ignore the weight of the rope, the output power for a weight so lifted
is constant when the rotation rate of the motor is constant.
There will be power variation, however, for a synchronous AC motor under load, and
there will be rate (rpm) variation for a universal or DC motor under load,
so that the problem solution might have some other operating condition than
the 4000 rpm and 150 kW given, depending on what 'an industrial motor' means.
A: Maybe the way the problem is fomulated is questionnable ? (sorry for my very poor english !) The power outpout of a motor is not a fixed data : it depend of the charge torque. For a given charge, the  electromagnetic torque adust to compensate the charge torque (if possible). In permanent regime, the two torques are opposite and the speed is constant.
A: You don't state whether it's a DC or AC motor, but the explanation is simple for the DC motor case: a turning DC motor in a permanent magnetic field, induces a voltage in the coils proportional to the turning speed (acting like a generator). This voltage (called the back-EMF) opposes the voltage applied to the motor minus the loss due to the resistance of the coils.
Given no other losses or effects, the closed circuit will be governed by Vdrive - Idrive * R = back_EMF. Vdrive and R are constants.
So, given a particular Vdrive, the motor is self-regulating - if you try to hold it (apply load), the backEMF will decrease (as it's proportional to the rotation speed only), then the Idrive*R term will have to increase, i.e. current increases and increases the torque which will then tend to maintain the speed. And vice versa.
