Does the motor speed always constant?

Theoretically, Yes, if the conditions under which it's running are constant (on squirrel cage induction motors). In reality, it varies due to a variety of factors. These can include changes in load, input power, environmental conditions, etc. Speed control really has 2 effects - providing a wider range of speed capabilities with the same motor (adjustability) as well as smoothing out (or at least attempting to smooth out) the effects that would change the motor speed and transients in loading, input voltage, etc.

Practically, NO, for electric motors connected to the AC power line. The speed of an AC induction motor is a function of the AC frequency, the number of poles in the motor, and the load on the motor. Under no-load conditions, the speed is quite constant. Under load, the speed will vary since the motor torque comes from the lag in the phase current. Unless the motor runs slower than synchronous speed, it cannot produce the torque necessary to do its work. Synchronous motors run at constant speed.

Practically, if the AC induction motor is driven by a Variable Frequency Drive, then there is a closed loop for speed control, and the speed can be made to be constant.
An AC motor speed = 120f/p P which is number of poles is fixed and f being the mains operating frequency again not changing the motor speed does no change.

However considering the slip factor a 1500 rpm 4 pole motor will run at around 1450 rpm. A VFD has the following benefits:
a. Acts as a soft starter
b. Can be part of a closed loop speed and process control system
c. Maintains power factor to almost unity
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