Is the rated speed of an induction motor the synchronous speed?
An induction motor never spins at synchronous speed.

A motor spinning at constant speed must be producing enough accelerating torque to overcome the decelerating torque of the load. Since an induction motor must slip to provide torque, the full-load speed will always be less than synchronous speed.

Even an un-loaded motor with nothing on the shaft still has a decelerating torque, due to the friction of the bearings and so on.

Therefore, we have:


Rated speed is the speed of the motor at rated output (shaft) power.
No-load speed is the speed of the motor with no load on the shaft. Note the motor bearings still impose a nominal load due to friction.
Synchronous speed is the speed the motor would spin at if it had absolutely zero load (frictionless bearings and no other losses of any kind.)


Related: What does the speed of regulation mean?



For reference, here is a typical motor characteristic graph.



Note synchronous speed of 3,000 RPM, no-load speed also about 3,000 RPM, and full load speed 2,932 RPM.