Calculated Speed of Synchronous Speed

NO! There is no simple single equation for the rotor shaft speed. Use a little common sense here and you'll see my point.

Lets take a simple thought experiment to prove my point. Take a 1HP two pole synchronous induction motor and securely bolt the stator chassis to a structural steel I-beam. Weld a 1 inch thick steel plate to the I-beam immediately in front of the rotor shaft. Now weld the rotor to the same steel plate. When the thermally protected three phase motor gets the correct 60 Hz voltage applied to all three stator windings and before the thermal breaker trips, what will be the rotor speed? Assume that none of the welds or bolts break.

Normal slip in induction motors is around 2-5%, I think. For example, a 1750 rpm motor (synchronous rpm 1800) has (1800-1750)/1800 x 100 ≈ 2.78% slip. The actual slip is proportional to (actual load)/(rated load), and can be computed from the motor's nameplate.

The induction motor characteristics curves include one that gives the slip-torque relationship.

On this curve if you plot the load speed-torque line then you get the Operating point.

This the the load torque and the slip of the motor.

The Speed of the motor at this point is

N = N_syn (1-s/100)

Percentage Slip is given by,

S(%) = [{(Synchronous Speed,N_s - Full Load Rated Speed, N_r)}/Synchronous Speed,N_s]X100

Rated Full Load Speed, N_r = [{(1-S)XN_s}/100]

Also, If the Rated Torque, Τ_r and Rated Power Developed, P_r are known, then Rated speed can be given by:

Nr = [(60XP_r) / (2X∏XT_r)]

Vinu_Answers Sure_Answers

Only if rated power developed stands for the mechaninal power on the output motor shaft....

Excuse me but the OP asked how to convert the synchronous speed to the rotor speed. People here have been relating the rotor speed versus synchronous speed to the torque produced. This is not what the OP asked for. At best once one has a torque for the motor one can make a differential equation that requires the moment of inertia on the motor shaft along with any drag components to describe a curve with no unique answer for rotor speed. But we also have no idea which induction motor design class is the rotor to know which curve should be used. The curve [Figure 7-26] of a Class D and a Class A rotor design will put the maximum torque at either 20% or 85% of the synchronous speed.

Synchronous motors have no "slip".

Induction motors have "slip".

Your question is not correct as it stands.