Induction motors are selected on torque requirements at a specific speed not kilowatts. The mechanical power is the rated power of the motor at the shaft not the input of the electrical energy. The difference between the two is the efficiency. The torque is the mechanical output power in kW x 9950 divided by the speed in rpm. When the induction motor starts the speed is zero and the current maximum. Power of the flux interaction causes the shaft to move slightly. This small number is divided into the power which is coming from the "locked rotor" current about 6 - 9 times FLC limited by impedance only. Big power and high torque (200 – 350% higher than FLT).
As the shaft begins to turn, the torque drops as the speed increases. The current drops as the back EMF takes a large and larger role. Failure of the shaft to move results in all the power being absorbed in the stator and the induction motor burns out. At about 80% the back EMP plays a smaller roll, the inertia increases, all the design issues around rotor bars and stator slots take over and the torque increase to a maximum at rated speed. If the speed increase above this, say the load begins to drive the motor its drive torque decrease. If it goes beyond 100% then it becomes an induction generator.
The typical T-N curve is common for all induction motors and it looked different in many cases for different values of different Motor parameters (as can be explained by Equivalent Circuit and equations). Physically, you have to first know how an induction motor rotates (although it is not having a separate Field supply). If you understand that without mathematical equation then you may also understand the T-N Curve. One or two things I like to point out but that too are not without mathematics (however of school level). T-N curve has two different zones :1- Stable zone , represented by a straight line from max Torque (at speed about 85 -90%) to Synch speed & 2, Unstable zone, which can be represented by a Hyperbola, from T=0 & Speed=0 to max Torque & corresponding speed. Draw these two graph with hand & you will see the T-N curve.
This shape is related with the change of frequency and reactance in the rotor because the rotor frequency depends on slip, that's the physics behind, the value of the impedance the rotor shows is variable, but in dynamics behavior the things are a little more complicate, because you cannot use the equivalent circuit model because it is in the frequency domain and should use the unified or park equations and you are working with a set of non linear differential equations that should be transformed to linear system for finding the solution and back again.