Difference between 50Hz and 60Hz to run a motor

The output power at any speed is fundamentally related to the torque that the motor can produce. Since the torque is determined by the machine parameters and these don't change (if the flux remains constant as described) - then the maximum torque and rated torque will not change. The torque is a product of conductor current and air-gap flux density, and these will of course remain constant whether the motor is driven at 25 Hz, 50 Hz, 60 Hz, or even 100 Hz. In practice - one can normally increase flux densities at lower frequencies, because the core losses are lower for the lower frequencies. The opposite also applies in that the flux densities at higher frequencies need to be reduced somewhat to not cause too high core losses.

Since Power = Torque x Speed, the same motor will be able to produce approximately twice the power at 100 Hz that it will at 50 Hz - assuming of course it can rotate safely at that speed.

The cooling operates better at higher speeds but this is not the reason why one can increase motor 'rated' output power as speed or applied supply frequency is increased. It is more fundamentally because Power = Torque x Speed, where a motor's torque is largely determined by its physical parameters.

It emphasizes also that it is very desirable to contact the motor manufacturer with questions like this. All of the general guidelines are correct and generally true but much depends on the motor design as stated. The manufacturer has design data making it very easy to answer this question, likely doing this every day. The difference between 50Hz and 60Hz power supply usually 20% - nominal power at 60Hz higher by 20% of 50Hz power supply. The data can be found on motor nameplate.

Being one of those people for many years I can say that induction motors are often designed conservatively, especially older motors. It is not uncommon for some motors to run satisfactorily at the same voltage both 50Hz and 60Hz. Yes performance is different but may be acceptable. Consider also that motors often run at less than full load in applications resulting in lower conduction losses and therefore higher induction losses may be OK. This difference between theory and practice can be a bit frustrating at times but we try to help customers while keeping them out of trouble. Note however when you start talking large changes in frequency like running at 100 Hz or 120 Hz there are many more things to consider and this definitely should not be attempted without consulting the manufacturer or a lot of experience allowing one to analyze the machine themselves.