Why motor, generator and transformer are designed for 50Hz or 60Hz?

In very general terms the silicon-iron magnetic cores of motors, generators, and transformers get smaller as the frequency they operate at gets higher. This of course assuming the energy delivered at the output of both cases is the same. Because of this, when you have 50 Hz core in a motor, generator or transformer, this core will generally bigger and heavier (more silicon-iron). This is because it needs to have more magnetic mass in order to be able to accommodate the extra volume of magnetic flux that will be needed to flow through it at these lower frequencies in comparison to those designs built for 60 Hz operation. This at first can feel kind of counter intuitive if not visualized correctly.

The way this makes intuitive sense in my mind is that the voltage generated is going to be L di/dt which means that to get the same voltage output level you will either need smaller volumes of currents changing directions really fast (like 60 Hz systems) or larger volumes of currents changing directions really slow (like 50 Hz systems).

This is when I like drawing an analogy of this effect by visualizing it similarly to a hose with water flowing through it. If you have slower but a greater volume of water flowing through the hose, the hose diameter would need to be larger to handle this large but slow flow (50 Hz core). If you have a lower volume of water flowing through the hose but moving back and forth (AC frequency), this hose could be smaller.

This is why most AC systems used for aviation are designed for 400 Hz operation, since the higher frequency of operation requires less magnetic mass at higher frequencies. This is why switching power supplies are much lighter than the conventional magnetic transformers since the flux density is attained through alternating voltages that are operating at much higher frequencies and require less magnetic permeability at the core. Because of this effect the cores can be much lighter and of lesser permeability.

So as a general rule can visualize 50 Hz systems as heavier beefier systems since they have to have to handle slower but larger lumbering power as opposed to faster and more agile but smaller in dimension of the 60 Hz systems assuming the same level of output power in both cases.

This is why there is a very important protection in transformers, generators and motors known as V/Hz. The purpose of V/Hz protection is for those situations where frequency could drop quickly and drastically (going slower) while still maintain trying to maintain the same output voltage. If this were to happen without proper monitoring protective devices, the magnetic flux density would rise rapidly during the drop in frequency and this extra magnetic flow, would "over flux" the magnetic core to such extent as to create extreme magnetic induction heating to the point of damaging the core by exceeding its designed temperatures.